Introduction
Let T T T A A A n × n n \times n n × n T T T M A − 1 M_{A^{-1}} M A − 1
M A − 1 f ( x ) = sup A Q ∋ x 1 ∣ Q ∣ ∫ Q ∣ f ∣ . M_{A^{-1}}f(x)=\sup_{AQ\ni x}\frac{1}{|Q|}\int_Q|f|. M A − 1 f ( x ) = A Q ∋ x sup ∣ Q ∣ 1 ∫ Q ∣ f ∣. where the supremum is taken over all cubes Q Q Q M A − 1 f ( x ) = M f ( A − 1 x ) M_{A^{-1}}f(x)=Mf(A^{-1}x) M A − 1 f ( x ) = M f ( A − 1 x ) M M M
T α , m f ( x ) = ∫ R n f ( y ) ∣ x − A 1 y ∣ β 1 ⋯ ∣ x − A m y ∣ β m d y T_{\alpha,m}f(x)=\int_{\mathbb{R}^n}\frac{f(y)}{|x-A_1y|^{\beta_1}\cdots|x-A_my|^{\beta_m}}\,dy T α , m f ( x ) = ∫ R n ∣ x − A 1 y ∣ β 1 ⋯ ∣ x − A m y ∣ β m f ( y ) d y where A i A_i A i A i − A j A_i-A_j A i − A j i ≠ j i\ne j i = j β 1 + ⋯ + β m = n − α \beta_1+\cdots+\beta_m=n-\alpha β 1 + ⋯ + β m = n − α 0 ≤ α < n 0\le\alpha<n 0 ≤ α < n T α , m T_{\alpha,m} T α , m M A i − 1 , α M_{A_i^{-1},\alpha} M A i − 1 , α [10 , 11 , 13 ] . This type of operators are widely studied in weighted Lebesgue spaces and unweighted variable Lebesgue spaces, see [11 , 16 ] and their references.
In [11 ] , the first author, Riveros and Vidal defined the class of weights A A , p \mathcal{A}_{A,p} A A , p w w w A A , p \mathcal{A}_{A,p} A A , p
[ w ] A A , p = sup Q ( 1 ∣ Q ∣ ∫ Q w A ) ( 1 ∣ Q ∣ ∫ Q w − 1 p − 1 ) p − 1 < ∞ [w]_{\mathcal{A}_{A,p}}=\sup_Q\left(\frac{1}{|Q|}\int_Qw_A\right)\left(\frac{1}{|Q|}\int_Qw^{-\frac{1}{p-1}}\right)^{p-1}<\infty [ w ] A A , p = Q sup ( ∣ Q ∣ 1 ∫ Q w A ) ( ∣ Q ∣ 1 ∫ Q w − p − 1 1 ) p − 1 < ∞ where the supremum is over all cubes Q Q Q A = I A=I A = I A A , p \mathcal{A}_{A,p} A A , p A p \mathcal{A}_p A p
2020 Mathematics Subject Classification. 42B25,42B35.
Key words and phrases. Maximal Operator, Variable Lebesgue Spaces, Weights.
The first author is partially supported by CONICET and SGCyT-UNS. The second author is partially supported by CONICET and SECYT-UNC.
They characterized the weights for which the maximal operator M A − 1 M_{A^{-1}} M A − 1 L p ( w ) L^p(w) L p ( w ) L p , ∞ ( w ) L^{p,\infty}(w) L p , ∞ ( w ) [9 ] , the first author studied the boundedness for M A − 1 M_{A^{-1}} M A − 1 L p ( w ) L^p(w) L p ( w ) A A A A k = I A^k=I A k = I
A A , p = { w ∈ A p : w ( A x ) ≲ w ( x ) a . e . x ∈ R n } \mathcal{A}_{A,p}=\{w\in A_p:w(Ax)\lesssim w(x)\ a.e.\ x\in\mathbb{R}^n\} A A , p = { w ∈ A p : w ( A x ) ≲ w ( x ) a . e . x ∈ R n } and the maximal operator M A − 1 M_{A^{-1}} M A − 1 L p ( w ) L^p(w) L p ( w ) w ∈ A A , p w\in\mathcal{A}_{A,p} w ∈ A A , p
In this paper, we study the boundedness of the maximal operator M A − 1 M_{A^{-1}} M A − 1 L w p ( ⋅ ) ( R n ) L_w^{p(\cdot)}(\mathbb{R}^n) L w p ( ⋅ ) ( R n ) ∥ f ∥ p ( ⋅ ) , w = ∥ f w ∥ p ( ⋅ ) \|f\|_{p(\cdot),w}=\|fw\|_{p(\cdot)} ∥ f ∥ p ( ⋅ ) , w = ∥ f w ∥ p ( ⋅ ) § 2 .
In [2 , 5 ] , the authors studied the boundedness of the Hardy-Littlewood maximal operator on L w p ( ⋅ ) ( R n ) L_w^{p(\cdot)}(\mathbb{R}^n) L w p ( ⋅ ) ( R n ) A p ( ⋅ ) A_{p(\cdot)} A p ( ⋅ )
[ w ] A p ( ⋅ ) = sup Q ∣ Q ∣ − 1 ∥ w χ Q ∥ p ( ⋅ ) ∥ w − 1 χ Q ∥ p ′ ( ⋅ ) < ∞ , [w]_{A_{p(\cdot)}}=\sup_Q |Q|^{-1}\|w\chi_Q\|_{p(\cdot)}\|w^{-1}\chi_Q\|_{p'(\cdot)}<\infty, [ w ] A p ( ⋅ ) = Q sup ∣ Q ∣ − 1 ∥ w χ Q ∥ p ( ⋅ ) ∥ w − 1 χ Q ∥ p ′ ( ⋅ ) < ∞ , where the supremum is over all cubes Q Q Q p ( ⋅ ) ∈ P log p(\cdot)\in\mathcal{P}^{\log} p ( ⋅ ) ∈ P l o g 1 < p − ≤ p + < ∞ 1<p_-\leq p_+<\infty 1 < p − ≤ p + < ∞ L w p ( ⋅ ) L_w^{p(\cdot)} L w p ( ⋅ ) w ∈ A p ( ⋅ ) w\in A_{p(\cdot)} w ∈ A p ( ⋅ )
For the study of the weighted boundedness of M A − 1 M_{A^{-1}} M A − 1 A A , p ( ⋅ ) \mathcal{A}_{A,p(\cdot)} A A , p ( ⋅ ) w w w A A , p ( ⋅ ) \mathcal{A}_{A,p(\cdot)} A A , p ( ⋅ )
[ w ] A A , p ( ⋅ ) = sup Q ∣ Q ∣ − 1 ∥ w A χ Q ∥ p ( ⋅ ) ∥ w − 1 χ Q ∥ p ′ ( ⋅ ) < ∞ , [w]_{\mathcal{A}_{A,p(\cdot)}}=\sup_Q |Q|^{-1}\|w_A\chi_Q\|_{p(\cdot)}\|w^{-1}\chi_Q\|_{p'(\cdot)}<\infty, [ w ] A A , p ( ⋅ ) = Q sup ∣ Q ∣ − 1 ∥ w A χ Q ∥ p ( ⋅ ) ∥ w − 1 χ Q ∥ p ′ ( ⋅ ) < ∞ , where the supremum is over all cubes Q Q Q A = I A=I A = I A A , p ( ⋅ ) \mathcal{A}_{A,p(\cdot)} A A , p ( ⋅ ) A p ( ⋅ ) A_{p(\cdot)} A p ( ⋅ ) p ( ⋅ ) ≡ p p(\cdot)\equiv p p ( ⋅ ) ≡ p A A , p ( ⋅ ) \mathcal{A}_{A,p(\cdot)} A A , p ( ⋅ ) A A , p \mathcal{A}_{A,p} A A , p
The first main result is the necessity of the class of weights in the boundedness of the maximal operator for any matrix invertible A A A
Theorem 1.1. Let w w w A A A p ( ⋅ ) ∈ P log p(\cdot)\in\mathcal{P}^{\log} p ( ⋅ ) ∈ P l o g p ( A x ) = p ( x ) p(Ax)=p(x) p ( A x ) = p ( x ) x ∈ R n x\in\mathbb{R}^n x ∈ R n M A − 1 M_{A^{-1}} M A − 1 L p ( ⋅ ) ( w ) L^{p(\cdot)}(w) L p ( ⋅ ) ( w ) w ∈ A A , p ( ⋅ ) w\in\mathcal{A}_{A,p(\cdot)} w ∈ A A , p ( ⋅ )
The second result is the necessity of relation between p ( A x ) p(Ax) p ( A x ) p ( x ) p(x) p ( x )
Theorem 1.2. Let A A A p ( ⋅ ) ∈ P ( R n ) p(\cdot)\in\mathcal{P}(\mathbb{R}^n) p ( ⋅ ) ∈ P ( R n ) x 0 ∈ R n x_0\in\mathbb{R}^n x 0 ∈ R n p ( ⋅ ) p(\cdot) p ( ⋅ ) x 0 x_0 x 0 A x 0 Ax_0 A x 0
p ( A x 0 ) > p ( x 0 ) . p(Ax_0)>p(x_0). p ( A x 0 ) > p ( x 0 ) . Then the operator M A − 1 M_{A^{-1}} M A − 1 L p ( ⋅ ) ( R n ) L^{p(\cdot)}(\mathbb{R}^n) L p ( ⋅ ) ( R n )
Remark 1.3. The previous theorem states that p ( A x ) ≤ p ( x ) p(Ax)\leq p(x) p ( A x ) ≤ p ( x ) x ∈ R n x\in\mathbb{R}^n x ∈ R n A − 1 A^{-1} A − 1 L p ( ⋅ ) L^{p(\cdot)} L p ( ⋅ ) p ( A x ) < p ( x ) p(Ax)<p(x) p ( A x ) < p ( x ) [16 ] the authors showed that if p ( ⋅ ) ∈ P log p(\cdot)\in\mathcal{P}^{\log} p ( ⋅ ) ∈ P l o g p ( A x ) ≤ p ( x ) p(Ax)\leq p(x) p ( A x ) ≤ p ( x ) x ∈ R n x\in\mathbb{R}^n x ∈ R n ∥ g ∘ A − 1 ∥ p ( ⋅ ) ≲ ∥ g ∥ p ( ⋅ ) \|g\circ A^{-1}\|_{p(\cdot)}\lesssim\|g\|_{p(\cdot)} ∥ g ∘ A − 1 ∥ p ( ⋅ ) ≲ ∥ g ∥ p ( ⋅ ) M A − 1 f = M f ∘ A − 1 M_{A^{-1}}f=Mf\circ A^{-1} M A − 1 f = M f ∘ A − 1 g = M f g=Mf g = M f p ( ⋅ ) ∈ P log p(\cdot)\in\mathcal{P}^{\log} p ( ⋅ ) ∈ P l o g p ( A x ) ≤ p ( x ) p(Ax)\leq p(x) p ( A x ) ≤ p ( x ) x ∈ R n x\in\mathbb{R}^n x ∈ R n M A − 1 M_{A^{-1}} M A − 1 L p ( ⋅ ) ( R n ) L^{p(\cdot)}(\mathbb{R}^n) L p ( ⋅ ) ( R n ) A = 2 I A=2I A = 2 I
For the boundedness of the maximal operator M A − 1 M_{A^{-1}} M A − 1
Theorem 1.4. Let A A A A k = I A^k=I A k = I k ∈ N k \in\mathbb{N} k ∈ N p ( ⋅ ) ∈ P log p(\cdot) \in\mathcal{P}^{\log} p ( ⋅ ) ∈ P l o g 1 < p − ≤ p + < ∞ 1 < p_- \leq p_+ < \infty 1 < p − ≤ p + < ∞ p ( A x ) = p ( x ) p(Ax)=p(x) p ( A x ) = p ( x ) x ∈ R n x \in\mathbb{R}^n x ∈ R n w ∈ A A , p ( ⋅ ) w \in A_{A,p(\cdot)} w ∈ A A , p ( ⋅ ) M A − 1 M_{A^{-1}} M A − 1 L w p ( ⋅ ) L^{p(\cdot)}_w L w p ( ⋅ )
Observe that this result characterizes the weights w w w M A − 1 M_{A^{-1}} M A − 1 L p ( ⋅ ) ( w ) L^{p(\cdot)}(w) L p ( ⋅ ) ( w ) [2 , 5 , 9 ] . An application directly related is the boundedness of the operators T α , m T_{\alpha,m} T α , m [11 ] and their references.
The remainder of this paper is organized as follows. In Section 2, we present some needed preliminaries results. In Section 3, we prove Theorem 1.1 and Theorem 1.2, which establish necessary conditions related to our hypotheses. In Section 4, we prove Theorem 1.4, which establishes the boundedness of M A − 1 M_{A^{-1}} M A − 1 L w p ( ⋅ ) L^{p(\cdot)}_w L w p ( ⋅ ) w ∈ A A , p ( ⋅ ) w \in A_{A,p(\cdot)} w ∈ A A , p ( ⋅ )
In Section 5, we study several properties of the weights in A A , p ( ⋅ ) A_{A,p(\cdot)} A A , p ( ⋅ )
Finally, in Section 6, we present a method for constructing weights in A p ( ⋅ ) A_{p(\cdot)} A p ( ⋅ ) A A , p ( ⋅ ) A_{A,p(\cdot)} A A , p ( ⋅ )
Throughout this paper, all notation is standard of will defined as needed. In order to emphasize that we are dealing with variable exponents, we will always write p ( ⋅ ) p(\cdot) p ( ⋅ ) p p p F ≲ G F \lesssim G F ≲ G c c c F ≤ c G F \leq cG F ≤ c G c c c A ≈ B A \approx B A ≈ B c 1 , c 2 c_1,c_2 c 1 , c 2 c 1 B ≤ A ≤ c 2 B c_1B \leq A \leq c_2B c 1 B ≤ A ≤ c 2 B
Preliminaries
Variable Lebesgue spaces
Let P ( R n ) \mathcal{P}(\mathbb{R}^n) P ( R n ) p ( ⋅ ) : R n → [ 1 , ∞ ) p(\cdot): \mathbb{R}^n \to[1,\infty) p ( ⋅ ) : R n → [ 1 , ∞ ) p ( ⋅ ) ∈ P ( R n ) p(\cdot) \in\mathcal{P}(\mathbb{R}^n) p ( ⋅ ) ∈ P ( R n ) Ω ⊂ R n \Omega\subset\mathbb{R}^n Ω ⊂ R n
p − ( Ω ) = ess inf x ∈ Ω p ( x ) , p + ( Ω ) = ess sup x ∈ Ω p ( x ) . p_-(\Omega)=\operatorname*{ess\,inf}_{x\in\Omega} p(x), \qquad p_+(\Omega)=\operatorname*{ess\,sup}_{x\in\Omega} p(x). p − ( Ω ) = x ∈ Ω ess inf p ( x ) , p + ( Ω ) = x ∈ Ω ess sup p ( x ) . When Ω = R n \Omega=\mathbb{R}^n Ω = R n p − p_- p − p + p_+ p +
For each p ( ⋅ ) ∈ P ( R n ) p(\cdot) \in\mathcal{P}(\mathbb{R}^n) p ( ⋅ ) ∈ P ( R n ) p ′ ( ⋅ ) p'(\cdot) p ′ ( ⋅ )
ρ p ( ⋅ ) ( f ) = ∫ R n ∣ f ( x ) ∣ p ( x ) d x . \rho_{p(\cdot)}(f)=\int_{\mathbb{R}^n} |f(x)|^{p(x)}\,dx. ρ p ( ⋅ ) ( f ) = ∫ R n ∣ f ( x ) ∣ p ( x ) d x . Given p ( ⋅ ) ∈ P ( R n ) p(\cdot) \in\mathcal{P}(\mathbb{R}^n) p ( ⋅ ) ∈ P ( R n ) L p ( ⋅ ) ( R n ) L^{p(\cdot)}(\mathbb{R}^n) L p ( ⋅ ) ( R n ) f f f R n \mathbb{R}^n R n λ > 0 \lambda>0 λ > 0 ρ p ( ⋅ ) ( f / λ ) < ∞ \rho_{p(\cdot)}(f/\lambda)<\infty ρ p ( ⋅ ) ( f / λ ) < ∞
∥ f ∥ p ( ⋅ ) = inf { λ > 0 : ρ p ( ⋅ ) ( f / λ ) ≤ 1 } . \|f\|_{p(\cdot)}=\inf\{\lambda>0:\rho_{p(\cdot)}(f/\lambda)\leq1\}. ∥ f ∥ p ( ⋅ ) = inf { λ > 0 : ρ p ( ⋅ ) ( f / λ ) ≤ 1 } . These spaces are known as variable Lebesgue spaces and generalize the classical Lebesgue spaces L p ( R n ) L^p(\mathbb{R}^n) L p ( R n ) [1 , 4 , 8 ] .
Definition 2.1. Given Ω ⊆ R n \Omega\subseteq\mathbb{R}^{n} Ω ⊆ R n r ( ⋅ ) : Ω → R r(\cdot):\Omega\to\mathbb{R} r ( ⋅ ) : Ω → R r ( ⋅ ) r(\cdot) r ( ⋅ ) log \log log H o ¨ l d e r H\ddot{\mathrm{o}}lder H o ¨ l d er r ( ⋅ ) ∈ L H 0 ( Ω ) r(\cdot) \in LH_{0}(\Omega) r ( ⋅ ) ∈ L H 0 ( Ω ) C 0 C_{0} C 0 x , y ∈ Ω x,y \in\Omega x , y ∈ Ω ∣ x − y ∣ < 1 2 |x-y| < \frac{1}{2} ∣ x − y ∣ < 2 1
∣ r ( x ) − r ( y ) ∣ ≤ C 0 − log ( ∣ x − y ∣ ) . (1) |r(x)-r(y)| \le\frac{C_{0}}{-\log(|x-y|)}. \tag*{(1)} ∣ r ( x ) − r ( y ) ∣ ≤ − log ( ∣ x − y ∣ ) C 0 . (1) We say that r ( ⋅ ) r(\cdot) r ( ⋅ ) log \log log H o ¨ l d e r H\ddot{\mathrm{o}}lder H o ¨ l d er r ( ⋅ ) ∈ L H ∞ ( Ω ) r(\cdot) \in LH_{\infty}(\Omega) r ( ⋅ ) ∈ L H ∞ ( Ω ) C ∞ C_{\infty} C ∞ r ∞ r_{\infty} r ∞
∣ r ( x ) − r ∞ ∣ ≤ C ∞ log ( e + ∣ x ∣ ) , (2) |r(x)-r_{\infty}| \le\frac{C_{\infty}}{\log(e+|x|)}, \tag*{(2)} ∣ r ( x ) − r ∞ ∣ ≤ log ( e + ∣ x ∣ ) C ∞ , (2) for all x ∈ Ω x \in\Omega x ∈ Ω
We denote by P log \mathcal{P}^{\log} P l o g r ( ⋅ ) ∈ L H 0 ( R n ) ∩ L H ∞ ( R n ) r(\cdot) \in LH_{0}(\mathbb{R}^{n}) \cap LH_{\infty}(\mathbb{R}^{n}) r ( ⋅ ) ∈ L H 0 ( R n ) ∩ L H ∞ ( R n )
In [3 ] the authors prove the following lemmas.
Lemma 2.2. Let r ( ⋅ ) ∈ L H 0 ( R n ) r(\cdot) \in LH_{0}(\mathbb{R}^{n}) r ( ⋅ ) ∈ L H 0 ( R n ) C > 0 C > 0 C > 0 n n n Q ⊂ R n Q \subset\mathbb{R}^{n} Q ⊂ R n x ∈ Q x \in Q x ∈ Q
∣ Q ∣ r ( x ) − r + ( Q ) ≤ C and ∣ Q ∣ r − ( Q ) − r ( x ) ≤ C . |Q|^{r(x)-r_{+}(Q)} \le C \qquad\text{and} \qquad|Q|^{r_{-}(Q)-r(x)} \le C. ∣ Q ∣ r ( x ) − r + ( Q ) ≤ C and ∣ Q ∣ r − ( Q ) − r ( x ) ≤ C . Lemma 2.3. Given a set Ω ⊆ R n \Omega\subseteq\mathbb{R}^{n} Ω ⊆ R n r ( ⋅ ) r(\cdot) r ( ⋅ ) s ( ⋅ ) s(\cdot) s ( ⋅ )
∣ s ( y ) − r ( y ) ∣ ≤ C log ( e + ∣ y ∣ ) |s(y)-r(y)| \le\frac{C}{\log(e+|y|)} ∣ s ( y ) − r ( y ) ∣ ≤ log ( e + ∣ y ∣ ) C and a non-negative measure μ \mu μ t ≥ 1 t \ge1 t ≥ 1 C > 0 C > 0 C > 0 F F F 0 ≤ F ( y ) ≤ 1 0 \le F(y) \le1 0 ≤ F ( y ) ≤ 1
∫ Ω F ( y ) s ( y ) d μ ( y ) ≤ C ( ∫ Ω F ( y ) r ( y ) d μ ( y ) + ∫ Ω ( e + ∣ y ∣ ) − N r − d μ ( y ) ) . \int_{\Omega} F(y)^{s(y)}\,d\mu(y) \le C\left(\int_{\Omega} F(y)^{r(y)}\,d\mu(y) + \int_{\Omega}(e+|y|)^{-Nr_{-}}\,d\mu(y)\right). ∫ Ω F ( y ) s ( y ) d μ ( y ) ≤ C ( ∫ Ω F ( y ) r ( y ) d μ ( y ) + ∫ Ω ( e + ∣ y ∣ ) − N r − d μ ( y ) ) . It is well known that the Hardy-Littlewood maximal operator is bounded on L p ( ⋅ ) ( R n ) L^{p(\cdot)}(\mathbb{R}^{n}) L p ( ⋅ ) ( R n ) r ( ⋅ ) ∈ P log r(\cdot) \in\mathcal{P}^{\log} r ( ⋅ ) ∈ P l o g
We recall several basic properties of variable Lebesgue spaces. We next state another useful property concerning the norm-modular comparison.
Lemma 2.4. Let p ( ⋅ ) ∈ P ( R n ) p(\cdot) \in\mathcal{P}(\mathbb{R}^{n}) p ( ⋅ ) ∈ P ( R n ) p + < ∞ p_{+} < \infty p + < ∞ Ω ⊂ R n \Omega\subset\mathbb{R}^{n} Ω ⊂ R n ∣ Ω ∣ > 0 |\Omega| > 0 ∣Ω∣ > 0
(1) If ∥ f χ Ω ∥ p ( ⋅ ) ≤ 1 \|f\chi_{\Omega}\|_{p(\cdot)} \le1 ∥ f χ Ω ∥ p ( ⋅ ) ≤ 1
∥ f χ Ω ∥ p ( ⋅ ) p + ( Ω ) ≤ ∫ Ω ∣ f ( x ) ∣ p ( x ) d x ≤ ∥ f χ Ω ∥ p ( ⋅ ) p − ( Ω ) . \|f\chi_{\Omega}\|_{p(\cdot)}^{p_{+}(\Omega)} \le\int_{\Omega}|f(x)|^{p(x)}\,dx \le\|f\chi_{\Omega}\|_{p(\cdot)}^{p_{-}(\Omega)}. ∥ f χ Ω ∥ p ( ⋅ ) p + ( Ω ) ≤ ∫ Ω ∣ f ( x ) ∣ p ( x ) d x ≤ ∥ f χ Ω ∥ p ( ⋅ ) p − ( Ω ) . (2) If ∥ f χ Ω ∥ p ( ⋅ ) ≥ 1 \|f\chi_{\Omega}\|_{p(\cdot)} \ge1 ∥ f χ Ω ∥ p ( ⋅ ) ≥ 1
∥ f χ Ω ∥ p ( ⋅ ) p − ( Ω ) ≤ ∫ Ω ∣ f ( x ) ∣ p ( x ) d x ≤ ∥ f χ Ω ∥ p ( ⋅ ) p + ( Ω ) . \|f\chi_{\Omega}\|_{p(\cdot)}^{p_{-}(\Omega)} \le\int_{\Omega}|f(x)|^{p(x)}\,dx \le\|f\chi_{\Omega}\|_{p(\cdot)}^{p_{+}(\Omega)}. ∥ f χ Ω ∥ p ( ⋅ ) p − ( Ω ) ≤ ∫ Ω ∣ f ( x ) ∣ p ( x ) d x ≤ ∥ f χ Ω ∥ p ( ⋅ ) p + ( Ω ) . In particular, if ∥ f ∥ p ( ⋅ ) ≤ 1 \|f\|_{p(\cdot)} \le1 ∥ f ∥ p ( ⋅ ) ≤ 1
∫ R n ∣ f ( x ) ∣ p ( x ) d x ≤ ∥ f ∥ p ( ⋅ ) . \int_{\mathbb{R}^{n}}|f(x)|^{p(x)}\,dx \le\|f\|_{p(\cdot)}. ∫ R n ∣ f ( x ) ∣ p ( x ) d x ≤ ∥ f ∥ p ( ⋅ ) . Lemma 2.5. Let p ( ⋅ ) ∈ P ( R n ) p(\cdot) \in\mathcal{P}(\mathbb{R}^{n}) p ( ⋅ ) ∈ P ( R n ) p + < ∞ p_{+}<\infty p + < ∞ ∥ f ∥ p ( ⋅ ) = 1 \|f\|_{p(\cdot)}=1 ∥ f ∥ p ( ⋅ ) = 1
∫ R n ∣ f ( x ) ∣ p ( x ) d x = 1. \int_{\mathbb{R}^{n}} |f(x)|^{p(x)}\,dx=1. ∫ R n ∣ f ( x ) ∣ p ( x ) d x = 1. As a consequence of Lemma 2.5, for every nonzero f ∈ L p ( ⋅ ) ( R n ) f\in L^{p(\cdot)}(\mathbb{R}^{n}) f ∈ L p ( ⋅ ) ( R n )
ρ p ( ⋅ ) ( f ∥ f ∥ p ( ⋅ ) ) = 1. \rho_{p(\cdot)}\left(\frac{f}{\|f\|_{p(\cdot)}}\right)=1. ρ p ( ⋅ ) ( ∥ f ∥ p ( ⋅ ) f ) = 1. The variable version of Hölder’s inequality is the following
Lemma 2.6 (Hölder’s inequality). Let p ( ⋅ ) ∈ P ( R n ) p(\cdot)\in\mathcal{P}(\mathbb{R}^{n}) p ( ⋅ ) ∈ P ( R n ) p ′ ( ⋅ ) p'(\cdot) p ′ ( ⋅ ) p ( ⋅ ) p(\cdot) p ( ⋅ ) C > 0 C>0 C > 0
∫ R n ∣ f ( x ) g ( x ) ∣ d x ≤ C ∥ f ∥ p ( ⋅ ) ∥ g ∥ p ′ ( ⋅ ) \int_{\mathbb{R}^{n}} |f(x)g(x)|\,dx\leq C\|f\|_{p(\cdot)}\|g\|_{p'(\cdot)} ∫ R n ∣ f ( x ) g ( x ) ∣ d x ≤ C ∥ f ∥ p ( ⋅ ) ∥ g ∥ p ′ ( ⋅ ) for every f ∈ L p ( ⋅ ) ( R n ) f\in L^{p(\cdot)}(\mathbb{R}^{n}) f ∈ L p ( ⋅ ) ( R n ) g ∈ L p ′ ( ⋅ ) ( R n ) g\in L^{p'(\cdot)}(\mathbb{R}^{n}) g ∈ L p ′ ( ⋅ ) ( R n )
The following lemma describes a property related to the homogeneity of the norm in the exponent.
Lemma 2.7. Let p ( ⋅ ) ∈ P ( R n ) p(\cdot)\in\mathcal{P}(\mathbb{R}^{n}) p ( ⋅ ) ∈ P ( R n ) s s s 1 / p − ≤ s < ∞ 1/p^{-}\leq s<\infty 1/ p − ≤ s < ∞
∥ ∣ f ∣ s ∥ p ( ⋅ ) = ∥ f ∥ s p ( ⋅ ) s . \||f|^{s}\|_{p(\cdot)}=\|f\|_{sp(\cdot)}^{s}. ∥∣ f ∣ s ∥ p ( ⋅ ) = ∥ f ∥ s p ( ⋅ ) s . In [16 ] the authors prove the following result.
Proposition 2.8. [16 ] Let A A A p ( ⋅ ) ∈ P ( R n ) p(\cdot)\in\mathcal{P}(\mathbb{R}^{n}) p ( ⋅ ) ∈ P ( R n ) p ( A x ) = p ( x ) p(Ax)=p(x) p ( A x ) = p ( x ) x x x
∥ f ∘ A ∥ p ( ⋅ ) ≤ max { 1 , ∣ det ( A ) ∣ 1 p − } ∥ f ∥ p ( ⋅ ) , \|f\circ A\|_{p(\cdot)}\leq\max\{1,|\det(A)|^{\frac{1}{p^{-}}}\}\|f\|_{p(\cdot)}, ∥ f ∘ A ∥ p ( ⋅ ) ≤ max { 1 , ∣ det ( A ) ∣ p − 1 } ∥ f ∥ p ( ⋅ ) , for all f ∈ L p ( ⋅ ) ( R n ) f\in L^{p(\cdot)}(\mathbb{R}^{n}) f ∈ L p ( ⋅ ) ( R n )
Weighted Variable Lebesgue spaces
A locally integrable function w : R n → [ 0 , ∞ ) w:\mathbb{R}^{n}\to[0,\infty) w : R n → [ 0 , ∞ ) p ( ⋅ ) ∈ P ( R n ) p(\cdot)\in\mathcal{P}(\mathbb{R}^{n}) p ( ⋅ ) ∈ P ( R n )
ρ p ( ⋅ ) , w ( f ) = ∫ R n ∣ f ( x ) ∣ p ( x ) w ( x ) d x . (3) \rho_{p(\cdot),w}(f)=\int_{\mathbb{R}^{n}}|f(x)|^{p(x)}w(x)\,dx. \tag*{(3)} ρ p ( ⋅ ) , w ( f ) = ∫ R n ∣ f ( x ) ∣ p ( x ) w ( x ) d x . (3) The space L w p ( ⋅ ) ( R n ) L_{w}^{p(\cdot)}(\mathbb{R}^{n}) L w p ( ⋅ ) ( R n ) f f f ρ p ( ⋅ ) , w ( f / λ ) < ∞ \rho_{p(\cdot),w}(f/\lambda)<\infty ρ p ( ⋅ ) , w ( f / λ ) < ∞ λ > 0 \lambda>0 λ > 0
The norm ∥ ⋅ ∥ p ( ⋅ ) , w \|\cdot\|_{p(\cdot),w} ∥ ⋅ ∥ p ( ⋅ ) , w L w p ( ⋅ ) ( R n ) L_{w}^{p(\cdot)}(\mathbb{R}^{n}) L w p ( ⋅ ) ( R n )
∥ f ∥ p ( ⋅ ) , w = ∥ f w ∥ p ( ⋅ ) (4) \|f\|_{p(\cdot),w}=\|fw\|_{p(\cdot)} \tag*{(4)} ∥ f ∥ p ( ⋅ ) , w = ∥ f w ∥ p ( ⋅ ) (4) Note that since w ∈ L l o c p ( ⋅ ) w\in L_{\mathrm{loc}}^{p(\cdot)} w ∈ L loc p ( ⋅ ) L w p ( ⋅ ) L_{w}^{p(\cdot)} L w p ( ⋅ ) L w p ( ⋅ ) ( Q ) L_{w}^{p(\cdot)}(Q) L w p ( ⋅ ) ( Q ) Q Q Q L w p ( ⋅ ) ( Q ) L_{w}^{p(\cdot)}(Q) L w p ( ⋅ ) ( Q ) L w − 1 p ′ ( ⋅ ) ( Q ) L_{w^{-1}}^{p'(\cdot)}(Q) L w − 1 p ′ ( ⋅ ) ( Q ) [8 Theorem 2.7.4] . The following norm conjugate formula holds for L w p ( ⋅ ) ( Q ) L_{w}^{p(\cdot)}(Q) L w p ( ⋅ ) ( Q ) [8 Corollary 2.7.5] ,
1 2 ∥ f ∥ L w p ( ⋅ ) ( Q ) ≤ sup ∥ g ∥ L w − 1 p ′ ( ⋅ ) ( Q ) ≤ 1 ∫ Q ∣ f ( x ) ∣ ∣ g ( x ) ∣ d x ≤ 2 ∥ f ∥ L w p ( ⋅ ) ( Q ) . \frac{1}{2}\|f\|_{L_{w}^{p(\cdot)}(Q)}
\leq
\sup_{\|g\|_{L_{w^{-1}}^{p'(\cdot)}(Q)}\leq1}
\int_{Q}|f(x)||g(x)|\,dx
\leq
2\|f\|_{L_{w}^{p(\cdot)}(Q)}. 2 1 ∥ f ∥ L w p ( ⋅ ) ( Q ) ≤ ∥ g ∥ L w − 1 p ′ ( ⋅ ) ( Q ) ≤ 1 sup ∫ Q ∣ f ( x ) ∣∣ g ( x ) ∣ d x ≤ 2∥ f ∥ L w p ( ⋅ ) ( Q ) . Therefore, by monotone convergence the norm conjugate formula also holds for L w p ( ⋅ ) ( R n ) L^{p(\cdot)}_w(\mathbb{R}^n) L w p ( ⋅ ) ( R n ) L w − 1 p ′ ( ⋅ ) ( R n ) L^{p'(\cdot)}_{w^{-1}}(\mathbb{R}^n) L w − 1 p ′ ( ⋅ ) ( R n ) L w p ( ⋅ ) ( R n ) L^{p(\cdot)}_w(\mathbb{R}^n) L w p ( ⋅ ) ( R n )
In [6 ] , Cruz-Uribe and Penrod proved the a variable reverse Hölder, in particular showed that the class A p ( ⋅ ) A_{p(\cdot)} A p ( ⋅ )
Proposition 2.9. [6 ] Given p ( ⋅ ) ∈ P log p(\cdot) \in P^{\log} p ( ⋅ ) ∈ P l o g with p + < ∞ p_+ < \infty p + < ∞ and w ∈ A p ( ⋅ ) w \in A_{p(\cdot)} w ∈ A p ( ⋅ ) then
i ) i) i ) r > 1 r > 1 r > 1 s ∈ [ 1 , r ] s \in[1,r] s ∈ [ 1 , r ] w ∈ A s p ( ⋅ ) w \in A_{sp(\cdot)} w ∈ A s p ( ⋅ )
i i ) ii) ii ) r > 1 r > 1 r > 1 s ∈ [ 1 , r ] s \in[1,r] s ∈ [ 1 , r ] q ( ⋅ ) = s p ′ ( ⋅ ) q(\cdot) = sp'(\cdot) q ( ⋅ ) = s p ′ ( ⋅ ) w ∈ A q ( ⋅ ) w \in A_{q(\cdot)} w ∈ A q ( ⋅ )
Maximal operator M A − 1 M_{A^{-1}} M A − 1
In the context of classical Lebesgue spaces, this type of maximal operator was widely studied, we refer to [9 , 10 , 11 ] .
In [11 ] , the authors introduce the class of weight A A , p \mathcal{A}_{A,p} A A , p w w w A A , p \mathcal{A}_{A,p} A A , p
[ w ] A A , p = sup Q ( 1 ∣ Q ∣ ∫ Q w ( A x ) d x ) ( 1 ∣ Q ∣ ∫ Q w ( x ) − 1 p − 1 d x ) p − 1 < ∞ [w]_{\mathcal{A}_{A,p}}=\sup_Q\left(\frac{1}{|Q|}\int_Q w(Ax)\,dx\right)\left(\frac{1}{|Q|}\int_Q w(x)^{-\frac{1}{p-1}}\,dx\right)^{p-1}<\infty [ w ] A A , p = Q sup ( ∣ Q ∣ 1 ∫ Q w ( A x ) d x ) ( ∣ Q ∣ 1 ∫ Q w ( x ) − p − 1 1 d x ) p − 1 < ∞ where the supremum is over all cubes Q Q Q
They showed that M A − 1 M_{A^{-1}} M A − 1 L p ( w ) L^p(w) L p ( w ) L p , ∞ ( w ) L^{p,\infty}(w) L p , ∞ ( w ) w ∈ A A , p w \in\mathcal{A}_{A,p} w ∈ A A , p
Now, we present several properties of this class of weights
Proposition 2.10. [10 ] Let A A A be a invertible matrix and w w w be a non-negative function. Then, M A − 1 w ( x ) = M ( w A ) ( x ) M_{A^{-1}}w(x)=M(w_A)(x) M A − 1 w ( x ) = M ( w A ) ( x )
Proposition 2.11. [11 ] Let A , B A,B A , B be invertible matrices.
i ) i) i ) w ∈ A A , p w \in\mathcal{A}_{A,p} w ∈ A A , p w ( A x ) ≤ [ w ] A A , p w ( x ) w(Ax) \le[w]_{\mathcal{A}_{A,p}}w(x) w ( A x ) ≤ [ w ] A A , p w ( x ) x ∈ R n x \in\mathbb{R}^n x ∈ R n
i i ) ii) ii ) w ∈ A A , p w \in\mathcal{A}_{A,p} w ∈ A A , p w ∈ A B , p w \in\mathcal{A}_{B,p} w ∈ A B , p w ∈ A A B , p w \in\mathcal{A}_{AB,p} w ∈ A A B , p
Recently, in [9 ] the first author proved that for some particular matrix A A A A A , p \mathcal{A}_{A,p} A A , p
Proposition 2.12. [9 ] Let A A A be an invertible matrix such that A k = I A^k=I A k = I for some k ∈ N k \in\mathbb{N} k ∈ N Then A A , p = { w ∈ A p : w ( A x ) ≲ w ( x ) a.e. x ∈ R n } \mathcal{A}_{A,p}=\{w \in A_p:w(Ax)\lesssim w(x)\ \text{a.e. }x\in\mathbb{R}^n\} A A , p = { w ∈ A p : w ( A x ) ≲ w ( x ) a.e. x ∈ R n }
Moreover, in [9 ] the following characterization for the weights was proved
Theorem 2.13. [9 ] Let A A A be an invertible matrix such that A k = I A^k=I A k = I for some k ∈ N k \in\mathbb{N} k ∈ N Then M A − 1 M_{A^{-1}} M A − 1 is bounded on L p ( w ) L^p(w) L p ( w ) if and only if w ∈ A A , p w \in\mathcal{A}_{A,p} w ∈ A A , p
Necessary condition
In this section, we prove the necessary conditions Theorem 1.1 and Theorem 1.2.
Proof of Theorem 1.1. Since M A − 1 M_{A^{-1}} M A − 1 L p ( ⋅ ) ( w ) L^{p(\cdot)}(w) L p ( ⋅ ) ( w ) f ∈ L p ( ⋅ ) ( w ) f \in L^{p(\cdot)}(w) f ∈ L p ( ⋅ ) ( w )
∥ M A − 1 f ∥ L p ( ⋅ ) ( w ) ≲ ∥ f ∥ L p ( ⋅ ) ( w ) \|M_{A^{-1}}f\|_{L^{p(\cdot)}(w)}\lesssim\|f\|_{L^{p(\cdot)}(w)} ∥ M A − 1 f ∥ L p ( ⋅ ) ( w ) ≲ ∥ f ∥ L p ( ⋅ ) ( w ) Since p ( A x ) = p ( x ) p(Ax)=p(x) p ( A x ) = p ( x ) x x x Proposition 2.8 we have
∥ M A − 1 f ∥ L p ( ⋅ ) ( w ) = ∥ w M A − 1 f ∥ L p ( ⋅ ) ≳ ∥ [ w M A − 1 f ] ∘ A ∥ L p ( ⋅ ) = ∥ w A M f ∥ L p ( ⋅ ) \|M_{A^{-1}}f\|_{L^{p(\cdot)}(w)}=\|wM_{A^{-1}}f\|_{L^{p(\cdot)}}\gtrsim\|[wM_{A^{-1}}f]\circ A\|_{L^{p(\cdot)}}=\|w_AMf\|_{L^{p(\cdot)}} ∥ M A − 1 f ∥ L p ( ⋅ ) ( w ) = ∥ w M A − 1 f ∥ L p ( ⋅ ) ≳ ∥ [ w M A − 1 f ] ∘ A ∥ L p ( ⋅ ) = ∥ w A M f ∥ L p ( ⋅ ) For any cube Q ⊂ R n Q\subset\mathbb{R}^n Q ⊂ R n M f ≥ ( 1 ∣ Q ∣ ∫ Q ∣ f ∣ ) χ Q Mf\ge\left(\frac{1}{|Q|}\int_Q|f|\right)\chi_Q M f ≥ ( ∣ Q ∣ 1 ∫ Q ∣ f ∣ ) χ Q
∥ w A M f ∥ L p ( ⋅ ) ≥ ( 1 ∣ Q ∣ ∫ Q ∣ f ∣ ) ∥ χ Q w A ∥ p ( ⋅ ) \|w_AMf\|_{L^{p(\cdot)}}\ge\left(\frac{1}{|Q|}\int_Q|f|\right)\|\chi_Qw_A\|_{p(\cdot)} ∥ w A M f ∥ L p ( ⋅ ) ≥ ( ∣ Q ∣ 1 ∫ Q ∣ f ∣ ) ∥ χ Q w A ∥ p ( ⋅ ) Now, taking
f ( x ) = λ ( w − 1 ( x ) λ ) p ′ ( x ) χ Q ( x ) f(x)=\lambda\left(\frac{w^{-1}(x)}{\lambda}\right)^{p'(x)}\chi_Q(x) f ( x ) = λ ( λ w − 1 ( x ) ) p ′ ( x ) χ Q ( x ) with λ \lambda λ
1 ∣ Q ∣ ∫ Q ∣ f ∣ = λ ∣ Q ∣ ∫ Q ( w − 1 ( x ) λ ) p ′ ( x ) d x = λ ∣ Q ∣ ρ p ′ ( ⋅ ) ( w − 1 χ Q λ ) \frac{1}{|Q|}\int_Q |f|=\frac{\lambda}{|Q|}\int_Q\left(\frac{w^{-1}(x)}{\lambda}\right)^{p'(x)}\,dx=\frac{\lambda}{|Q|}\rho_{p'(\cdot)}\left(\frac{w^{-1}\chi_Q}{\lambda}\right) ∣ Q ∣ 1 ∫ Q ∣ f ∣ = ∣ Q ∣ λ ∫ Q ( λ w − 1 ( x ) ) p ′ ( x ) d x = ∣ Q ∣ λ ρ p ′ ( ⋅ ) ( λ w − 1 χ Q ) and
ρ p ( ⋅ ) ( f w ) = ∫ Q ( w − p ′ ( x ) ( x ) w ( x ) λ p ′ ( x ) − 1 ) p ( x ) d x = ∫ Q ( w − 1 ( x ) λ ) p ′ ( x ) d x = ρ p ′ ( ⋅ ) ( w − 1 χ Q λ ) . \begin{aligned}
\rho_{p(\cdot)}(fw)=\int_Q\left(\frac{w^{-p'(x)}(x)w(x)}{\lambda^{p'(x)-1}}\right)^{p(x)}\,dx=\int_Q\left(\frac{w^{-1}(x)}{\lambda}\right)^{p'(x)}\,dx \\
&=\rho_{p'(\cdot)}\left(\frac{w^{-1}\chi_Q}{\lambda}\right).
\end{aligned} ρ p ( ⋅ ) ( f w ) = ∫ Q ( λ p ′ ( x ) − 1 w − p ′ ( x ) ( x ) w ( x ) ) p ( x ) d x = ∫ Q ( λ w − 1 ( x ) ) p ′ ( x ) d x = ρ p ′ ( ⋅ ) ( λ w − 1 χ Q ) . Now, taking λ = ∥ w − 1 χ Q ∥ p ′ ( ⋅ ) \lambda=\|w^{-1}\chi_Q\|_{p'(\cdot)} λ = ∥ w − 1 χ Q ∥ p ′ ( ⋅ ) ρ p ( ⋅ ) ( f w ) = ρ p ′ ( ⋅ ) ( w − 1 χ Q ∥ w − 1 χ Q ∥ p ′ ( ⋅ ) ) = 1 \rho_{p(\cdot)}(fw)=\rho_{p'(\cdot)}\left(\frac{w^{-1}\chi_Q}{\|w^{-1}\chi_Q\|_{p'(\cdot)}}\right)=1 ρ p ( ⋅ ) ( f w ) = ρ p ′ ( ⋅ ) ( ∥ w − 1 χ Q ∥ p ′ ( ⋅ ) w − 1 χ Q ) = 1 ∥ f w ∥ p ( ⋅ ) ≤ 1 \|fw\|_{p(\cdot)}\le1 ∥ f w ∥ p ( ⋅ ) ≤ 1
On the other hand,
1 ≳ ∥ M A − 1 f ∥ L p ( ⋅ ) ( w ) ≥ ( 1 ∣ Q ∣ ∫ Q ∣ f ∣ ) ∥ χ Q w A ∥ p ( ⋅ ) = λ ∣ Q ∣ ρ p ′ ( ⋅ ) ( w − 1 χ Q λ ) ∥ χ Q w A ∥ p ( ⋅ ) = ∥ w − 1 χ Q ∥ p ′ ( ⋅ ) ∣ Q ∣ ρ p ′ ( ⋅ ) ( w − 1 χ Q ∥ w − 1 χ Q ∥ p ′ ( ⋅ ) ) ∥ χ Q w A ∥ p ( ⋅ ) = ∥ w − 1 χ Q ∥ p ′ ( ⋅ ) ∣ Q ∣ ∥ χ Q w A ∥ p ( ⋅ ) . \begin{aligned}
1\gtrsim\|M_{A^{-1}}f\|_{L^{p(\cdot)}(w)}\geq\left(\frac{1}{|Q|}\int_Q |f|\right)\|\chi_Qw_A\|_{p(\cdot)} \\
&=\frac{\lambda}{|Q|}\rho_{p'(\cdot)}\left(\frac{w^{-1}\chi_Q}{\lambda}\right)\|\chi_Qw_A\|_{p(\cdot)} \\
&=\frac{\|w^{-1}\chi_Q\|_{p'(\cdot)}}{|Q|}\rho_{p'(\cdot)}\left(\frac{w^{-1}\chi_Q}{\|w^{-1}\chi_Q\|_{p'(\cdot)}}\right)\|\chi_Qw_A\|_{p(\cdot)} \\
&=\frac{\|w^{-1}\chi_Q\|_{p'(\cdot)}}{|Q|}\|\chi_Qw_A\|_{p(\cdot)}.
\end{aligned} 1 ≳ ∥ M A − 1 f ∥ L p ( ⋅ ) ( w ) ≥ ( ∣ Q ∣ 1 ∫ Q ∣ f ∣ ) ∥ χ Q w A ∥ p ( ⋅ ) = ∣ Q ∣ λ ρ p ′ ( ⋅ ) ( λ w − 1 χ Q ) ∥ χ Q w A ∥ p ( ⋅ ) = ∣ Q ∣ ∥ w − 1 χ Q ∥ p ′ ( ⋅ ) ρ p ′ ( ⋅ ) ( ∥ w − 1 χ Q ∥ p ′ ( ⋅ ) w − 1 χ Q ) ∥ χ Q w A ∥ p ( ⋅ ) = ∣ Q ∣ ∥ w − 1 χ Q ∥ p ′ ( ⋅ ) ∥ χ Q w A ∥ p ( ⋅ ) . Now, taking account the estimates above we obtain
∥ w − 1 χ Q ∥ p ′ ( ⋅ ) ∣ Q ∣ ∥ χ Q w A ∥ p ( ⋅ ) ≤ ∥ M A − 1 f ∥ L p ( ⋅ ) ( w ) ≤ ∥ M A − 1 ∥ L p ( ⋅ ) ( w ) → L p ( ⋅ ) ( w ) ∥ f w ∥ p ( ⋅ ) ≤ ∥ M A − 1 ∥ L p ( ⋅ ) ( w ) → L p ( ⋅ ) ( w ) \begin{aligned}
\frac{\|w^{-1}\chi_Q\|_{p'(\cdot)}}{|Q|}\|\chi_Qw_A\|_{p(\cdot)}
\leq\|M_{A^{-1}}f\|_{L^{p(\cdot)}(w)} \\
&\leq\|M_{A^{-1}}\|_{L^{p(\cdot)}(w)\to L^{p(\cdot)}(w)}\|fw\|_{p(\cdot)} \\
&\leq\|M_{A^{-1}}\|_{L^{p(\cdot)}(w)\to L^{p(\cdot)}(w)}
\end{aligned} ∣ Q ∣ ∥ w − 1 χ Q ∥ p ′ ( ⋅ ) ∥ χ Q w A ∥ p ( ⋅ ) ≤ ∥ M A − 1 f ∥ L p ( ⋅ ) ( w ) ≤ ∥ M A − 1 ∥ L p ( ⋅ ) ( w ) → L p ( ⋅ ) ( w ) ∥ f w ∥ p ( ⋅ ) ≤ ∥ M A − 1 ∥ L p ( ⋅ ) ( w ) → L p ( ⋅ ) ( w ) for any cube Q Q Q [ w ] A A , p ( ⋅ ) ≤ ∥ M A − 1 ∥ L p ( ⋅ ) ( w ) → L p ( ⋅ ) ( w ) [w]_{A_{A,p(\cdot)}}\leq\|M_{A^{-1}}\|_{L^{p(\cdot)}(w)\to L^{p(\cdot)}(w)} [ w ] A A , p ( ⋅ ) ≤ ∥ M A − 1 ∥ L p ( ⋅ ) ( w ) → L p ( ⋅ ) ( w ) w ∈ A A , p ( ⋅ ) w\in A_{A,p(\cdot)} w ∈ A A , p ( ⋅ )
Proof of Theorem 1.2. Set
p 0 = p ( x 0 ) , p 1 = p ( A x 0 ) , p_0=p(x_0),\qquad p_1=p(Ax_0), p 0 = p ( x 0 ) , p 1 = p ( A x 0 ) , so that p 0 < p 1 p_0<p_1 p 0 < p 1 ε > 0 \varepsilon>0 ε > 0
p 0 + ε < p 1 − ε . p_0+\varepsilon<p_1-\varepsilon. p 0 + ε < p 1 − ε . By the continuity of p p p x 0 x_0 x 0 r 0 > 0 r_0>0 r 0 > 0 0 < r < r 0 0<r<r_0 0 < r < r 0 x ∈ Q r : = Q ( x 0 , r ) x\in Q_r:=Q(x_0,r) x ∈ Q r := Q ( x 0 , r )
p 0 − ε ≤ p ( x ) ≤ p 0 + ε . (5) p_0-\varepsilon\leq p(x)\leq p_0+\varepsilon. \tag*{(5)} p 0 − ε ≤ p ( x ) ≤ p 0 + ε . (5) Similarly, by the continuity of p p p A x 0 Ax_0 A x 0 r 0 r_0 r 0
p 1 − ε ≤ p ( x ) ≤ p 1 + ε , x ∈ A Q r . (6) p_1-\varepsilon\le p(x) \le p_1+\varepsilon,\qquad x\in AQ_r. \tag*{(6)} p 1 − ε ≤ p ( x ) ≤ p 1 + ε , x ∈ A Q r . (6) Indeed, A Q r AQ_r A Q r A x 0 Ax_0 A x 0 r → 0 r\to0 r → 0
For r r r
∣ Q r ∣ < 1 and ∣ A Q r ∣ < 1. |Q_r|<1\qquad\text{and}\qquad|AQ_r|<1. ∣ Q r ∣ < 1 and ∣ A Q r ∣ < 1. First, observe that
χ Q r ∈ L p ( ⋅ ) ( R n ) , \chi_{Q_r}\in L^{p(\cdot)}(\mathbb{R}^n), χ Q r ∈ L p ( ⋅ ) ( R n ) , since
ρ p ( ⋅ ) ( χ Q r ) = ∫ R n ∣ χ Q r ( x ) ∣ p ( x ) d x = ∣ Q r ∣ < ∞ . \rho_{p(\cdot)}(\chi_{Q_r})=\int_{\mathbb{R}^n}|\chi_{Q_r}(x)|^{p(x)}\,dx=|Q_r|<\infty. ρ p ( ⋅ ) ( χ Q r ) = ∫ R n ∣ χ Q r ( x ) ∣ p ( x ) d x = ∣ Q r ∣ < ∞. We now estimate ∥ χ Q r ∥ p ( ⋅ ) \|\chi_{Q_r}\|_{p(\cdot)} ∥ χ Q r ∥ p ( ⋅ )
λ = ∣ Q r ∣ 1 / ( p 0 + ε ) . \lambda=|Q_r|^{1/(p_0+\varepsilon)}. λ = ∣ Q r ∣ 1/ ( p 0 + ε ) . Since ∣ Q r ∣ < 1 |Q_r|<1 ∣ Q r ∣ < 1 0 < λ < 1 0<\lambda<1 0 < λ < 1 (5) ,
λ − p ( x ) ≤ λ − ( p 0 + ε ) = ∣ Q r ∣ − 1 , x ∈ Q r . \lambda^{-p(x)}\le\lambda^{-(p_0+\varepsilon)}=|Q_r|^{-1},\qquad x\in Q_r. λ − p ( x ) ≤ λ − ( p 0 + ε ) = ∣ Q r ∣ − 1 , x ∈ Q r . Hence
∫ Q r λ − p ( x ) d x ≤ ∣ Q r ∣ ∣ Q r ∣ − 1 = 1 , \int_{Q_r}\lambda^{-p(x)}\,dx\le|Q_r||Q_r|^{-1}=1, ∫ Q r λ − p ( x ) d x ≤ ∣ Q r ∣∣ Q r ∣ − 1 = 1 , and therefore
∥ χ Q r ∥ p ( ⋅ ) ≤ ∣ Q r ∣ 1 / ( p 0 + ε ) . (7) \|\chi_{Q_r}\|_{p(\cdot)}\le|Q_r|^{1/(p_0+\varepsilon)}. \tag*{(7)} ∥ χ Q r ∥ p ( ⋅ ) ≤ ∣ Q r ∣ 1/ ( p 0 + ε ) . (7) On the other hand, take
λ = ∣ Q r ∣ 1 / ( p 0 − ε ) . \lambda=|Q_r|^{1/(p_0-\varepsilon)}. λ = ∣ Q r ∣ 1/ ( p 0 − ε ) . Again, 0 < λ < 1 0<\lambda<1 0 < λ < 1 (5) ,
λ − p ( x ) ≥ λ − ( p 0 − ε ) = ∣ Q r ∣ − 1 , x ∈ Q r . \lambda^{-p(x)}\ge\lambda^{-(p_0-\varepsilon)}=|Q_r|^{-1},\qquad x\in Q_r. λ − p ( x ) ≥ λ − ( p 0 − ε ) = ∣ Q r ∣ − 1 , x ∈ Q r . Thus
∫ Q r λ − p ( x ) d x ≥ 1 , \int_{Q_r}\lambda^{-p(x)}\,dx\ge1, ∫ Q r λ − p ( x ) d x ≥ 1 , which implies
∥ χ Q r ∥ p ( ⋅ ) ≥ ∣ Q r ∣ 1 / ( p 0 − ε ) . (8) \|\chi_{Q_r}\|_{p(\cdot)}\ge|Q_r|^{1/(p_0-\varepsilon)}. \tag*{(8)} ∥ χ Q r ∥ p ( ⋅ ) ≥ ∣ Q r ∣ 1/ ( p 0 − ε ) . (8) Consequently,
∣ Q r ∣ 1 / ( p 0 − ε ) ≤ ∥ χ Q r ∥ p ( ⋅ ) ≤ ∣ Q r ∣ 1 / ( p 0 + ε ) . (9) |Q_r|^{1/(p_0-\varepsilon)}\le\|\chi_{Q_r}\|_{p(\cdot)}\le|Q_r|^{1/(p_0+\varepsilon)}. \tag*{(9)} ∣ Q r ∣ 1/ ( p 0 − ε ) ≤ ∥ χ Q r ∥ p ( ⋅ ) ≤ ∣ Q r ∣ 1/ ( p 0 + ε ) . (9) Similarly, using (6) , we obtain
∣ A Q r ∣ 1 / ( p 1 − ε ) ≤ ∥ χ A Q r ∥ p ( ⋅ ) ≤ ∣ A Q r ∣ 1 / ( p 1 + ε ) . (10) |AQ_r|^{1/(p_1-\varepsilon)}\le\|\chi_{AQ_r}\|_{p(\cdot)}\le|AQ_r|^{1/(p_1+\varepsilon)}. \tag*{(10)} ∣ A Q r ∣ 1/ ( p 1 − ε ) ≤ ∥ χ A Q r ∥ p ( ⋅ ) ≤ ∣ A Q r ∣ 1/ ( p 1 + ε ) . (10) Now, if x ∈ A Q r x\in AQ_r x ∈ A Q r M A − 1 M_{A^{-1}} M A − 1
M A − 1 χ Q r ( x ) ≥ 1 ∣ Q r ∣ ∫ Q r χ Q r ( y ) d y = 1. M_{A^{-1}}\chi_{Q_r}(x)\ge\frac{1}{|Q_r|}\int_{Q_r}\chi_{Q_r}(y)\,dy=1. M A − 1 χ Q r ( x ) ≥ ∣ Q r ∣ 1 ∫ Q r χ Q r ( y ) d y = 1. Therefore,
M A − 1 χ Q r ≥ χ A Q r , M_{A^{-1}}\chi_{Q_r}\ge\chi_{AQ_r}, M A − 1 χ Q r ≥ χ A Q r , and hence
∥ M A − 1 χ Q r ∥ p ( ⋅ ) ≥ ∥ χ A Q r ∥ p ( ⋅ ) . (11) \|M_{A^{-1}}\chi_{Q_r}\|_{p(\cdot)}\ge\|\chi_{AQ_r}\|_{p(\cdot)}. \tag*{(11)} ∥ M A − 1 χ Q r ∥ p ( ⋅ ) ≥ ∥ χ A Q r ∥ p ( ⋅ ) . (11) Combining (9) , (10) , and (11) , we obtain
∥ M A − 1 χ Q r ∥ p ( ⋅ ) ∥ χ Q r ∥ p ( ⋅ ) ≥ ∣ A Q r ∣ 1 / ( p 1 − ε ) ∣ Q r ∣ 1 / ( p 0 + ε ) . \frac{\lVert M_{A^{-1}}\chi_{Q_r}\rVert_{p(\cdot)}}{\lVert\chi_{Q_r}\rVert_{p(\cdot)}} \ge\frac{\lvert A Q_r\rvert^{1/(p_1-\varepsilon)}}{\lvert Q_r\rvert^{1/(p_0+\varepsilon)}}. ∥ χ Q r ∥ p ( ⋅ ) ∥ M A − 1 χ Q r ∥ p ( ⋅ ) ≥ ∣ Q r ∣ 1/ ( p 0 + ε ) ∣ A Q r ∣ 1/ ( p 1 − ε ) . Since A A A
∣ A Q r ∣ = ∣ det A ∣ ∣ Q r ∣ , \lvert A Q_r\rvert= \lvert\det A\rvert\lvert Q_r\rvert, ∣ A Q r ∣ = ∣ det A ∣ ∣ Q r ∣ , and there
∥ M A − 1 χ Q r ∥ p ( ⋅ ) ∥ χ Q r ∥ p ( ⋅ ) ≥ ∣ det A ∣ 1 / ( p 1 − ε ) ∣ Q r ∣ 1 p 1 − ε − 1 p 0 + ε . (12) \frac{\lVert M_{A^{-1}}\chi_{Q_r}\rVert_{p(\cdot)}}{\lVert\chi_{Q_r}\rVert_{p(\cdot)}} \ge\lvert\det A\rvert^{1/(p_1-\varepsilon)}\lvert Q_r\rvert^{\frac{1}{p_1-\varepsilon}-\frac{1}{p_0+\varepsilon}}. \tag*{(12)} ∥ χ Q r ∥ p ( ⋅ ) ∥ M A − 1 χ Q r ∥ p ( ⋅ ) ≥ ∣ det A ∣ 1/ ( p 1 − ε ) ∣ Q r ∣ p 1 − ε 1 − p 0 + ε 1 . (12) By our choice of ε \varepsilon ε
p 0 + ε < p 1 − ε , p_0+\varepsilon< p_1-\varepsilon, p 0 + ε < p 1 − ε , and hence
1 p 1 − ε − 1 p 0 + ε < 0. \frac{1}{p_1-\varepsilon}-\frac{1}{p_0+\varepsilon}<0. p 1 − ε 1 − p 0 + ε 1 < 0. Since ∣ Q r ∣ → 0 \lvert Q_r\rvert\to0 ∣ Q r ∣ → 0 r → 0 r\to0 r → 0 (12) that
∥ M A − 1 χ Q r ∥ p ( ⋅ ) ∥ χ Q r ∥ p ( ⋅ ) ⟶ + ∞ as r → 0. (13) \frac{\lVert M_{A^{-1}}\chi_{Q_r}\rVert_{p(\cdot)}}{\lVert\chi_{Q_r}\rVert_{p(\cdot)}}\longrightarrow+\infty\quad\text{as }r\to0. \tag*{(13)} ∥ χ Q r ∥ p ( ⋅ ) ∥ M A − 1 χ Q r ∥ p ( ⋅ ) ⟶ + ∞ as r → 0. (13) Suppose, by contradiction, that M A − 1 M_{A^{-1}} M A − 1 L p ( ⋅ ) ( R n ) L^{p(\cdot)}(\mathbb{R}^n) L p ( ⋅ ) ( R n ) C > 0 C>0 C > 0 f f f
∥ M A − 1 f ∥ p ( ⋅ ) ≤ C ∥ f ∥ p ( ⋅ ) \lVert M_{A^{-1}}f\rVert_{p(\cdot)}\le C\lVert f\rVert_{p(\cdot)} ∥ M A − 1 f ∥ p ( ⋅ ) ≤ C ∥ f ∥ p ( ⋅ ) for every f ∈ L p ( ⋅ ) ( R n ) f\in L^{p(\cdot)}(\mathbb{R}^n) f ∈ L p ( ⋅ ) ( R n ) f = χ Q r f=\chi_{Q_r} f = χ Q r
∥ M A − 1 χ Q r ∥ p ( ⋅ ) ∥ χ Q r ∥ p ( ⋅ ) ≤ C \frac{\lVert M_{A^{-1}}\chi_{Q_r}\rVert_{p(\cdot)}}{\lVert\chi_{Q_r}\rVert_{p(\cdot)}}\le C ∥ χ Q r ∥ p ( ⋅ ) ∥ M A − 1 χ Q r ∥ p ( ⋅ ) ≤ C for every sufficiently small r r r (13) . Therefore,
M A − 1 is not bounded on L p ( ⋅ ) ( R n ) . M_{A^{-1}}\text{ is not bounded on }L^{p(\cdot)}(\mathbb{R}^n). M A − 1 is not bounded on L p ( ⋅ ) ( R n ) . Sufficient condition
For the proof of the sufficient condition we need to define the following operators. Let p ( ⋅ ) ∈ P ( R n ) p(\cdot)\in\mathcal{P}(\mathbb{R}^n) p ( ⋅ ) ∈ P ( R n ) Q ⊂ R n Q\subset\mathbb{R}^n Q ⊂ R n
f p ( ⋅ ) , Q : = ∥ χ Q f ∥ p ( ⋅ ) ∥ χ Q ∥ p ( ⋅ ) . (14) f_{p(\cdot),Q}:=\frac{\lVert\chi_Q f\rVert_{p(\cdot)}}{\lVert\chi_Q\rVert_{p(\cdot)}}. \tag*{(14)} f p ( ⋅ ) , Q := ∥ χ Q ∥ p ( ⋅ ) ∥ χ Q f ∥ p ( ⋅ ) . (14) and the maximal operator
M p ( ⋅ ) ( f ) ( x ) = sup Q ∋ x f p ( ⋅ ) , Q . (15) M_{p(\cdot)}(f)(x)=\sup_{Q\ni x} f_{p(\cdot),Q}. \tag*{(15)} M p ( ⋅ ) ( f ) ( x ) = Q ∋ x sup f p ( ⋅ ) , Q . (15) Auxiliary results
In [2 ] D. Cruz Uribe et al. studied this average and proved the following result
Lemma 4.1. [2 ] Let p ( ⋅ ) ∈ P ( R n ) p(\cdot) \in\mathcal{P}(\mathbb{R}^{n}) p ( ⋅ ) ∈ P ( R n ) p + < ∞ p_{+}<\infty p + < ∞ 1 ∈ A p ( ⋅ ) 1\in A_{p(\cdot)} 1 ∈ A p ( ⋅ ) w w w w ∈ A p ( ⋅ ) w\in A_{p(\cdot)} w ∈ A p ( ⋅ )
1 ≲ w p ( ⋅ ) , Q ( w − 1 ) p ′ ( ⋅ ) , Q ≲ [ w ] A p ( ⋅ ) . 1\lesssim w_{p(\cdot),Q}(w^{-1})_{p'(\cdot),Q}\lesssim[w]_{A_{p(\cdot)}}. 1 ≲ w p ( ⋅ ) , Q ( w − 1 ) p ′ ( ⋅ ) , Q ≲ [ w ] A p ( ⋅ ) . for all cubes Q Q Q R n \mathbb{R}^{n} R n
Diening et al. proved the following result for the maximal operator defined in (15) .
Theorem 4.2. [8 ] Let p ( ⋅ ) , q ( ⋅ ) , s ( ⋅ ) ∈ P log ( R n ) p(\cdot),q(\cdot),s(\cdot)\in\mathcal{P}^{\log}(\mathbb{R}^{n}) p ( ⋅ ) , q ( ⋅ ) , s ( ⋅ ) ∈ P l o g ( R n ) p + < ∞ p_{+}<\infty p + < ∞ p ( ⋅ ) = q ( ⋅ ) s ( ⋅ ) p(\cdot)=q(\cdot)s(\cdot) p ( ⋅ ) = q ( ⋅ ) s ( ⋅ ) s − > 1 s_{-}>1 s − > 1 M q ( ⋅ ) M_{q(\cdot)} M q ( ⋅ ) L p ( ⋅ ) ( R n ) L^{p(\cdot)}(\mathbb{R}^{n}) L p ( ⋅ ) ( R n )
We now prove the following result for our class of weights A A , p ( ⋅ ) A_{A,p(\cdot)} A A , p ( ⋅ )
Proposition 4.3. Let p ( ⋅ ) ∈ P ( R n ) p(\cdot)\in\mathcal{P}(\mathbb{R}^{n}) p ( ⋅ ) ∈ P ( R n ) p + < ∞ p_{+}<\infty p + < ∞ A A A B B B p ( B x ) = p ( x ) p(Bx)=p(x) p ( B x ) = p ( x ) w ∈ A A , p ( ⋅ ) ∩ A B , p ( ⋅ ) w\in A_{A,p(\cdot)}\cap A_{B,p(\cdot)} w ∈ A A , p ( ⋅ ) ∩ A B , p ( ⋅ ) w ∈ A A B , p ( ⋅ ) w\in A_{AB,p(\cdot)} w ∈ A A B , p ( ⋅ ) [ w ] A A B , p ( ⋅ ) ≤ [ w ] A A , p ( ⋅ ) [ w ] A B , p ( ⋅ ) [w]_{A_{AB,p(\cdot)}}\le[w]_{A_{A,p(\cdot)}}[w]_{A_{B,p(\cdot)}} [ w ] A A B , p ( ⋅ ) ≤ [ w ] A A , p ( ⋅ ) [ w ] A B , p ( ⋅ )
Proof. Let Q Q Q w A B ( x ) = w ( A B x ) = w A ( B x ) w_{AB}(x)=w(ABx)=w_A(Bx) w A B ( x ) = w ( A B x ) = w A ( B x )
∣ Q ∣ − 1 ∥ w A B χ Q ∥ p ( ⋅ ) ∥ w − 1 χ Q ∥ p ′ ( ⋅ ) |Q|^{-1}\|w_{AB}\chi_Q\|_{p(\cdot)}\|w^{-1}\chi_Q\|_{p'(\cdot)} ∣ Q ∣ − 1 ∥ w A B χ Q ∥ p ( ⋅ ) ∥ w − 1 χ Q ∥ p ′ ( ⋅ ) ≤ ∣ Q ∣ − 1 ∥ w A B χ Q ∥ p ( ⋅ ) K p ( ⋅ ) ∥ w B − 1 χ Q ∥ p ′ ( ⋅ ) ∥ w B χ Q ∥ p ( ⋅ ) ∣ Q ∣ ∥ w − 1 χ Q ∥ p ′ ( ⋅ ) . \le|Q|^{-1}\|w_{AB}\chi_Q\|_{p(\cdot)}\frac{K_{p(\cdot)}\|w_B^{-1}\chi_Q\|_{p'(\cdot)}\|w_B\chi_Q\|_{p(\cdot)}}{|Q|}\|w^{-1}\chi_Q\|_{p'(\cdot)}. ≤ ∣ Q ∣ − 1 ∥ w A B χ Q ∥ p ( ⋅ ) ∣ Q ∣ K p ( ⋅ ) ∥ w B − 1 χ Q ∥ p ′ ( ⋅ ) ∥ w B χ Q ∥ p ( ⋅ ) ∥ w − 1 χ Q ∥ p ′ ( ⋅ ) . Observe that
∥ w A B χ Q ∥ p ( ⋅ ) = ∥ ( w ∘ A B ) χ Q ∥ p ( ⋅ ) = ∥ ( ( w A χ B − 1 Q ) ∘ B ) ∥ p ( ⋅ ) \|w_{AB}\chi_Q\|_{p(\cdot)}=\|(w\circ AB)\chi_Q\|_{p(\cdot)}=\|((w_A\chi_{B^{-1}Q})\circ B)\|_{p(\cdot)} ∥ w A B χ Q ∥ p ( ⋅ ) = ∥ ( w ∘ A B ) χ Q ∥ p ( ⋅ ) = ∥ (( w A χ B − 1 Q ) ∘ B ) ∥ p ( ⋅ ) and
∥ w B − 1 χ Q ∥ p ′ ( ⋅ ) = ∥ ( w − 1 ∘ B ) χ Q ∥ p ′ ( ⋅ ) = ∥ ( ( w − 1 χ B − 1 Q ) ∘ B ) ∥ p ′ ( ⋅ ) . \|w_B^{-1}\chi_Q\|_{p'(\cdot)}=\|(w^{-1}\circ B)\chi_Q\|_{p'(\cdot)}=\|((w^{-1}\chi_{B^{-1}Q})\circ B)\|_{p'(\cdot)}. ∥ w B − 1 χ Q ∥ p ′ ( ⋅ ) = ∥ ( w − 1 ∘ B ) χ Q ∥ p ′ ( ⋅ ) = ∥ (( w − 1 χ B − 1 Q ) ∘ B ) ∥ p ′ ( ⋅ ) . We take a cube R R R B − 1 Q ⊂ R B^{-1}Q\subset R B − 1 Q ⊂ R ∣ Q ∣ ≈ ∣ R ∣ |Q|\approx|R| ∣ Q ∣ ≈ ∣ R ∣ p ( B x ) = p ( x ) p(Bx)=p(x) p ( B x ) = p ( x ) A A , p ( ⋅ ) A_{A,p(\cdot)} A A , p ( ⋅ ) A B , p ( ⋅ ) A_{B,p(\cdot)} A B , p ( ⋅ )
∣ Q ∣ − 1 ∥ w A B χ Q ∥ p ( ⋅ ) ∥ w − 1 χ Q ∥ p ′ ( ⋅ ) |Q|^{-1}\|w_{AB}\chi_Q\|_{p(\cdot)}\|w^{-1}\chi_Q\|_{p'(\cdot)} ∣ Q ∣ − 1 ∥ w A B χ Q ∥ p ( ⋅ ) ∥ w − 1 χ Q ∥ p ′ ( ⋅ ) ≲ ∣ Q ∣ − 1 ∥ ( ( w A χ B − 1 Q ) ∘ B ) ∥ p ( ⋅ ) ∥ ( ( w − 1 χ B − 1 Q ) ∘ B ) ∥ p ′ ( ⋅ ) ∥ w B χ Q ∥ p ( ⋅ ) ∣ Q ∣ ∥ w − 1 χ Q ∥ p ′ ( ⋅ ) \lesssim|Q|^{-1}\|((w_A\chi_{B^{-1}Q})\circ B)\|_{p(\cdot)}
\frac{\|((w^{-1}\chi_{B^{-1}Q})\circ B)\|_{p'(\cdot)}\|w_B\chi_Q\|_{p(\cdot)}}{|Q|}
\|w^{-1}\chi_Q\|_{p'(\cdot)} ≲ ∣ Q ∣ − 1 ∥ (( w A χ B − 1 Q ) ∘ B ) ∥ p ( ⋅ ) ∣ Q ∣ ∥ (( w − 1 χ B − 1 Q ) ∘ B ) ∥ p ′ ( ⋅ ) ∥ w B χ Q ∥ p ( ⋅ ) ∥ w − 1 χ Q ∥ p ′ ( ⋅ ) ≲ ∣ Q ∣ − 1 ∥ w A χ B − 1 Q ∥ p ( ⋅ ) ∥ w − 1 χ B − 1 Q ∥ p ′ ( ⋅ ) ∥ w B χ Q ∥ p ( ⋅ ) ∣ Q ∣ ∥ w − 1 χ Q ∥ p ′ ( ⋅ ) \lesssim|Q|^{-1}\|w_A\chi_{B^{-1}Q}\|_{p(\cdot)}
\frac{\|w^{-1}\chi_{B^{-1}Q}\|_{p'(\cdot)}\|w_B\chi_Q\|_{p(\cdot)}}{|Q|}
\|w^{-1}\chi_Q\|_{p'(\cdot)} ≲ ∣ Q ∣ − 1 ∥ w A χ B − 1 Q ∥ p ( ⋅ ) ∣ Q ∣ ∥ w − 1 χ B − 1 Q ∥ p ′ ( ⋅ ) ∥ w B χ Q ∥ p ( ⋅ ) ∥ w − 1 χ Q ∥ p ′ ( ⋅ ) ≲ [ ∣ Q ∣ − 1 ∥ w A χ B − 1 Q ∥ p ( ⋅ ) ∥ w − 1 χ B − 1 Q ∥ p ′ ( ⋅ ) ] [ ∣ Q ∣ − 1 ∥ w B χ Q ∥ p ( ⋅ ) ∥ w − 1 χ Q ∥ p ′ ( ⋅ ) ] \lesssim
\left[|Q|^{-1}\|w_A\chi_{B^{-1}Q}\|_{p(\cdot)}
\|w^{-1}\chi_{B^{-1}Q}\|_{p'(\cdot)}\right]
\left[|Q|^{-1}\|w_B\chi_Q\|_{p(\cdot)}
\|w^{-1}\chi_Q\|_{p'(\cdot)}\right] ≲ [ ∣ Q ∣ − 1 ∥ w A χ B − 1 Q ∥ p ( ⋅ ) ∥ w − 1 χ B − 1 Q ∥ p ′ ( ⋅ ) ] [ ∣ Q ∣ − 1 ∥ w B χ Q ∥ p ( ⋅ ) ∥ w − 1 χ Q ∥ p ′ ( ⋅ ) ] ≲ [ ∣ R ∣ − 1 ∥ w A χ R ∥ p ( ⋅ ) ∥ w − 1 χ R ∥ p ′ ( ⋅ ) ] [ ∣ Q ∣ − 1 ∥ w B χ Q ∥ p ( ⋅ ) ∥ w − 1 χ Q ∥ p ′ ( ⋅ ) ] \lesssim
\left[|R|^{-1}\|w_A\chi_R\|_{p(\cdot)}
\|w^{-1}\chi_R\|_{p'(\cdot)}\right]
\left[|Q|^{-1}\|w_B\chi_Q\|_{p(\cdot)}
\|w^{-1}\chi_Q\|_{p'(\cdot)}\right] ≲ [ ∣ R ∣ − 1 ∥ w A χ R ∥ p ( ⋅ ) ∥ w − 1 χ R ∥ p ′ ( ⋅ ) ] [ ∣ Q ∣ − 1 ∥ w B χ Q ∥ p ( ⋅ ) ∥ w − 1 χ Q ∥ p ′ ( ⋅ ) ] ≲ [ w ] A A , p ( ⋅ ) [ w ] A B , p ( ⋅ ) < ∞ . \lesssim[w]_{A_{A,p(\cdot)}}[w]_{A_{B,p(\cdot)}}<\infty. ≲ [ w ] A A , p ( ⋅ ) [ w ] A B , p ( ⋅ ) < ∞. This imply that [ w ] A A B , p ( ⋅ ) ≤ [ w ] A A , p ( ⋅ ) [ w ] A B , p ( ⋅ ) [w]_{A_{AB,p(\cdot)}}\le[w]_{A_{A,p(\cdot)}}[w]_{A_{B,p(\cdot)}} [ w ] A A B , p ( ⋅ ) ≤ [ w ] A A , p ( ⋅ ) [ w ] A B , p ( ⋅ ) w ∈ A A B , p ( ⋅ ) w\in A_{AB,p(\cdot)} w ∈ A A B , p ( ⋅ )
A consequence of the last proposition, we obtain the following result.
Corollary 4.4. Let p ( ⋅ ) ∈ P ( R n ) p(\cdot) \in\mathcal{P}(\mathbb{R}^{n}) p ( ⋅ ) ∈ P ( R n ) p + < ∞ p_{+} < \infty p + < ∞ p ( A x ) = p ( x ) p(Ax) = p(x) p ( A x ) = p ( x ) A A A A k = I A^{k} = I A k = I k ∈ N k \in\mathbb{N} k ∈ N A A , p ( ⋅ ) ⊂ A p ( ⋅ ) \mathcal{A}_{A,p(\cdot)} \subset\mathcal{A}_{p(\cdot)} A A , p ( ⋅ ) ⊂ A p ( ⋅ ) [ w ] A p ( ⋅ ) ≤ [ w ] A A , p ( ⋅ ) [w]_{\mathcal{A}_{p(\cdot)}} \leq[w]_{\mathcal{A}_{A,p(\cdot)}} [ w ] A p ( ⋅ ) ≤ [ w ] A A , p ( ⋅ )
The following lemma guaranties conditions for that w A ∈ A p ( ⋅ ) w_A \in\mathcal{A}_{p(\cdot)} w A ∈ A p ( ⋅ )
Lemma 4.5. Let A A A ∣ det ( A ) ∣ = 1 |\det(A)| = 1 ∣ det ( A ) ∣ = 1 p ( ⋅ ) ∈ P ( R n ) p(\cdot) \in\mathcal{P}(\mathbb{R}^{n}) p ( ⋅ ) ∈ P ( R n ) p + < ∞ p_{+} < \infty p + < ∞ p ( A x ) = p ( x ) p(Ax) = p(x) p ( A x ) = p ( x ) x ∈ R n x \in\mathbb{R}^{n} x ∈ R n w ∈ A p ( ⋅ ) w \in\mathcal{A}_{p(\cdot)} w ∈ A p ( ⋅ ) w A ∈ A p ( ⋅ ) w_A \in\mathcal{A}_{p(\cdot)} w A ∈ A p ( ⋅ ) [ w A ] A p ( ⋅ ) ≤ [ w ] A p ( ⋅ ) ∥ A ∥ n [w_A]_{\mathcal{A}_{p(\cdot)}} \leq[w]_{\mathcal{A}_{p(\cdot)}}\|A\|^{n} [ w A ] A p ( ⋅ ) ≤ [ w ] A p ( ⋅ ) ∥ A ∥ n
Proof. Since ∣ det ( A ) ∣ = 1 |\det(A)| = 1 ∣ det ( A ) ∣ = 1 w ∈ A p ( ⋅ ) w \in\mathcal{A}_{p(\cdot)} w ∈ A p ( ⋅ ) p ( A x ) = p ( x ) p(Ax) = p(x) p ( A x ) = p ( x ) x ∈ R n x \in\mathbb{R}^{n} x ∈ R n ∥ w A χ B ∥ p ( ⋅ ) \|w_A\chi_B\|_{p(\cdot)} ∥ w A χ B ∥ p ( ⋅ ) B = B ( x 0 , r ) B = B(x_0,r) B = B ( x 0 , r ) A B = { A y : y ∈ B } AB = \{Ay:y \in B\} A B = { A y : y ∈ B }
ρ p ( ⋅ ) ( w A χ B λ ) = ∫ B ( w A ( x ) λ ) p ( x ) d x = ∫ B ( w ( A x ) λ ) p ( x ) d x \rho_{p(\cdot)}\left(\frac{w_A\chi_B}{\lambda}\right)=\int_B\left(\frac{w_A(x)}{\lambda}\right)^{p(x)}dx=\int_B\left(\frac{w(Ax)}{\lambda}\right)^{p(x)}dx ρ p ( ⋅ ) ( λ w A χ B ) = ∫ B ( λ w A ( x ) ) p ( x ) d x = ∫ B ( λ w ( A x ) ) p ( x ) d x = ∫ A B ( w ( y ) λ ) p ( A − 1 y ) d y = ∫ A B ( w ( y ) λ ) p ( y ) d y . =\int_{AB}\left(\frac{w(y)}{\lambda}\right)^{p(A^{-1}y)}dy=\int_{AB}\left(\frac{w(y)}{\lambda}\right)^{p(y)}dy. = ∫ A B ( λ w ( y ) ) p ( A − 1 y ) d y = ∫ A B ( λ w ( y ) ) p ( y ) d y . Since A B ⊂ B ( A x 0 , ∥ A ∥ r ) = B A AB \subset B(Ax_0,\|A\|r)=B_A A B ⊂ B ( A x 0 , ∥ A ∥ r ) = B A ∥ A ∥ \|A\| ∥ A ∥
ρ p ( ⋅ ) ( w A χ B λ ) ≤ ∫ B A ( w ( y ) λ ) p ( y ) d y = ρ p ( ⋅ ) ( w χ B A λ ) . \rho_{p(\cdot)}\left(\frac{w_A\chi_B}{\lambda}\right)\leq\int_{B_A}\left(\frac{w(y)}{\lambda}\right)^{p(y)}dy=\rho_{p(\cdot)}\left(\frac{w\chi_{B_A}}{\lambda}\right). ρ p ( ⋅ ) ( λ w A χ B ) ≤ ∫ B A ( λ w ( y ) ) p ( y ) d y = ρ p ( ⋅ ) ( λ w χ B A ) . So, taking λ = ∥ w χ B A ∥ p ( ⋅ ) \lambda=\|w\chi_{B_A}\|_{p(\cdot)} λ = ∥ w χ B A ∥ p ( ⋅ ) ρ p ( ⋅ ) ( w χ B A λ ) ≤ 1 \rho_{p(\cdot)}\left(\frac{w\chi_{B_A}}{\lambda}\right)\leq1 ρ p ( ⋅ ) ( λ w χ B A ) ≤ 1 ρ p ( ⋅ ) ( w A χ B λ ) ≤ 1 \rho_{p(\cdot)}\left(\frac{w_A\chi_B}{\lambda}\right)\leq1 ρ p ( ⋅ ) ( λ w A χ B ) ≤ 1 L p ( ⋅ ) ( R n ) L^{p(\cdot)}(\mathbb{R}^{n}) L p ( ⋅ ) ( R n ) ∥ w A χ B ∥ p ( ⋅ ) ≤ ∥ w χ B A ∥ p ( ⋅ ) \|w_A\chi_B\|_{p(\cdot)}\leq\|w\chi_{B_A}\|_{p(\cdot)} ∥ w A χ B ∥ p ( ⋅ ) ≤ ∥ w χ B A ∥ p ( ⋅ )
Analogously, since p ′ ( A x ) = p ′ ( x ) p'(Ax)=p'(x) p ′ ( A x ) = p ′ ( x ) x ∈ R n x\in\mathbb{R}^{n} x ∈ R n ∥ w A − 1 χ B ∥ p ′ ( ⋅ ) ≤ ∥ w − 1 χ B A ∥ p ′ ( ⋅ ) \|w_A^{-1}\chi_B\|_{p'(\cdot)}\leq\|w^{-1}\chi_{B_A}\|_{p'(\cdot)} ∥ w A − 1 χ B ∥ p ′ ( ⋅ ) ≤ ∥ w − 1 χ B A ∥ p ′ ( ⋅ ) A p ( ⋅ ) \mathcal{A}_{p(\cdot)} A p ( ⋅ )
∥ w χ B A ∥ p ( ⋅ ) ∥ w − 1 χ B A ∥ p ′ ( ⋅ ) ≤ [ w ] A p ( ⋅ ) ∣ B A ∣ . \|w\chi_{B_A}\|_{p(\cdot)}\|w^{-1}\chi_{B_A}\|_{p'(\cdot)}\leq[w]_{\mathcal{A}_{p(\cdot)}}|B_A|. ∥ w χ B A ∥ p ( ⋅ ) ∥ w − 1 χ B A ∥ p ′ ( ⋅ ) ≤ [ w ] A p ( ⋅ ) ∣ B A ∣. Since ∣ B A ∣ = ∥ A ∥ ∣ B ∣ |B_A|=\|A\||B| ∣ B A ∣ = ∥ A ∥∣ B ∣
∥ w A χ B ∥ p ( ⋅ ) ∥ w A − 1 χ B ∥ p ′ ( ⋅ ) ≤ [ w ] A p ( ⋅ ) ∥ A ∥ n ∣ B ∣ . \|w_A\chi_B\|_{p(\cdot)}\|w_A^{-1}\chi_B\|_{p'(\cdot)}\leq[w]_{\mathcal{A}_{p(\cdot)}}\|A\|^{n}|B|. ∥ w A χ B ∥ p ( ⋅ ) ∥ w A − 1 χ B ∥ p ′ ( ⋅ ) ≤ [ w ] A p ( ⋅ ) ∥ A ∥ n ∣ B ∣. Therefore w A ∈ A p ( ⋅ ) w_A\in\mathcal{A}_{p(\cdot)} w A ∈ A p ( ⋅ ) [ w A ] A p ( ⋅ ) ≤ [ w ] A p ( ⋅ ) ∥ A ∥ n [w_A]_{\mathcal{A}_{p(\cdot)}}\leq[w]_{\mathcal{A}_{p(\cdot)}}\|A\|^{n} [ w A ] A p ( ⋅ ) ≤ [ w ] A p ( ⋅ ) ∥ A ∥ n
Proof of Theorem 1.4
Proof of Theorem 1.4. If M A − 1 M_{A^{-1}} M A − 1 L w p ( ⋅ ) L_w^{p(\cdot)} L w p ( ⋅ ) w ∈ A A , p ( ⋅ ) w\in\mathcal{A}_{A,p(\cdot)} w ∈ A A , p ( ⋅ ) p ( ⋅ ) ∈ P log ( R n ) p(\cdot)\in\mathcal{P}^{\log}(\mathbb{R}^{n}) p ( ⋅ ) ∈ P l o g ( R n ) 1 < p − ≤ p + < ∞ 1<p_{-}\leq p_{+}<\infty 1 < p − ≤ p + < ∞ p ( A x ) = p ( x ) p(Ax)=p(x) p ( A x ) = p ( x ) x ∈ R n x\in\mathbb{R}^{n} x ∈ R n w ∈ A A , p ( ⋅ ) w\in\mathcal{A}_{A,p(\cdot)} w ∈ A A , p ( ⋅ ) f ∈ L p ( ⋅ ) ( w ) f\in L^{p(\cdot)}(w) f ∈ L p ( ⋅ ) ( w ) b ≥ 2 n + 1 b\geq2^{n+1} b ≥ 2 n + 1 Ω A , k = { x ∈ R n : M A − 1 f ( x ) > b k } \Omega_{A,k}=\{x\in\mathbb{R}^{n}:M_{A^{-1}}f(x)>b^{k}\} Ω A , k = { x ∈ R n : M A − 1 f ( x ) > b k } D A , k = { x ∈ R n : b k < M A − 1 f ( x ) ≤ b k + 1 } D_{A,k}=\{x\in\mathbb{R}^{n}:b^{k}<M_{A^{-1}}f(x)\leq b^{k+1}\} D A , k = { x ∈ R n : b k < M A − 1 f ( x ) ≤ b k + 1 } Ω A , k = A ( { x ∈ R n : M A − 1 f ( x ) > b k } ) = A ( Ω k ) \Omega_{A,k}=A(\{x\in\mathbb{R}^{n}:M_{A^{-1}}f(x)>b^{k}\})=A(\Omega_k) Ω A , k = A ({ x ∈ R n : M A − 1 f ( x ) > b k }) = A ( Ω k ) D A , k = A ( D k ) D_{A,k}=A(D_k) D A , k = A ( D k )
By the Calderón-Zygmund decomposition, we have Q j , k Q_{j,k} Q j , k
b k < 1 ∣ Q j , k ∣ ∫ Q j , k ∣ f ( x ) ∣ d x ≤ b k + 1 b^{k}<\frac{1}{|Q_{j,k}|}\int_{Q_{j,k}}|f(x)|\,dx\leq b^{k+1} b k < ∣ Q j , k ∣ 1 ∫ Q j , k ∣ f ( x ) ∣ d x ≤ b k + 1 and
Ω A ⊂ ⋃ j A Q j , k 3 \Omega_A\subset\bigcup_j AQ_{j,k}^{3} Ω A ⊂ j ⋃ A Q j , k 3 where Q j , k 3 Q^{3}_{j,k} Q j , k 3 Q j , k Q_{j,k} Q j , k F i , k = Q i , k ∖ Ω k + 1 F_{i,k}=Q_{i,k}\setminus\Omega_{k+1} F i , k = Q i , k ∖ Ω k + 1 { F i , k } i , k \{F_{i,k}\}_{i,k} { F i , k } i , k ∣ Q i , k ∣ ≤ 2 ∣ F i , k ∣ |Q_{i,k}|\leq2|F_{i,k}| ∣ Q i , k ∣ ≤ 2∣ F i , k ∣
Now, if g ∈ L p ′ ( ⋅ ) ( w − 1 ) g\in L^{p'(\cdot)}(w^{-1}) g ∈ L p ′ ( ⋅ ) ( w − 1 )
∫ R n M A − 1 f ( x ) g ( x ) d x ≤ ∑ k ∫ D A , k b k + 1 g ( x ) d x ≤ b k + 1 ∑ j , k ∫ A Q j , k 3 g ( x ) d x ≤ b ∣ det ( A ) ∣ ∑ j , k ( 1 ∣ Q j , k ∣ ∫ Q j , k ∣ f ∣ ) ∫ Q j , k 3 g A ( x ) d x \begin{aligned}
\int_{\mathbb{R}^{n}}M_{A^{-1}}f(x)g(x)\,dx\leq\sum_{k}\int_{D_{A,k}}b^{k+1}g(x)\,dx\leq b^{k+1}\sum_{j,k}\int_{AQ^{3}_{j,k}}g(x)\,dx \\
&\leq b|\det(A)|\sum_{j,k}\left(\frac{1}{|Q_{j,k}|}\int_{Q_{j,k}}|f|\right)\int_{Q^{3}_{j,k}}g_{A}(x)\,dx
\end{aligned} ∫ R n M A − 1 f ( x ) g ( x ) d x ≤ k ∑ ∫ D A , k b k + 1 g ( x ) d x ≤ b k + 1 j , k ∑ ∫ A Q j , k 3 g ( x ) d x ≤ b ∣ det ( A ) ∣ j , k ∑ ( ∣ Q j , k ∣ 1 ∫ Q j , k ∣ f ∣ ) ∫ Q j , k 3 g A ( x ) d x ≲ ∑ j , k ( 1 ∣ Q j , k 3 ∣ ∫ Q j , k 3 ∣ f ∣ ) ( 1 ∣ Q j , k 3 ∣ ∫ Q j , k 3 g A ) ∣ F j , k ∣ \lesssim\sum_{j,k}\left(\frac{1}{|Q^{3}_{j,k}|}\int_{Q^{3}_{j,k}}|f|\right)\left(\frac{1}{|Q^{3}_{j,k}|}\int_{Q^{3}_{j,k}}g_{A}\right)|F_{j,k}| ≲ j , k ∑ ( ∣ Q j , k 3 ∣ 1 ∫ Q j , k 3 ∣ f ∣ ) ( ∣ Q j , k 3 ∣ 1 ∫ Q j , k 3 g A ) ∣ F j , k ∣ By Corollary 4.4 we have w ∈ A p ( ⋅ ) w\in A_{p(\cdot)} w ∈ A p ( ⋅ ) A p ( ⋅ ) A_{p(\cdot)} A p ( ⋅ ) Proposition 2.9 , there exists s s s min { 1 p − , 1 p − ′ } < s < 1 \min\left\{\frac{1}{p_{-}},\frac{1}{p'_{-}}\right\}<s<1 min { p − 1 , p − ′ 1 } < s < 1 w 1 s ∈ A s p ( ⋅ ) w^{\frac{1}{s}}\in A_{sp(\cdot)} w s 1 ∈ A s p ( ⋅ ) w − 1 s ∈ A s p ′ ( ⋅ ) w^{-\frac{1}{s}}\in A_{sp'(\cdot)} w − s 1 ∈ A s p ′ ( ⋅ )
We define the exponents u ′ ( ⋅ ) = 1 s ( s p ( ⋅ ) ) ′ u'(\cdot)=\frac{1}{s}(sp(\cdot))' u ′ ( ⋅ ) = s 1 ( s p ( ⋅ ) ) ′ v ( ⋅ ) = 1 s ( s p ′ ( ⋅ ) ) ′ v(\cdot)=\frac{1}{s}(sp'(\cdot))' v ( ⋅ ) = s 1 ( s p ′ ( ⋅ ) ) ′ p ( ⋅ ) ∈ P log ( R n ) p(\cdot)\in\mathcal{P}^{\log}(\mathbb{R}^{n}) p ( ⋅ ) ∈ P l o g ( R n ) u ( ⋅ ) , v ( ⋅ ) ∈ P log ( R n ) u(\cdot),v(\cdot)\in\mathcal{P}^{\log}(\mathbb{R}^{n}) u ( ⋅ ) , v ( ⋅ ) ∈ P l o g ( R n )
Observe that
p ′ ( ⋅ ) v ′ ( ⋅ ) = p ′ ( ⋅ ) ( 1 − s ) + 1 ≥ ( p ′ ) − ( 1 − s ) + 1 > 1 , \frac{p'(\cdot)}{v'(\cdot)}=p'(\cdot)(1-s)+1\geq(p')_{-}(1-s)+1>1, v ′ ( ⋅ ) p ′ ( ⋅ ) = p ′ ( ⋅ ) ( 1 − s ) + 1 ≥ ( p ′ ) − ( 1 − s ) + 1 > 1 , p ( ⋅ ) u ( ⋅ ) = p ( ⋅ ) ( 1 − s ) + 1 ≥ p − ( 1 − s ) + 1 > 1. \frac{p(\cdot)}{u(\cdot)}=p(\cdot)(1-s)+1\geq p_{-}(1-s)+1>1. u ( ⋅ ) p ( ⋅ ) = p ( ⋅ ) ( 1 − s ) + 1 ≥ p − ( 1 − s ) + 1 > 1. Then, by Theorem 4.2 we have M u ( ⋅ ) M_{u(\cdot)} M u ( ⋅ ) L p ( ⋅ ) ( R n ) L^{p(\cdot)}(\mathbb{R}^{n}) L p ( ⋅ ) ( R n ) M v ′ ( ⋅ ) M_{v'(\cdot)} M v ′ ( ⋅ ) L p ′ ( ⋅ ) ( R n ) L^{p'(\cdot)}(\mathbb{R}^{n}) L p ′ ( ⋅ ) ( R n )
Then by Hölder
∫ R n M A − 1 f ( x ) g ( x ) d x ≲ ∑ j , k ( 1 ∣ Q j , k 3 ∣ ∫ Q j , k 3 ∣ f ∣ ) ( 1 ∣ Q j , k 3 ∣ ∫ Q j , k 3 g A ) ∣ F j , k ∣ \int_{\mathbb{R}^{n}}M_{A^{-1}}f(x)g(x)\,dx\lesssim\sum_{j,k}\left(\frac{1}{|Q^{3}_{j,k}|}\int_{Q^{3}_{j,k}}|f|\right)\left(\frac{1}{|Q^{3}_{j,k}|}\int_{Q^{3}_{j,k}}g_{A}\right)|F_{j,k}| ∫ R n M A − 1 f ( x ) g ( x ) d x ≲ j , k ∑ ( ∣ Q j , k 3 ∣ 1 ∫ Q j , k 3 ∣ f ∣ ) ( ∣ Q j , k 3 ∣ 1 ∫ Q j , k 3 g A ) ∣ F j , k ∣ ≲ ∑ j , k ∣ F j , k ∣ 1 ∣ Q j , k 3 ∣ ∥ f w χ Q j , k 3 ∥ u ( ⋅ ) ∥ w − 1 χ Q j , k 3 ∥ u ′ ( ⋅ ) 1 ∣ Q j , k 3 ∣ ∥ g A w A − 1 χ Q j , k 3 ∥ v ′ ( ⋅ ) ∥ w A χ Q j , k 3 ∥ v ( ⋅ ) \lesssim\sum_{j,k}|F_{j,k}|\frac{1}{|Q^{3}_{j,k}|}\|fw\chi_{Q^{3}_{j,k}}\|_{u(\cdot)}\|w^{-1}\chi_{Q^{3}_{j,k}}\|_{u'(\cdot)}\frac{1}{|Q^{3}_{j,k}|}\|g_{A}w_{A}^{-1}\chi_{Q^{3}_{j,k}}\|_{v'(\cdot)}\|w_{A}\chi_{Q^{3}_{j,k}}\|_{v(\cdot)} ≲ j , k ∑ ∣ F j , k ∣ ∣ Q j , k 3 ∣ 1 ∥ f w χ Q j , k 3 ∥ u ( ⋅ ) ∥ w − 1 χ Q j , k 3 ∥ u ′ ( ⋅ ) ∣ Q j , k 3 ∣ 1 ∥ g A w A − 1 χ Q j , k 3 ∥ v ′ ( ⋅ ) ∥ w A χ Q j , k 3 ∥ v ( ⋅ ) ≲ ∑ j , k ∣ F j , k ∣ ∥ f w χ Q j , k 3 ∥ u ( ⋅ ) ∥ w − 1 χ Q j , k 3 ∥ u ′ ( ⋅ ) ∥ χ Q j , k 3 ∥ u ( ⋅ ) ∥ χ Q j , k 3 ∥ u ′ ( ⋅ ) ∥ g A w A − 1 χ Q j , k 3 ∥ v ′ ( ⋅ ) ∥ w A χ Q j , k 3 ∥ v ( ⋅ ) ∥ χ Q j , k 3 ∥ v ′ ( ⋅ ) ∥ χ Q j , k 3 ∥ v ( ⋅ ) . \lesssim\sum_{j,k}|F_{j,k}|\frac{\|fw\chi_{Q^{3}_{j,k}}\|_{u(\cdot)}\|w^{-1}\chi_{Q^{3}_{j,k}}\|_{u'(\cdot)}}{\|\chi_{Q^{3}_{j,k}}\|_{u(\cdot)}\|\chi_{Q^{3}_{j,k}}\|_{u'(\cdot)}}\frac{\|g_{A}w_{A}^{-1}\chi_{Q^{3}_{j,k}}\|_{v'(\cdot)}\|w_{A}\chi_{Q^{3}_{j,k}}\|_{v(\cdot)}}{\|\chi_{Q^{3}_{j,k}}\|_{v'(\cdot)}\|\chi_{Q^{3}_{j,k}}\|_{v(\cdot)}}. ≲ j , k ∑ ∣ F j , k ∣ ∥ χ Q j , k 3 ∥ u ( ⋅ ) ∥ χ Q j , k 3 ∥ u ′ ( ⋅ ) ∥ f w χ Q j , k 3 ∥ u ( ⋅ ) ∥ w − 1 χ Q j , k 3 ∥ u ′ ( ⋅ ) ∥ χ Q j , k 3 ∥ v ′ ( ⋅ ) ∥ χ Q j , k 3 ∥ v ( ⋅ ) ∥ g A w A − 1 χ Q j , k 3 ∥ v ′ ( ⋅ ) ∥ w A χ Q j , k 3 ∥ v ( ⋅ ) . We now claim that
∥ w − 1 χ Q j , k 3 ∥ u ′ ( ⋅ ) ∥ w A χ Q j , k 3 ∥ v ( ⋅ ) ∥ χ Q j , k 3 ∥ u ′ ( ⋅ ) ∥ χ Q j , k 3 ∥ v ( ⋅ ) ≲ C w (16) \frac{\|w^{-1}\chi_{Q^{3}_{j,k}}\|_{u'(\cdot)}\|w_{A}\chi_{Q^{3}_{j,k}}\|_{v(\cdot)}}{\|\chi_{Q^{3}_{j,k}}\|_{u'(\cdot)}\|\chi_{Q^{3}_{j,k}}\|_{v(\cdot)}}\lesssim C_{w} \tag*{(16)} ∥ χ Q j , k 3 ∥ u ′ ( ⋅ ) ∥ χ Q j , k 3 ∥ v ( ⋅ ) ∥ w − 1 χ Q j , k 3 ∥ u ′ ( ⋅ ) ∥ w A χ Q j , k 3 ∥ v ( ⋅ ) ≲ C w (16) Therefore
∫ R n M A − 1 f ( x ) g ( x ) d x ≲ C w ∑ j , k ∣ F j , k ∣ ∥ f w χ Q j , k 3 ∥ u ( ⋅ ) ∥ χ Q j , k 3 ∥ u ( ⋅ ) ∥ g A w A − 1 χ Q j , k 3 ∥ v ′ ( ⋅ ) ∥ χ Q j , k 3 ∥ v ′ ( ⋅ ) \int_{\mathbb{R}^{n}}M_{A^{-1}}f(x)g(x)\,dx\lesssim C_{w}\sum_{j,k}|F_{j,k}|\frac{\|fw\chi_{Q^{3}_{j,k}}\|_{u(\cdot)}}{\|\chi_{Q^{3}_{j,k}}\|_{u(\cdot)}}\frac{\|g_{A}w_{A}^{-1}\chi_{Q^{3}_{j,k}}\|_{v'(\cdot)}}{\|\chi_{Q^{3}_{j,k}}\|_{v'(\cdot)}} ∫ R n M A − 1 f ( x ) g ( x ) d x ≲ C w j , k ∑ ∣ F j , k ∣ ∥ χ Q j , k 3 ∥ u ( ⋅ ) ∥ f w χ Q j , k 3 ∥ u ( ⋅ ) ∥ χ Q j , k 3 ∥ v ′ ( ⋅ ) ∥ g A w A − 1 χ Q j , k 3 ∥ v ′ ( ⋅ ) ≲ C w ∑ j , k ∫ F j , k M u ( ⋅ ) ( f w ) M v ′ ( ⋅ ) ( g A w A − 1 ) d x ≲ C w ∫ R n M u ( ⋅ ) ( f w ) M v ′ ( ⋅ ) ( g A w A − 1 ) d x ≲ C w ∥ M u ( ⋅ ) ( f w ) ∥ p ( ⋅ ) ∥ M v ′ ( ⋅ ) ( g A w A − 1 ) ∥ p ′ ( ⋅ ) \begin{aligned}
&\lesssim C_w\sum_{j,k}\int_{F_{j,k}} M_{u(\cdot)}(fw)M_{v'(\cdot)}(g_Aw_A^{-1})\,dx\\
&\lesssim C_w\int_{\mathbb{R}^n} M_{u(\cdot)}(fw)M_{v'(\cdot)}(g_Aw_A^{-1})\,dx\\
&\lesssim C_w\lVert M_{u(\cdot)}(fw)\rVert_{p(\cdot)}\lVert M_{v'(\cdot)}(g_Aw_A^{-1})\rVert_{p'(\cdot)}
\end{aligned} ≲ C w j , k ∑ ∫ F j , k M u ( ⋅ ) ( f w ) M v ′ ( ⋅ ) ( g A w A − 1 ) d x ≲ C w ∫ R n M u ( ⋅ ) ( f w ) M v ′ ( ⋅ ) ( g A w A − 1 ) d x ≲ C w ∥ M u ( ⋅ ) ( f w ) ∥ p ( ⋅ ) ∥ M v ′ ( ⋅ ) ( g A w A − 1 ) ∥ p ′ ( ⋅ ) Note that since p ( A x ) = p ( x ) p(Ax)=p(x) p ( A x ) = p ( x ) x x x
∥ g A w A − 1 ∥ p ′ ( ⋅ ) ≲ ∥ g w − 1 ∥ p ′ ( ⋅ ) . \lVert g_Aw_A^{-1}\rVert_{p'(\cdot)}\lesssim\lVert gw^{-1}\rVert_{p'(\cdot)}. ∥ g A w A − 1 ∥ p ′ ( ⋅ ) ≲ ∥ g w − 1 ∥ p ′ ( ⋅ ) . this imply g A w A − 1 ∈ L p ′ ( ⋅ ) ( R n ) g_Aw_A^{-1}\in L^{p'(\cdot)}(\mathbb{R}^n) g A w A − 1 ∈ L p ′ ( ⋅ ) ( R n )
Now, since M u ( ⋅ ) M_{u(\cdot)} M u ( ⋅ ) L p ( ⋅ ) ( R n ) L^{p(\cdot)}(\mathbb{R}^n) L p ( ⋅ ) ( R n ) M v ′ ( ⋅ ) M_{v'(\cdot)} M v ′ ( ⋅ ) L p ′ ( ⋅ ) ( R n ) L^{p'(\cdot)}(\mathbb{R}^n) L p ′ ( ⋅ ) ( R n )
∫ R n M A − 1 f ( x ) g ( x ) d x ≲ C w ∥ f w ∥ p ( ⋅ ) ∥ g w − 1 ∥ p ′ ( ⋅ ) . \int_{\mathbb{R}^n}M_{A^{-1}}f(x)g(x)\,dx\lesssim C_w\lVert fw\rVert_{p(\cdot)}\lVert gw^{-1}\rVert_{p'(\cdot)}. ∫ R n M A − 1 f ( x ) g ( x ) d x ≲ C w ∥ f w ∥ p ( ⋅ ) ∥ g w − 1 ∥ p ′ ( ⋅ ) . Then, by duality argument we have the desired estimate.
Finally to complete the proof, we must prove that the claim (4.3) holds. Observe that (4.3) is
( w − 1 ) u ′ ( ⋅ ) , Q ( w A ) v ( ⋅ ) , Q ≲ C w . (w^{-1})_{u'(\cdot),Q}(w_A)_{v(\cdot),Q}\lesssim C_w. ( w − 1 ) u ′ ( ⋅ ) , Q ( w A ) v ( ⋅ ) , Q ≲ C w . with Q = Q j , k 3 Q=Q^3_{j,k} Q = Q j , k 3
By Lemma 4.5 we have w A ∈ A p ( ⋅ ) w_A\in\mathcal{A}_{p(\cdot)} w A ∈ A p ( ⋅ ) w A − 1 ∈ A p ′ ( ⋅ ) w_A^{-1}\in\mathcal{A}_{p'(\cdot)} w A − 1 ∈ A p ′ ( ⋅ ) w A 1 / s ∈ A s p ( ⋅ ) w_A^{1/s}\in\mathcal{A}_{sp(\cdot)} w A 1/ s ∈ A s p ( ⋅ ) w A − 1 / s ∈ A s p ′ ( ⋅ ) w_A^{-1/s}\in\mathcal{A}_{sp'(\cdot)} w A − 1/ s ∈ A s p ′ ( ⋅ )
( w 1 / s ) s p ( ⋅ ) , Q ( w − 1 / s ) [ s p ( ⋅ ) ] ′ , Q ≲ [ w 1 / s ] A s p ( ⋅ ) (w^{1/s})_{sp(\cdot),Q}(w^{-1/s})_{[sp(\cdot)]',Q}\lesssim[w^{1/s}]_{\mathcal{A}_{sp(\cdot)}} ( w 1/ s ) s p ( ⋅ ) , Q ( w − 1/ s ) [ s p ( ⋅ ) ] ′ , Q ≲ [ w 1/ s ] A s p ( ⋅ ) and
( w A − 1 / s ) s p ′ ( ⋅ ) , Q ( w A 1 / s ) [ s p ′ ( ⋅ ) ] ′ , Q ≲ [ w A − 1 / s ] A s p ′ ( ⋅ ) . (w_A^{-1/s})_{sp'(\cdot),Q}(w_A^{1/s})_{[sp'(\cdot)]',Q}\lesssim[w_A^{-1/s}]_{\mathcal{A}_{sp'(\cdot)}}. ( w A − 1/ s ) s p ′ ( ⋅ ) , Q ( w A 1/ s ) [ s p ′ ( ⋅ ) ] ′ , Q ≲ [ w A − 1/ s ] A s p ′ ( ⋅ ) . So, by Lemma 2.7, we obtain
( w ) p ( ⋅ ) , Q ( w − 1 ) u ′ ( ⋅ ) , Q ≲ [ w 1 / s ] A s p ( ⋅ ) s (w)_{p(\cdot),Q}(w^{-1})_{u'(\cdot),Q}\lesssim[w^{1/s}]_{\mathcal{A}_{sp(\cdot)}}^s ( w ) p ( ⋅ ) , Q ( w − 1 ) u ′ ( ⋅ ) , Q ≲ [ w 1/ s ] A s p ( ⋅ ) s and
( w A − 1 ) p ′ ( ⋅ ) , Q ( w A ) v ( ⋅ ) , Q ≲ [ w A − 1 / s ] A s p ′ ( ⋅ ) s . (w_A^{-1})_{p'(\cdot),Q}(w_A)_{v(\cdot),Q}\lesssim[w_A^{-1/s}]_{\mathcal{A}_{sp'(\cdot)}}^s. ( w A − 1 ) p ′ ( ⋅ ) , Q ( w A ) v ( ⋅ ) , Q ≲ [ w A − 1/ s ] A s p ′ ( ⋅ ) s . Then using Lemma 4.1 for w , w A ∈ A p ( ⋅ ) w,w_A\in\mathcal{A}_{p(\cdot)} w , w A ∈ A p ( ⋅ ) A A , p ( ⋅ ) \mathcal{A}_{A,p(\cdot)} A A , p ( ⋅ )
( w − 1 ) u ′ ( ⋅ ) , Q ( w A ) v ( ⋅ ) , Q ≲ ( w − 1 ) u ′ ( ⋅ ) , Q ( w ) p ( ⋅ ) , Q ( w − 1 ) p ′ ( ⋅ ) , Q ( w A ) p ( ⋅ ) , Q ( w A − 1 ) p ( ⋅ ) , Q ( w A ) v ( ⋅ ) , Q ≤ [ w 1 / s ] A s p ( ⋅ ) s [ w ] A A , p ( ⋅ ) [ w A − 1 / s ] A s p ′ ( ⋅ ) s = : C w . \begin{aligned}
(w^{-1})_{u'(\cdot),Q}(w_A)_{v(\cdot),Q}
&\lesssim(w^{-1})_{u'(\cdot),Q}(w)_{p(\cdot),Q}(w^{-1})_{p'(\cdot),Q}(w_A)_{p(\cdot),Q}(w_A^{-1})_{p(\cdot),Q}(w_A)_{v(\cdot),Q}\\
&\le[w^{1/s}]_{\mathcal{A}_{sp(\cdot)}}^s[w]_{\mathcal{A}_{A,p(\cdot)}}[w_A^{-1/s}]_{\mathcal{A}_{sp'(\cdot)}}^s=:C_w.
\end{aligned} ( w − 1 ) u ′ ( ⋅ ) , Q ( w A ) v ( ⋅ ) , Q ≲ ( w − 1 ) u ′ ( ⋅ ) , Q ( w ) p ( ⋅ ) , Q ( w − 1 ) p ′ ( ⋅ ) , Q ( w A ) p ( ⋅ ) , Q ( w A − 1 ) p ( ⋅ ) , Q ( w A ) v ( ⋅ ) , Q ≤ [ w 1/ s ] A s p ( ⋅ ) s [ w ] A A , p ( ⋅ ) [ w A − 1/ s ] A s p ′ ( ⋅ ) s =: C w . Properties of the class A A , p ( ⋅ ) \mathcal{A}_{A,p(\cdot)} A A , p ( ⋅ )
In this section, we prove several properties of the weight class A A , p ( ⋅ ) \mathcal{A}_{A,p(\cdot)} A A , p ( ⋅ ) [ w ] A p ≥ 1 [w]_{A_p} \ge1 [ w ] A p ≥ 1 [ w ] A p ( ⋅ ) K p ( ⋅ ) ≥ 1 [w]_{A_{p(\cdot)}}K_{p(\cdot)} \ge1 [ w ] A p ( ⋅ ) K p ( ⋅ ) ≥ 1 [ w ] A A , p ( ⋅ ) [w]_{\mathcal{A}_{A,p(\cdot)}} [ w ] A A , p ( ⋅ )
Let w ∈ A A , p ( ⋅ ) w \in\mathcal{A}_{A,p(\cdot)} w ∈ A A , p ( ⋅ ) Q Q Q A Q ∩ Q ≠ ∅ AQ \cap Q \ne\emptyset A Q ∩ Q = ∅
∣ A Q ∩ Q ∣ ≤ K p ( ⋅ ) ∥ w χ A Q ∥ p ( ⋅ ) ∥ w − 1 χ Q ∥ p ′ ( ⋅ ) ≤ K p ( ⋅ ) max { 1 , ∣ det ( A − 1 ) ∣ } ∥ w A χ Q ∥ p ( ⋅ ) ∥ w − 1 χ Q ∥ p ′ ( ⋅ ) ≤ K p ( ⋅ ) max { 1 , ∣ det ( A − 1 ) ∣ } [ w ] A A , p ( ⋅ ) ∣ Q ∣ . \begin{aligned}
|AQ \cap Q| &\le K_{p(\cdot)}\|w\chi_{AQ}\|_{p(\cdot)}\|w^{-1}\chi_Q\|_{p'(\cdot)} \\
&\le K_{p(\cdot)}\max\{1,|\det(A^{-1})|\}\|w_A\chi_Q\|_{p(\cdot)}\|w^{-1}\chi_Q\|_{p'(\cdot)} \\
&\le K_{p(\cdot)}\max\{1,|\det(A^{-1})|\}[w]_{\mathcal{A}_{A,p(\cdot)}}|Q|.
\end{aligned} ∣ A Q ∩ Q ∣ ≤ K p ( ⋅ ) ∥ w χ A Q ∥ p ( ⋅ ) ∥ w − 1 χ Q ∥ p ′ ( ⋅ ) ≤ K p ( ⋅ ) max { 1 , ∣ det ( A − 1 ) ∣ } ∥ w A χ Q ∥ p ( ⋅ ) ∥ w − 1 χ Q ∥ p ′ ( ⋅ ) ≤ K p ( ⋅ ) max { 1 , ∣ det ( A − 1 ) ∣ } [ w ] A A , p ( ⋅ ) ∣ Q ∣. Then, we obtain
[ w ] A A , p ( ⋅ ) K p ( ⋅ ) max { 1 , ∣ det ( A − 1 ) ∣ } ≥ sup Q = Q ( 0 , ℓ ) ∣ A Q ∩ Q ∣ ∣ Q ∣ . [w]_{\mathcal{A}_{A,p(\cdot)}}K_{p(\cdot)}\max\{1,|\det(A^{-1})|\}\ge\sup_{Q=Q(0,\ell)}\frac{|AQ\cap Q|}{|Q|}. [ w ] A A , p ( ⋅ ) K p ( ⋅ ) max { 1 , ∣ det ( A − 1 ) ∣ } ≥ Q = Q ( 0 , ℓ ) sup ∣ Q ∣ ∣ A Q ∩ Q ∣ . where the supremum is taken over all cubes Q ( 0 , ℓ ) Q(0,\ell) Q ( 0 , ℓ ) ℓ \ell ℓ
Observe that B ( 0 , ℓ ) ⊂ Q ( 0 , 2 ℓ ) B(0,\ell)\subset Q(0,2\ell) B ( 0 , ℓ ) ⊂ Q ( 0 , 2 ℓ ) B ~ = B ( 0 , ℓ max { ∥ A − 1 ∥ , 1 } ) \widetilde{B}=B\left(0,\frac{\ell}{\max\{\|A^{-1}\|,1\}}\right) B = B ( 0 , m a x { ∥ A − 1 ∥ , 1 } ℓ ) B ~ ⊂ A Q ∩ Q \widetilde{B}\subset AQ\cap Q B ⊂ A Q ∩ Q
∣ A Q ∩ Q ∣ ∣ Q ∣ ≥ ∣ B ~ ∣ ∣ Q ∣ = 1 2 n max { ∥ A − 1 ∥ n , 1 } . \frac{|AQ\cap Q|}{|Q|}\ge\frac{|\widetilde{B}|}{|Q|}=\frac{1}{2^n\max\{\|A^{-1}\|^n,1\}}. ∣ Q ∣ ∣ A Q ∩ Q ∣ ≥ ∣ Q ∣ ∣ B ∣ = 2 n max { ∥ A − 1 ∥ n , 1 } 1 . Observe that this lower bound depends only of the matrix A A A
We define
[ [ w ] ] A A , p ( ⋅ ) : = 2 n max { ∥ A − 1 ∥ 2 n , 1 } [ w ] A A , p ( ⋅ ) K p ( ⋅ ) . [[w]]_{\mathcal{A}_{A,p(\cdot)}}:=2^n\max\{\|A^{-1}\|^{2n},1\}[w]_{\mathcal{A}_{A,p(\cdot)}}K_{p(\cdot)}. [[ w ] ] A A , p ( ⋅ ) := 2 n max { ∥ A − 1 ∥ 2 n , 1 } [ w ] A A , p ( ⋅ ) K p ( ⋅ ) . Note that [ [ w ] ] A A , p ( ⋅ ) ≥ 1 [[w]]_{\mathcal{A}_{A,p(\cdot)}}\ge1 [[ w ] ] A A , p ( ⋅ ) ≥ 1 [ [ w ] ] A A , p ( ⋅ ) ≥ [ w ] A A , p ( ⋅ ) [[w]]_{\mathcal{A}_{A,p(\cdot)}}\ge[w]_{\mathcal{A}_{A,p(\cdot)}} [[ w ] ] A A , p ( ⋅ ) ≥ [ w ] A A , p ( ⋅ ) [ w ] A A , p ( ⋅ ) ≥ 1 [w]_{\mathcal{A}_{A,p(\cdot)}}\ge1 [ w ] A A , p ( ⋅ ) ≥ 1
The following results are in the spirit of the lemmas present in [2 , 6 ] .
Lemma 5.1. Let A A A p ( ⋅ ) ∈ P ( R n ) p(\cdot)\in\mathcal{P}(\mathbb{R}^n) p ( ⋅ ) ∈ P ( R n ) w ∈ A A , p ( ⋅ ) w\in\mathcal{A}_{A,p(\cdot)} w ∈ A A , p ( ⋅ ) Q Q Q E ⊂ Q E\subset Q E ⊂ Q
∣ E ∣ ∣ Q ∣ ≤ K p ( ⋅ ) [ w ] A A , p ( ⋅ ) ∥ w χ E ∥ p ( ⋅ ) ∥ w A χ Q ∥ p ( ⋅ ) . \frac{|E|}{|Q|}\le K_{p(\cdot)}[w]_{\mathcal{A}_{A,p(\cdot)}}\frac{\|w\chi_E\|_{p(\cdot)}}{\|w_A\chi_Q\|_{p(\cdot)}}. ∣ Q ∣ ∣ E ∣ ≤ K p ( ⋅ ) [ w ] A A , p ( ⋅ ) ∥ w A χ Q ∥ p ( ⋅ ) ∥ w χ E ∥ p ( ⋅ ) . Proof. Let E ⊂ Q E\subset Q E ⊂ Q w ∈ A A , p ( ⋅ ) w\in\mathcal{A}_{A,p(\cdot)} w ∈ A A , p ( ⋅ ) A A , p ( ⋅ ) \mathcal{A}_{A,p(\cdot)} A A , p ( ⋅ )
∣ E ∣ = ∫ χ E w w − 1 χ Q ≤ K p ( ⋅ ) ∥ w χ E ∥ p ( ⋅ ) ∥ w − 1 χ Q ∥ p ′ ( ⋅ ) ≤ K p ( ⋅ ) [ w ] A A , p ( ⋅ ) ∥ w χ E ∥ p ( ⋅ ) ∥ w A χ Q ∥ p ( ⋅ ) − 1 ∣ Q ∣ . \begin{aligned}
|E|=\int\chi_Eww^{-1}\chi_Q\le K_{p(\cdot)}\|w\chi_E\|_{p(\cdot)}\|w^{-1}\chi_Q\|_{p'(\cdot)} \\
&\le K_{p(\cdot)}[w]_{\mathcal{A}_{A,p(\cdot)}}\|w\chi_E\|_{p(\cdot)}\|w_A\chi_Q\|_{p(\cdot)}^{-1}|Q|.
\end{aligned} ∣ E ∣ = ∫ χ E w w − 1 χ Q ≤ K p ( ⋅ ) ∥ w χ E ∥ p ( ⋅ ) ∥ w − 1 χ Q ∥ p ′ ( ⋅ ) ≤ K p ( ⋅ ) [ w ] A A , p ( ⋅ ) ∥ w χ E ∥ p ( ⋅ ) ∥ w A χ Q ∥ p ( ⋅ ) − 1 ∣ Q ∣. The following lemma is an adapted version of the Lemma 2.2 to the A Q AQ A Q
Lemma 5.2. Let p ( ⋅ ) ∈ L H 0 ( R n ) p(\cdot) \in LH_0(\mathbb{R}^n) p ( ⋅ ) ∈ L H 0 ( R n ) p + < ∞ p_+ < \infty p + < ∞ A A A n × n n \times n n × n p ( A x ) = p ( x ) p(Ax)=p(x) p ( A x ) = p ( x ) x ∈ R n x \in\mathbb{R}^n x ∈ R n Q ⊂ R n Q \subset\mathbb{R}^n Q ⊂ R n x ∈ Q x \in Q x ∈ Q
∣ A Q ∣ p − ( Q ) − p + ( Q ) ≤ C A , |AQ|^{p_-(Q)-p_+(Q)} \le C_A, ∣ A Q ∣ p − ( Q ) − p + ( Q ) ≤ C A , where the constant C A C_A C A Q Q Q
C A = C L H 0 max { 1 , ∣ det A ∣ − p + } . C_A=C_{LH_0}\max\{1,|\det A|^{-p_+}\}. C A = C L H 0 max { 1 , ∣ det A ∣ − p + } . Proof. Since p − ( Q ) − p + ( Q ) = p − ( Q ) − p ( x ) + p ( x ) − p + ( Q ) p_-(Q)-p_+(Q)=p_-(Q)-p(x)+p(x)-p_+(Q) p − ( Q ) − p + ( Q ) = p − ( Q ) − p ( x ) + p ( x ) − p + ( Q )
∣ A Q ∣ p ( x ) − p + ( Q ) ≤ C A . |AQ|^{p(x)-p_+(Q)} \le C_A. ∣ A Q ∣ p ( x ) − p + ( Q ) ≤ C A . The proof of ∣ A Q ∣ p − ( Q ) − p ( x ) ≤ C A |AQ|^{p_-(Q)-p(x)}\le C_A ∣ A Q ∣ p − ( Q ) − p ( x ) ≤ C A
Let Q ⊂ R n Q\subset\mathbb{R}^n Q ⊂ R n x ∈ Q x\in Q x ∈ Q A A A
∣ A Q ∣ p ( x ) − p + ( Q ) = ∣ det A ∣ p ( x ) − p + ( Q ) ∣ Q ∣ p ( x ) − p + ( Q ) . |AQ|^{p(x)-p_+(Q)}=|\det A|^{p(x)-p_+(Q)}|Q|^{p(x)-p_+(Q)}. ∣ A Q ∣ p ( x ) − p + ( Q ) = ∣ det A ∣ p ( x ) − p + ( Q ) ∣ Q ∣ p ( x ) − p + ( Q ) . Since p ( ⋅ ) ∈ L H 0 ( R n ) p(\cdot)\in LH_0(\mathbb{R}^n) p ( ⋅ ) ∈ L H 0 ( R n ) Lemma 2.2 , there exists a constant C L H 0 > 0 C_{LH_0}>0 C L H 0 > 0 Q Q Q
∣ Q ∣ p ( x ) − p + ( Q ) ≤ C L H 0 . |Q|^{p(x)-p_+(Q)}\le C_{LH_0}. ∣ Q ∣ p ( x ) − p + ( Q ) ≤ C L H 0 . Therefore,
∣ A Q ∣ p ( x ) − p + ( A Q ) ≤ C L H 0 ∣ det A ∣ p ( x ) − p + ( Q ) . |AQ|^{p(x)-p_+(AQ)}\le C_{LH_0}|\det A|^{p(x)-p_+(Q)}. ∣ A Q ∣ p ( x ) − p + ( A Q ) ≤ C L H 0 ∣ det A ∣ p ( x ) − p + ( Q ) . It remains to estimate the factor involving the determinant. Since p + < ∞ p_+<\infty p + < ∞
− p + ≤ p ( x ) − p + ( Q ) ≤ 0. -p_+\le p(x)-p_+(Q)\le0. − p + ≤ p ( x ) − p + ( Q ) ≤ 0. We distinguish two cases. If ∣ det A ∣ ≥ 1 |\det A|\ge1 ∣ det A ∣ ≥ 1 p ( x ) − p + ( Q ) ≤ 0 p(x)-p_+(Q)\le0 p ( x ) − p + ( Q ) ≤ 0
∣ det A ∣ p ( x ) − p + ( Q ) ≤ 1. |\det A|^{p(x)-p_+(Q)}\le1. ∣ det A ∣ p ( x ) − p + ( Q ) ≤ 1. If 0 < ∣ det A ∣ < 1 0<|\det A|<1 0 < ∣ det A ∣ < 1 p ( x ) − p + ( Q ) ≥ − p + p(x)-p_+(Q)\ge-p_+ p ( x ) − p + ( Q ) ≥ − p +
∣ det A ∣ p ( x ) − p + ( Q ) ≤ ∣ det A ∣ − p + . |\det A|^{p(x)-p_+(Q)}\le|\det A|^{-p_+}. ∣ det A ∣ p ( x ) − p + ( Q ) ≤ ∣ det A ∣ − p + . Thus, in both cases,
∣ det A ∣ p ( x ) − p + ( Q ) ≤ max { 1 , ∣ det A ∣ − p + } . |\det A|^{p(x)-p_+(Q)}\le\max\{1,|\det A|^{-p_+}\}. ∣ det A ∣ p ( x ) − p + ( Q ) ≤ max { 1 , ∣ det A ∣ − p + } . Consequently,
∣ A Q ∣ p ( x ) − p + ( Q ) ≤ C L H 0 max { 1 , ∣ det A ∣ − p + } = : C A |AQ|^{p(x)-p_+(Q)}\le C_{LH_0}\max\{1,|\det A|^{-p_+}\}=:C_A ∣ A Q ∣ p ( x ) − p + ( Q ) ≤ C L H 0 max { 1 , ∣ det A ∣ − p + } =: C A where C A C_A C A A A A p ( ⋅ ) p(\cdot) p ( ⋅ ) L H 0 LH_0 L H 0 Q Q Q x x x
Lemma 5.3. Given A A A p ( ⋅ ) ∈ P log ( R n ) p(\cdot)\in P^{\log}(\mathbb{R}^n) p ( ⋅ ) ∈ P l o g ( R n ) p + < ∞ p_+<\infty p + < ∞ p ( A x ) = p ( x ) p(Ax)=p(x) p ( A x ) = p ( x ) x x x w ∈ A A , p ( ⋅ ) w\in A_{A,p(\cdot)} w ∈ A A , p ( ⋅ ) Q Q Q
∥ w χ A Q ∥ p ( ⋅ ) p − ( Q ) − p + ( Q ) ≤ L 1 [ [ w ] ] A A , p ( ⋅ ) p + − p − \|w\chi_{AQ}\|_{p(\cdot)}^{p_-(Q)-p_+(Q)}\le L_1[[w]]_{A_{A,p(\cdot)}}^{p_+-p_-} ∥ w χ A Q ∥ p ( ⋅ ) p − ( Q ) − p + ( Q ) ≤ L 1 [[ w ] ] A A , p ( ⋅ ) p + − p − where
L 1 = C ( A , n , p ( ⋅ ) , C ∞ ) C A max { 1 , ∥ w χ Q 0 ∥ p ( ⋅ ) p − − p + } L_1=C(A,n,p(\cdot),C_\infty)C_A\max\{1,\|w\chi_{Q_0}\|_{p(\cdot)}^{p_--p_+}\} L 1 = C ( A , n , p ( ⋅ ) , C ∞ ) C A max { 1 , ∥ w χ Q 0 ∥ p ( ⋅ ) p − − p + } and Q 0 = Q ( 0 , 2 e ) Q_0=Q(0,2e) Q 0 = Q ( 0 , 2 e )
Proof. The proof follows the ideas of [2 , 6 ] with several changes that involve matrix A A A w ∈ A A , p ( ⋅ ) w \in A_{A,p(\cdot)} w ∈ A A , p ( ⋅ ) Q Q Q Q 0 = Q ( 0 , e ) Q_0=Q(0,e) Q 0 = Q ( 0 , e ) ∥ w χ A Q ∥ p ( ⋅ ) ≥ 1 \lVert w\chi_{AQ}\rVert_{p(\cdot)}\ge1 ∥ w χ A Q ∥ p ( ⋅ ) ≥ 1 ∥ w χ A Q ∥ p ( ⋅ ) < 1 \lVert w\chi_{AQ}\rVert_{p(\cdot)}<1 ∥ w χ A Q ∥ p ( ⋅ ) < 1
Now, we split the proof in several cases according the diameter of A Q AQ A Q Q 0 Q_0 Q 0 A Q AQ A Q Q 0 Q_0 Q 0
The first case, suppose diam ( A Q ) ≤ ℓ ( Q 0 ) \operatorname{diam}(AQ)\le\ell(Q_0) diam ( A Q ) ≤ ℓ ( Q 0 ) dist ( A Q , Q 0 ) ≤ ℓ ( Q 0 ) \operatorname{dist}(AQ,Q_0)\le\ell(Q_0) dist ( A Q , Q 0 ) ≤ ℓ ( Q 0 ) A Q ⊂ Q 0 5 AQ\subset Q_0^5 A Q ⊂ Q 0 5
∣ A Q ∣ ≤ K p ( ⋅ ) [ w ] A A , p ( ⋅ ) ∥ w χ A Q ∥ p ( ⋅ ) ∥ w A χ 5 Q 0 ∥ p ( ⋅ ) − 1 ≤ ∣ 5 Q 0 ∣ [ [ w ] ] A A , p ( ⋅ ) ∥ w χ A Q ∥ p ( ⋅ ) ∥ w A χ Q 0 ∥ p ( ⋅ ) − 1 . |AQ|\le K_{p(\cdot)}[w]_{A_{A,p(\cdot)}}\lVert w\chi_{AQ}\rVert_{p(\cdot)}\lVert w_A\chi_{5Q_0}\rVert_{p(\cdot)}^{-1}
\le|5Q_0|[[w]]_{A_{A,p(\cdot)}}\lVert w\chi_{AQ}\rVert_{p(\cdot)}\lVert w_A\chi_{Q_0}\rVert_{p(\cdot)}^{-1}. ∣ A Q ∣ ≤ K p ( ⋅ ) [ w ] A A , p ( ⋅ ) ∥ w χ A Q ∥ p ( ⋅ ) ∥ w A χ 5 Q 0 ∥ p ( ⋅ ) − 1 ≤ ∣5 Q 0 ∣ [[ w ] ] A A , p ( ⋅ ) ∥ w χ A Q ∥ p ( ⋅ ) ∥ w A χ Q 0 ∥ p ( ⋅ ) − 1 . Thus, by Lemma 5.2
∥ w χ A Q ∥ p ( ⋅ ) p − ( Q ) − p + ( Q ) ≤ ( ∣ 5 Q 0 ∣ ∣ A Q ∣ ) p + ( Q ) − p − ( Q ) [ [ w ] ] A A , p ( ⋅ ) p + ( Q ) − p − ( Q ) ∥ w A χ Q 0 ∥ p ( ⋅ ) p − ( Q ) − p + ( Q ) ≤ ∣ 5 Q 0 ∣ p + − p − C A [ [ w ] ] A A , p ( ⋅ ) p + − p − ∥ w A χ Q 0 ∥ p ( ⋅ ) p − ( Q ) − p + ( Q ) . \begin{aligned}
\lVert w\chi_{AQ}\rVert_{p(\cdot)}^{p_-(Q)-p_+(Q)}
\le\left(\frac{|5Q_0|}{|AQ|}\right)^{p_+(Q)-p_-(Q)}
[[w]]_{A_{A,p(\cdot)}}^{p_+(Q)-p_-(Q)}
\lVert w_A\chi_{Q_0}\rVert_{p(\cdot)}^{p_-(Q)-p_+(Q)} \\
&\le|5Q_0|^{p_+-p_-}C_A[[w]]_{A_{A,p(\cdot)}}^{p_+-p_-}
\lVert w_A\chi_{Q_0}\rVert_{p(\cdot)}^{p_-(Q)-p_+(Q)}.
\end{aligned} ∥ w χ A Q ∥ p ( ⋅ ) p − ( Q ) − p + ( Q ) ≤ ( ∣ A Q ∣ ∣5 Q 0 ∣ ) p + ( Q ) − p − ( Q ) [[ w ] ] A A , p ( ⋅ ) p + ( Q ) − p − ( Q ) ∥ w A χ Q 0 ∥ p ( ⋅ ) p − ( Q ) − p + ( Q ) ≤ ∣5 Q 0 ∣ p + − p − C A [[ w ] ] A A , p ( ⋅ ) p + − p − ∥ w A χ Q 0 ∥ p ( ⋅ ) p − ( Q ) − p + ( Q ) . For the second case, suppose diam ( A Q ) ≤ ℓ ( Q 0 ) \operatorname{diam}(AQ)\le\ell(Q_0) diam ( A Q ) ≤ ℓ ( Q 0 ) dist ( A Q , Q 0 ) > ℓ ( Q 0 ) \operatorname{dist}(AQ,Q_0)>\ell(Q_0) dist ( A Q , Q 0 ) > ℓ ( Q 0 ) Q ~ = Q ( 0 , 3 dist ( A Q , Q 0 ) ) \widetilde{Q}=Q(0,3\operatorname{dist}(AQ,Q_0)) Q = Q ( 0 , 3 dist ( A Q , Q 0 )) A Q , Q 0 ⊂ Q ~ AQ,Q_0\subset\widetilde{Q} A Q , Q 0 ⊂ Q ℓ ( Q ~ ) ≤ 3 dist ( A Q , 0 ) \ell(\widetilde{Q})\le3\operatorname{dist}(AQ,0) ℓ ( Q ) ≤ 3 dist ( A Q , 0 )
∣ A Q ∣ ≤ K p ( ⋅ ) [ w ] A A , p ( ⋅ ) ∥ w χ A Q ∥ p ( ⋅ ) ∣ Q ~ ∣ ∥ w A χ Q ~ ∥ p ( ⋅ ) − 1 ≤ [ [ w ] ] A A , p ( ⋅ ) ∥ w χ A Q ∥ p ( ⋅ ) ∣ Q ~ ∣ ∥ w A χ Q 0 ∥ p ( ⋅ ) − 1 . |AQ|\le K_{p(\cdot)}[w]_{A_{A,p(\cdot)}}\lVert w\chi_{AQ}\rVert_{p(\cdot)}|\widetilde{Q}|\lVert w_A\chi_{\widetilde{Q}}\rVert_{p(\cdot)}^{-1}
\le[[w]]_{A_{A,p(\cdot)}}\lVert w\chi_{AQ}\rVert_{p(\cdot)}|\widetilde{Q}|\lVert w_A\chi_{Q_0}\rVert_{p(\cdot)}^{-1}. ∣ A Q ∣ ≤ K p ( ⋅ ) [ w ] A A , p ( ⋅ ) ∥ w χ A Q ∥ p ( ⋅ ) ∣ Q ∣ ∥ w A χ Q ∥ p ( ⋅ ) − 1 ≤ [[ w ] ] A A , p ( ⋅ ) ∥ w χ A Q ∥ p ( ⋅ ) ∣ Q ∣ ∥ w A χ Q 0 ∥ p ( ⋅ ) − 1 . Thus,
∥ w χ A Q ∥ p ( ⋅ ) p − ( Q ) − p + ( Q ) ≤ [ [ w ] ] A A , p ( ⋅ ) p + ( Q ) − p − ( Q ) ∥ w A χ Q 0 ∥ p ( ⋅ ) p − ( Q ) − p + ( Q ) ( ∣ Q ~ ∣ ∣ A Q ∣ ) p + ( Q ) − p − ( Q ) ≤ [ [ w ] ] A A , p ( ⋅ ) p + − p − ∥ w A χ Q 0 ∥ p ( ⋅ ) p − ( Q ) − p + ( Q ) ∣ Q ~ ∣ p + ( Q ) − p − ( Q ) C A . \begin{aligned}
\lVert w\chi_{AQ}\rVert_{p(\cdot)}^{p_-(Q)-p_+(Q)}
\le[[w]]_{A_{A,p(\cdot)}}^{p_+(Q)-p_-(Q)}
\lVert w_A\chi_{Q_0}\rVert_{p(\cdot)}^{p_-(Q)-p_+(Q)}
\left(\frac{|\widetilde{Q}|}{|AQ|}\right)^{p_+(Q)-p_-(Q)} \\
&\le[[w]]_{A_{A,p(\cdot)}}^{p_+-p_-}
\lVert w_A\chi_{Q_0}\rVert_{p(\cdot)}^{p_-(Q)-p_+(Q)}
|\widetilde{Q}|^{p_+(Q)-p_-(Q)}C_A.
\end{aligned} ∥ w χ A Q ∥ p ( ⋅ ) p − ( Q ) − p + ( Q ) ≤ [[ w ] ] A A , p ( ⋅ ) p + ( Q ) − p − ( Q ) ∥ w A χ Q 0 ∥ p ( ⋅ ) p − ( Q ) − p + ( Q ) ( ∣ A Q ∣ ∣ Q ∣ ) p + ( Q ) − p − ( Q ) ≤ [[ w ] ] A A , p ( ⋅ ) p + − p − ∥ w A χ Q 0 ∥ p ( ⋅ ) p − ( Q ) − p + ( Q ) ∣ Q ∣ p + ( Q ) − p − ( Q ) C A . The estimate of ∣ Q ~ ∣ p + ( Q ) − p − ( Q ) |\widetilde{Q}|^{p_+(Q)-p_-(Q)} ∣ Q ∣ p + ( Q ) − p − ( Q ) [6 ] . That is,
∣ Q ~ ∣ p + ( Q ) − p − ( Q ) ≤ 3 n ( p + − p − ) e 2 n C ∞ |\widetilde{Q}|^{p_+(Q)-p_-(Q)}\le3^{n(p_+-p_-)}e^{2nC_\infty} ∣ Q ∣ p + ( Q ) − p − ( Q ) ≤ 3 n ( p + − p − ) e 2 n C ∞ then
∥ w χ A Q ∥ p ( ⋅ ) p − ( Q ) − p + ( Q ) ≤ 2 n ( p + − p − ) e 2 n C ∞ C D [ [ w ] ] A A , p ( ⋅ ) p + − p − ∥ w A χ Q 0 ∥ p ( ⋅ ) p − ( Q ) − p + ( Q ) . \lVert w\chi_{AQ}\rVert_{p(\cdot)}^{p_-(Q)-p_+(Q)}
\le2^{n(p_+-p_-)}e^{2nC_\infty}C_D[[w]]_{A_{A,p(\cdot)}}^{p_+-p_-}
\lVert w_A\chi_{Q_0}\rVert_{p(\cdot)}^{p_-(Q)-p_+(Q)}. ∥ w χ A Q ∥ p ( ⋅ ) p − ( Q ) − p + ( Q ) ≤ 2 n ( p + − p − ) e 2 n C ∞ C D [[ w ] ] A A , p ( ⋅ ) p + − p − ∥ w A χ Q 0 ∥ p ( ⋅ ) p − ( Q ) − p + ( Q ) . For the third case, suppose diam ( A Q ) > ℓ ( Q 0 ) \operatorname{diam}(AQ)>\ell(Q_0) diam ( A Q ) > ℓ ( Q 0 ) dist ( A Q , Q 0 ) ≤ diam ( A Q ) \operatorname{dist}(AQ,Q_0)\le\operatorname{diam}(AQ) dist ( A Q , Q 0 ) ≤ diam ( A Q ) A Q , Q 0 ⊂ Q A = Q ( A c Q , 5 diam ( A Q ) ) AQ,Q_0\subset Q_A=Q(Ac_Q,5\operatorname{diam}(AQ)) A Q , Q 0 ⊂ Q A = Q ( A c Q , 5 diam ( A Q ))
∣ A Q ∣ ≤ K p ( ⋅ ) ∣ Q A ∣ [ w ] A A , p ( ⋅ ) ∥ w χ A Q ∥ p ( ⋅ ) ∥ w A χ Q A ∥ p ( ⋅ ) − 1 ≤ ∣ Q A ∣ [ [ w ] ] A A , p ( ⋅ ) ∥ w χ A Q ∥ p ( ⋅ ) ∥ w A χ Q 0 ∥ p ( ⋅ ) − 1 . |AQ|\le K_{p(\cdot)}|Q_A|[w]_{A_{A,p(\cdot)}}\lVert w\chi_{AQ}\rVert_{p(\cdot)}\lVert w_A\chi_{Q_A}\rVert_{p(\cdot)}^{-1}
\le|Q_A|[[w]]_{A_{A,p(\cdot)}}\lVert w\chi_{AQ}\rVert_{p(\cdot)}\lVert w_A\chi_{Q_0}\rVert_{p(\cdot)}^{-1}. ∣ A Q ∣ ≤ K p ( ⋅ ) ∣ Q A ∣ [ w ] A A , p ( ⋅ ) ∥ w χ A Q ∥ p ( ⋅ ) ∥ w A χ Q A ∥ p ( ⋅ ) − 1 ≤ ∣ Q A ∣ [[ w ] ] A A , p ( ⋅ ) ∥ w χ A Q ∥ p ( ⋅ ) ∥ w A χ Q 0 ∥ p ( ⋅ ) − 1 . Observe that 1 ≤ ∣ Q A ∣ ∣ A Q ∣ ≤ 5 n ∥ A ∥ n n ∣ det ( A ) ∣ = : d A 1\le\dfrac{|Q_A|}{|AQ|}\le\dfrac{5^n\lVert A\rVert^n\sqrt{n}}{|\det(A)|}=:d_A 1 ≤ ∣ A Q ∣ ∣ Q A ∣ ≤ ∣ det ( A ) ∣ 5 n ∥ A ∥ n n =: d A
∥ w χ A Q ∥ p ( ⋅ ) p − ( Q ) − p + ( Q ) ≤ d A n ( p + − p − ) [ [ w ] ] A A , p ( ⋅ ) p + − p − ∥ w A χ Q 0 ∥ p ( ⋅ ) p − ( Q ) − p + ( Q ) . \lVert w\chi_{AQ}\rVert_{p(\cdot)}^{p_-(Q)-p_+(Q)}
\le d_A^{n(p_+-p_-)}[[w]]_{A_{A,p(\cdot)}}^{p_+-p_-}
\lVert w_A\chi_{Q_0}\rVert_{p(\cdot)}^{p_-(Q)-p_+(Q)}. ∥ w χ A Q ∥ p ( ⋅ ) p − ( Q ) − p + ( Q ) ≤ d A n ( p + − p − ) [[ w ] ] A A , p ( ⋅ ) p + − p − ∥ w A χ Q 0 ∥ p ( ⋅ ) p − ( Q ) − p + ( Q ) . For the last case, suppose diam ( A Q ) > ℓ ( Q 0 ) \operatorname{diam}(AQ)>\ell(Q_0) diam ( A Q ) > ℓ ( Q 0 ) dist ( A Q , Q 0 ) > diam ( A Q ) \operatorname{dist}(AQ,Q_0)>\operatorname{diam}(AQ) dist ( A Q , Q 0 ) > diam ( A Q ) Q ~ = Q ( 0 , 3 dist ( A Q , Q 0 ) ) \widetilde{Q}=Q(0,3\operatorname{dist}(AQ,Q_0)) Q = Q ( 0 , 3 dist ( A Q , Q 0 )) A Q , Q 0 ⊂ Q ~ AQ,Q_0\subset\widetilde{Q} A Q , Q 0 ⊂ Q
∣ A Q ∣ ≤ ∣ Q ~ ∣ [ [ w ] ] A , p ( ⋅ ) ∥ w χ A Q ∥ p ( ⋅ ) ∥ w A χ Q 0 ∥ p ( ⋅ ) − 1 . |AQ|\leq|\widetilde{Q}|[[w]]_{A,p(\cdot)}\|w\chi_{AQ}\|_{p(\cdot)}\|w_A\chi_{Q_0}\|_{p(\cdot)}^{-1}. ∣ A Q ∣ ≤ ∣ Q ∣ [[ w ] ] A , p ( ⋅ ) ∥ w χ A Q ∥ p ( ⋅ ) ∥ w A χ Q 0 ∥ p ( ⋅ ) − 1 . Thus,
∥ w χ A Q ∥ p ( ⋅ ) p − ( Q ) − p + ( Q ) ≤ [ [ w ] ] A , p ( ⋅ ) p + ( Q ) − p − ( Q ) ∥ w A χ Q 0 ∥ p ( ⋅ ) p − ( Q ) − p + ( Q ) ( ∣ Q ~ ∣ ∣ A Q ∣ ) p + ( Q ) − p − ( Q ) ≤ [ [ w ] ] A , p ( ⋅ ) p + − p − ∥ w A χ Q 0 ∥ p ( ⋅ ) p − ( Q ) − p + ( Q ) C A 2 n ( p + − p − ) e 2 n C ∞ . \begin{aligned}
\|w\chi_{AQ}\|_{p(\cdot)}^{p_-(Q)-p_+(Q)}
\leq[[w]]_{A,p(\cdot)}^{p_+(Q)-p_-(Q)}
\|w_A\chi_{Q_0}\|_{p(\cdot)}^{p_-(Q)-p_+(Q)}
\left(\frac{|\widetilde{Q}|}{|AQ|}\right)^{p_+(Q)-p_-(Q)} \\
&\leq[[w]]_{A,p(\cdot)}^{p_+-p_-}
\|w_A\chi_{Q_0}\|_{p(\cdot)}^{p_-(Q)-p_+(Q)}
C_A2^{n(p_+-p_-)}e^{2nC_\infty}.
\end{aligned} ∥ w χ A Q ∥ p ( ⋅ ) p − ( Q ) − p + ( Q ) ≤ [[ w ] ] A , p ( ⋅ ) p + ( Q ) − p − ( Q ) ∥ w A χ Q 0 ∥ p ( ⋅ ) p − ( Q ) − p + ( Q ) ( ∣ A Q ∣ ∣ Q ∣ ) p + ( Q ) − p − ( Q ) ≤ [[ w ] ] A , p ( ⋅ ) p + − p − ∥ w A χ Q 0 ∥ p ( ⋅ ) p − ( Q ) − p + ( Q ) C A 2 n ( p + − p − ) e 2 n C ∞ . Summary of the estimate proved above, for a cube Q Q Q
∥ w χ A Q ∥ p ( ⋅ ) p − ( Q ) − p + ( Q ) ≤ C 1 ( n , p ( ⋅ ) , C ∞ ) C A [ [ w ] ] A , p ( ⋅ ) p + − p − ∥ w A χ Q 0 ∥ p ( ⋅ ) p − ( Q ) − p + ( Q ) . \|w\chi_{AQ}\|_{p(\cdot)}^{p_-(Q)-p_+(Q)}
\leq C_1(n,p(\cdot),C_\infty)C_A[[w]]_{A,p(\cdot)}^{p_+-p_-}\|w_A\chi_{Q_0}\|_{p(\cdot)}^{p_-(Q)-p_+(Q)}. ∥ w χ A Q ∥ p ( ⋅ ) p − ( Q ) − p + ( Q ) ≤ C 1 ( n , p ( ⋅ ) , C ∞ ) C A [[ w ] ] A , p ( ⋅ ) p + − p − ∥ w A χ Q 0 ∥ p ( ⋅ ) p − ( Q ) − p + ( Q ) . where C 1 ( n , p ( ⋅ ) , C ∞ ) C_1(n,p(\cdot),C_\infty) C 1 ( n , p ( ⋅ ) , C ∞ ) ∥ w A χ Q 0 ∥ p ( ⋅ ) p + ( Q ) − p − ( Q ) \|w_A\chi_{Q_0}\|_{p(\cdot)}^{p_+(Q)-p_-(Q)} ∥ w A χ Q 0 ∥ p ( ⋅ ) p + ( Q ) − p − ( Q ) ∥ w A χ Q 0 ∥ p ( ⋅ ) < 1 \|w_A\chi_{Q_0}\|_{p(\cdot)}<1 ∥ w A χ Q 0 ∥ p ( ⋅ ) < 1 ∥ w A χ Q 0 ∥ p ( ⋅ ) p − ( Q ) − p + ( Q ) ≤ ∥ w A χ Q 0 ∥ p ( ⋅ ) p − − p + \|w_A\chi_{Q_0}\|_{p(\cdot)}^{p_-(Q)-p_+(Q)}\leq\|w_A\chi_{Q_0}\|_{p(\cdot)}^{p_--p_+} ∥ w A χ Q 0 ∥ p ( ⋅ ) p − ( Q ) − p + ( Q ) ≤ ∥ w A χ Q 0 ∥ p ( ⋅ ) p − − p + ∥ w A χ Q 0 ∥ p ( ⋅ ) ≥ 1 \|w_A\chi_{Q_0}\|_{p(\cdot)}\geq1 ∥ w A χ Q 0 ∥ p ( ⋅ ) ≥ 1 ∥ w A χ Q 0 ∥ p ( ⋅ ) p − ( Q ) − p + ( Q ) ≤ 1 \|w_A\chi_{Q_0}\|_{p(\cdot)}^{p_-(Q)-p_+(Q)}\leq1 ∥ w A χ Q 0 ∥ p ( ⋅ ) p − ( Q ) − p + ( Q ) ≤ 1
∥ w A χ Q 0 ∥ p ( ⋅ ) p − ( Q ) − p + ( Q ) ≤ max { 1 , ∥ w A χ Q 0 ∥ p ( ⋅ ) p − − p + } . \|w_A\chi_{Q_0}\|_{p(\cdot)}^{p_-(Q)-p_+(Q)}
\leq\max\left\{1,\|w_A\chi_{Q_0}\|_{p(\cdot)}^{p_--p_+}\right\}. ∥ w A χ Q 0 ∥ p ( ⋅ ) p − ( Q ) − p + ( Q ) ≤ max { 1 , ∥ w A χ Q 0 ∥ p ( ⋅ ) p − − p + } . Hence,
∥ w χ Q ∥ p ( ⋅ ) p − ( Q ) − p + ( Q ) ≤ C ( A , n , p ( ⋅ ) , C ∞ ) [ [ w ] ] A , p ( ⋅ ) p + − p − max { 1 , ∥ w A χ Q 0 ∥ p ( ⋅ ) p − − p + } \|w\chi_Q\|_{p(\cdot)}^{p_-(Q)-p_+(Q)}
\leq C(A,n,p(\cdot),C_\infty)[[w]]_{A,p(\cdot)}^{p_+-p_-}
\max\left\{1,\|w_A\chi_{Q_0}\|_{p(\cdot)}^{p_--p_+}\right\} ∥ w χ Q ∥ p ( ⋅ ) p − ( Q ) − p + ( Q ) ≤ C ( A , n , p ( ⋅ ) , C ∞ ) [[ w ] ] A , p ( ⋅ ) p + − p − max { 1 , ∥ w A χ Q 0 ∥ p ( ⋅ ) p − − p + } where
C ( A , n , p ( ⋅ ) , C ∞ ) = C 1 ( n , p ( ⋅ ) , C ∞ ) C A . C(A,n,p(\cdot),C_\infty)=C_1(n,p(\cdot),C_\infty)C_A. C ( A , n , p ( ⋅ ) , C ∞ ) = C 1 ( n , p ( ⋅ ) , C ∞ ) C A . Lemma 5.4
Lemma 5.4. Let A A A p ( ⋅ ) ∈ P log ( R n ) p(\cdot)\in\mathcal{P}^{\log}(\mathbb{R}^n) p ( ⋅ ) ∈ P l o g ( R n ) p + < ∞ p_+<\infty p + < ∞ p ( A x ) = p ( x ) p(Ax)=p(x) p ( A x ) = p ( x ) x x x w ∈ A A , p ( ⋅ ) w\in A_{A,p(\cdot)} w ∈ A A , p ( ⋅ ) Q Q Q
ρ ( w A χ Q ) ≤ L 2 [ [ w ] ] A , p ( ⋅ ) max { 2 p + − p − , 2 C ∞ p + p − + p + } ρ ( w χ Q ) . \rho(w_A\chi_Q)\leq L_2[[w]]_{A,p(\cdot)}^{\max\left\{2p_+-p_-,2C_\infty\frac{p_+}{p_-+p_+}\right\}}\rho(w\chi_Q). ρ ( w A χ Q ) ≤ L 2 [[ w ] ] A , p ( ⋅ ) m a x { 2 p + − p − , 2 C ∞ p − + p + p + } ρ ( w χ Q ) . where
L 2 = C ( A , n , p ( ⋅ ) ) max { ∥ w A χ Q 0 ∥ p ( ⋅ ) p − − p + , ∥ w A χ Q 0 ∥ p ( ⋅ ) p + − p − } L_2=C(A,n,p(\cdot))\max\left\{\|w_A\chi_{Q_0}\|_{p(\cdot)}^{p_--p_+},\|w_A\chi_{Q_0}\|_{p(\cdot)}^{p_+-p_-}\right\} L 2 = C ( A , n , p ( ⋅ )) max { ∥ w A χ Q 0 ∥ p ( ⋅ ) p − − p + , ∥ w A χ Q 0 ∥ p ( ⋅ ) p + − p − } and Q 0 = Q ( 0 , 2 e ) Q_0=Q(0,2e) Q 0 = Q ( 0 , 2 e )
Proof. Fix a cube Q Q Q Q 0 = Q ( 0 , 2 e ) Q_0=Q(0,2e) Q 0 = Q ( 0 , 2 e ) ∥ w A χ Q ∥ p ( ⋅ ) \|w_A\chi_Q\|_{p(\cdot)} ∥ w A χ Q ∥ p ( ⋅ ) ∥ w χ Q ∥ p ( ⋅ ) \|w\chi_Q\|_{p(\cdot)} ∥ w χ Q ∥ p ( ⋅ ) ∥ w A χ Q ∥ p ( ⋅ ) , ∥ w χ Q ∥ p ( ⋅ ) ≤ 1 \|w_A\chi_Q\|_{p(\cdot)},\|w\chi_Q\|_{p(\cdot)}\leq1 ∥ w A χ Q ∥ p ( ⋅ ) , ∥ w χ Q ∥ p ( ⋅ ) ≤ 1
1 = ∣ Q ∣ ∣ Q ∣ ≤ K p ( ⋅ ) [ w ] A , p ( ⋅ ) ∥ w χ Q ∥ p ( ⋅ ) ∥ w A χ Q ∥ p ( ⋅ ) ≤ [ [ w ] ] A , p ( ⋅ ) ∥ w χ Q ∥ p ( ⋅ ) ∥ w A χ Q ∥ p ( ⋅ ) . 1=\frac{|Q|}{|Q|}\leq K_{p(\cdot)}[w]_{A,p(\cdot)}
\frac{\|w\chi_Q\|_{p(\cdot)}}{\|w_A\chi_Q\|_{p(\cdot)}}
\leq[[w]]_{A,p(\cdot)}
\frac{\|w\chi_Q\|_{p(\cdot)}}{\|w_A\chi_Q\|_{p(\cdot)}}. 1 = ∣ Q ∣ ∣ Q ∣ ≤ K p ( ⋅ ) [ w ] A , p ( ⋅ ) ∥ w A χ Q ∥ p ( ⋅ ) ∥ w χ Q ∥ p ( ⋅ ) ≤ [[ w ] ] A , p ( ⋅ ) ∥ w A χ Q ∥ p ( ⋅ ) ∥ w χ Q ∥ p ( ⋅ ) . By Lemma 2.4 we have ∥ w χ Q ∥ p ( ⋅ ) ≤ ρ ( w χ Q ) 1 p + ( Q ) \|w\chi_Q\|_{p(\cdot)} \leq\rho(w\chi_Q)^{\frac{1}{p^{+}(Q)}} ∥ w χ Q ∥ p ( ⋅ ) ≤ ρ ( w χ Q ) p + ( Q ) 1 ∥ w A χ Q ∥ p ( ⋅ ) ≥ ρ ( w A χ Q ) 1 p − ( Q ) \|w_A\chi_Q\|_{p(\cdot)} \geq\rho(w_A\chi_Q)^{\frac{1}{p^{-}(Q)}} ∥ w A χ Q ∥ p ( ⋅ ) ≥ ρ ( w A χ Q ) p − ( Q ) 1
1 ≤ ( [ [ w ] ] A A , p ( ⋅ ) ∥ w χ Q ∥ p ( ⋅ ) ∥ w A χ Q ∥ p ( ⋅ ) ) p + ( Q ) p + ≤ [ [ w ] ] A A , p ( ⋅ ) p + ( Q ) p + ( ∥ w χ Q ∥ p ( ⋅ ) p + ( Q ) ∥ w A χ Q ∥ p ( ⋅ ) p − ( Q ) ) 1 p + ∥ w A χ Q ∥ p ( ⋅ ) p − ( Q ) − p + ( Q ) p + ≤ [ [ w ] ] A A , p ( ⋅ ) 1 p + ( ρ ( w χ Q ) ρ ( w A χ Q ) ) 1 p + ( ∥ w A χ Q ∥ p ( ⋅ ) p − ( Q ) − p + ( Q ) ) 1 p + . \begin{aligned}
1 \leq\left( [[w]]_{AA,p(\cdot)}\frac{\|w\chi_Q\|_{p(\cdot)}}{\|w_A\chi_Q\|_{p(\cdot)}}\right)^{\frac{p^{+}(Q)}{p^{+}}} \\
&\leq[[w]]_{AA,p(\cdot)}^{\frac{p^{+}(Q)}{p^{+}}}\left(\frac{\|w\chi_Q\|_{p(\cdot)}^{p^{+}(Q)}}{\|w_A\chi_Q\|_{p(\cdot)}^{p^{-}(Q)}}\right)^{\frac{1}{p^{+}}}\|w_A\chi_Q\|_{p(\cdot)}^{\frac{p^{-}(Q)-p^{+}(Q)}{p^{+}}} \\
&\leq[[w]]_{AA,p(\cdot)}^{\frac{1}{p^{+}}}\left(\frac{\rho(w\chi_Q)}{\rho(w_A\chi_Q)}\right)^{\frac{1}{p^{+}}}\left(\|w_A\chi_Q\|_{p(\cdot)}^{p^{-}(Q)-p^{+}(Q)}\right)^{\frac{1}{p^{+}}}.
\end{aligned} 1 ≤ ( [[ w ] ] AA , p ( ⋅ ) ∥ w A χ Q ∥ p ( ⋅ ) ∥ w χ Q ∥ p ( ⋅ ) ) p + p + ( Q ) ≤ [[ w ] ] AA , p ( ⋅ ) p + p + ( Q ) ∥ w A χ Q ∥ p ( ⋅ ) p − ( Q ) ∥ w χ Q ∥ p ( ⋅ ) p + ( Q ) p + 1 ∥ w A χ Q ∥ p ( ⋅ ) p + p − ( Q ) − p + ( Q ) ≤ [[ w ] ] AA , p ( ⋅ ) p + 1 ( ρ ( w A χ Q ) ρ ( w χ Q ) ) p + 1 ( ∥ w A χ Q ∥ p ( ⋅ ) p − ( Q ) − p + ( Q ) ) p + 1 . Then by Proposition 2.8 and Lemma 5.3 we have
∥ w A χ Q ∥ p ( ⋅ ) p − ( Q ) − p + ( Q ) = ∥ ( w χ A Q ) ∘ A ∥ p ( ⋅ ) p − ( Q ) − p + ( Q ) ≲ ∥ w χ A Q ∥ p ( ⋅ ) p − ( Q ) − p + ( Q ) ≤ C ( A , n , p ( ⋅ ) ) [ w ] A A , p ( ⋅ ) p + − p − . \begin{aligned}
\|w_A\chi_Q\|_{p(\cdot)}^{p^{-}(Q)-p^{+}(Q)}
= \|(w\chi_{AQ})\circ A\|_{p(\cdot)}^{p^{-}(Q)-p^{+}(Q)}
\lesssim\|w\chi_{AQ}\|_{p(\cdot)}^{p^{-}(Q)-p^{+}(Q)} \\
&\leq C(A,n,p(\cdot)) [w]_{AA,p(\cdot)}^{p^{+}-p^{-}}.
\end{aligned} ∥ w A χ Q ∥ p ( ⋅ ) p − ( Q ) − p + ( Q ) = ∥ ( w χ A Q ) ∘ A ∥ p ( ⋅ ) p − ( Q ) − p + ( Q ) ≲ ∥ w χ A Q ∥ p ( ⋅ ) p − ( Q ) − p + ( Q ) ≤ C ( A , n , p ( ⋅ )) [ w ] AA , p ( ⋅ ) p + − p − . Then,
ρ ( w A χ Q ) ≤ C ( A , n , p ( ⋅ ) ) [ [ w ] ] A A , p ( ⋅ ) 2 p + − p − ρ ( w χ Q ) . \rho(w_A\chi_Q) \leq C(A,n,p(\cdot))[[w]]_{AA,p(\cdot)}^{2p^{+}-p^{-}}\rho(w\chi_Q). ρ ( w A χ Q ) ≤ C ( A , n , p ( ⋅ )) [[ w ] ] AA , p ( ⋅ ) 2 p + − p − ρ ( w χ Q ) . The second case is ∥ w χ Q ∥ p ( ⋅ ) ≤ 1 < ∥ w A χ Q ∥ p ( ⋅ ) \|w\chi_Q\|_{p(\cdot)} \leq1 < \|w_A\chi_Q\|_{p(\cdot)} ∥ w χ Q ∥ p ( ⋅ ) ≤ 1 < ∥ w A χ Q ∥ p ( ⋅ )
1 = ∣ Q ∣ ∣ Q ∣ ≤ K p ( ⋅ ) [ w ] A A , p ( ⋅ ) ∥ w χ Q ∥ p ( ⋅ ) ∥ w A χ Q ∥ p ( ⋅ ) ≤ [ [ w ] ] A A , p ( ⋅ ) ( ρ ( w χ Q ) ρ ( w A χ Q ) ) 1 p + . 1=\frac{|Q|}{|Q|}\leq K_{p(\cdot)}[w]_{AA,p(\cdot)}\frac{\|w\chi_Q\|_{p(\cdot)}}{\|w_A\chi_Q\|_{p(\cdot)}}\leq[[w]]_{AA,p(\cdot)}\left(\frac{\rho(w\chi_Q)}{\rho(w_A\chi_Q)}\right)^{\frac{1}{p^{+}}}. 1 = ∣ Q ∣ ∣ Q ∣ ≤ K p ( ⋅ ) [ w ] AA , p ( ⋅ ) ∥ w A χ Q ∥ p ( ⋅ ) ∥ w χ Q ∥ p ( ⋅ ) ≤ [[ w ] ] AA , p ( ⋅ ) ( ρ ( w A χ Q ) ρ ( w χ Q ) ) p + 1 . The third case is ∥ w A χ Q ∥ p ( ⋅ ) ≤ 1 < ∥ w χ Q ∥ p ( ⋅ ) \|w_A\chi_Q\|_{p(\cdot)} \leq1 < \|w\chi_Q\|_{p(\cdot)} ∥ w A χ Q ∥ p ( ⋅ ) ≤ 1 < ∥ w χ Q ∥ p ( ⋅ )
1 = ∣ Q ∣ ∣ Q ∣ ≤ K p ( ⋅ ) [ w ] A A , p ( ⋅ ) ∥ w χ Q ∥ p ( ⋅ ) ∥ w A χ Q ∥ p ( ⋅ ) ≤ [ [ w ] ] A A , p ( ⋅ ) ( ρ ( w χ Q ) ρ ( w A χ Q ) ) 1 p − . 1=\frac{|Q|}{|Q|}\leq K_{p(\cdot)}[w]_{AA,p(\cdot)}\frac{\|w\chi_Q\|_{p(\cdot)}}{\|w_A\chi_Q\|_{p(\cdot)}}\leq[[w]]_{AA,p(\cdot)}\left(\frac{\rho(w\chi_Q)}{\rho(w_A\chi_Q)}\right)^{\frac{1}{p^{-}}}. 1 = ∣ Q ∣ ∣ Q ∣ ≤ K p ( ⋅ ) [ w ] AA , p ( ⋅ ) ∥ w A χ Q ∥ p ( ⋅ ) ∥ w χ Q ∥ p ( ⋅ ) ≤ [[ w ] ] AA , p ( ⋅ ) ( ρ ( w A χ Q ) ρ ( w χ Q ) ) p − 1 . The fourth and last case is ∥ w A χ Q ∥ p ( ⋅ ) , ∥ w χ Q ∥ p ( ⋅ ) > 1 \|w_A\chi_Q\|_{p(\cdot)},\|w\chi_Q\|_{p(\cdot)}>1 ∥ w A χ Q ∥ p ( ⋅ ) , ∥ w χ Q ∥ p ( ⋅ ) > 1 [6 Lemma 3.5] , we only mentioned the main changes.
Let λ = ∥ w A χ Q ∥ p ( ⋅ ) \lambda=\|w_A\chi_Q\|_{p(\cdot)} λ = ∥ w A χ Q ∥ p ( ⋅ ) d μ ( x ) = w A ( x ) p ( x ) d x d\mu(x)=w_A(x)^{p(x)}\,dx d μ ( x ) = w A ( x ) p ( x ) d x t > 0 t>0 t > 0
∫ Q λ − p ∞ w A ( x ) p ( x ) d x ≤ e n t C ∞ ∫ Q λ − p ( x ) w A ( x ) p ( x ) d x + ∫ Q w A ( x ) p ( x ) ( e + ∣ x ∣ ) t n p − d x = e n t C ∞ ρ ( w A χ Q λ ) + ∫ Q w A ( x ) p ( x ) ( e + ∣ x ∣ ) t n p − d x . \begin{aligned}
\int_Q \lambda^{-p^{\infty}}w_A(x)^{p(x)}\,dx
\leq e^{ntC_{\infty}}\int_Q\lambda^{-p(x)}w_A(x)^{p(x)}\,dx
+\int_Q\frac{w_A(x)^{p(x)}}{(e+|x|)^{tnp^{-}}}\,dx \\
&=e^{ntC_{\infty}}\rho\left(\frac{w_A\chi_Q}{\lambda}\right)
+\int_Q\frac{w_A(x)^{p(x)}}{(e+|x|)^{tnp^{-}}}\,dx.
\end{aligned} ∫ Q λ − p ∞ w A ( x ) p ( x ) d x ≤ e n t C ∞ ∫ Q λ − p ( x ) w A ( x ) p ( x ) d x + ∫ Q ( e + ∣ x ∣ ) t n p − w A ( x ) p ( x ) d x = e n t C ∞ ρ ( λ w A χ Q ) + ∫ Q ( e + ∣ x ∣ ) t n p − w A ( x ) p ( x ) d x . Observe that ρ ( w A χ Q λ ) = 1 \rho\left(\frac{w_A\chi_Q}{\lambda}\right)=1 ρ ( λ w A χ Q ) = 1 k ∈ N k\in\mathbb{N} k ∈ N Q k = Q ( 0 , 2 e k + 1 ) Q_k=Q(0,2e^{k+1}) Q k = Q ( 0 , 2 e k + 1 )
∫ R n w A ( x ) p ( x ) e + ∣ x ∣ t n p − d x ≤ e − t n p − + ∑ k = 1 ∞ e − k n t p − ∥ w A χ Q k ∥ p ( ⋅ ) p + . \int_{\mathbb{R}^n}\frac{w_A(x)^{p(x)}}{e+|x|^{tnp^{-}}}\,dx
\leq e^{-tnp^{-}}+\sum_{k=1}^{\infty}e^{-kntp^{-}}\|w_A\chi_{Q_k}\|_{p(\cdot)}^{p^{+}}. ∫ R n e + ∣ x ∣ t n p − w A ( x ) p ( x ) d x ≤ e − t n p − + k = 1 ∑ ∞ e − k n t p − ∥ w A χ Q k ∥ p ( ⋅ ) p + . Observe that by Lemma 5.1 applied to Q 0 ⊂ Q k Q_0\subset Q_k Q 0 ⊂ Q k
∥ w A χ Q k ∥ p ( ⋅ ) ≤ K p ( ⋅ ) [ w ] A A , p ( ⋅ ) ∣ Q k ∣ ∣ Q 0 ∣ ∥ w χ Q 0 ∥ p ( ⋅ ) ≤ [ [ w ] ] A A , p ( ⋅ ) e k n ∥ w χ Q 0 ∥ p ( ⋅ ) . \|w_A\chi_{Q_k}\|_{p(\cdot)}
\leq K_{p(\cdot)}[w]_{AA,p(\cdot)}\frac{|Q_k|}{|Q_0|}\|w\chi_{Q_0}\|_{p(\cdot)}
\leq[[w]]_{AA,p(\cdot)}e^{kn}\|w\chi_{Q_0}\|_{p(\cdot)}. ∥ w A χ Q k ∥ p ( ⋅ ) ≤ K p ( ⋅ ) [ w ] AA , p ( ⋅ ) ∣ Q 0 ∣ ∣ Q k ∣ ∥ w χ Q 0 ∥ p ( ⋅ ) ≤ [[ w ] ] AA , p ( ⋅ ) e k n ∥ w χ Q 0 ∥ p ( ⋅ ) . In particular, 1 = ∥ w A χ Q 0 ∥ p ( ⋅ ) ≤ [ [ w ] ] A A , p ( ⋅ ) ∥ w χ Q 0 ∥ p ( ⋅ ) 1=\|w_A\chi_{Q_0}\|_{p(\cdot)}\leq[[w]]_{AA,p(\cdot)}\|w\chi_{Q_0}\|_{p(\cdot)} 1 = ∥ w A χ Q 0 ∥ p ( ⋅ ) ≤ [[ w ] ] AA , p ( ⋅ ) ∥ w χ Q 0 ∥ p ( ⋅ )
Taking t 1 = p + p − + 1 n p − log ( 2 [ [ w ] ] A , p ( ⋅ ) p + ∥ w χ Q 0 ∥ p ( ⋅ ) p + ) t_1=\frac{p_+}{p_-}+\frac{1}{np_-}\log\left(2[[w]]_{A,p(\cdot)}^{p_+}\|w\chi_{Q_0}\|_{p(\cdot)}^{p_+}\right) t 1 = p − p + + n p − 1 log ( 2 [[ w ] ] A , p ( ⋅ ) p + ∥ w χ Q 0 ∥ p ( ⋅ ) p + )
∫ Q λ − p ∞ w A ( x ) p ( x ) d x ≤ e n t 1 C ∞ + [ e − t 1 n p − + [ [ w ] ] A , p ( ⋅ ) p + ∥ w χ Q 0 ∥ p ( ⋅ ) p + ∑ k = 1 ∞ e k n ( p + − t 1 p − ) ] \int_Q \lambda^{-p^\infty}w_A(x)^{p(x)}\,dx\le e^{nt_1C_\infty}+\left[e^{-t_1np_-}+[[w]]_{A,p(\cdot)}^{p_+}\|w\chi_{Q_0}\|_{p(\cdot)}^{p_+}\sum_{k=1}^{\infty}e^{kn(p_+-t_1p_-)}\right] ∫ Q λ − p ∞ w A ( x ) p ( x ) d x ≤ e n t 1 C ∞ + [ e − t 1 n p − + [[ w ] ] A , p ( ⋅ ) p + ∥ w χ Q 0 ∥ p ( ⋅ ) p + k = 1 ∑ ∞ e k n ( p + − t 1 p − ) ] ≤ 2 1 + C ∞ p − [ e n [ [ w ] ] A , p ( ⋅ ) ∥ w χ Q 0 ∥ p ( ⋅ ) ] C ∞ p + p − . \le2^{1+\frac{C_\infty}{p_-}}\left[e^n[[w]]_{A,p(\cdot)}\|w\chi_{Q_0}\|_{p(\cdot)}\right]^{C_\infty\frac{p_+}{p_-}}. ≤ 2 1 + p − C ∞ [ e n [[ w ] ] A , p ( ⋅ ) ∥ w χ Q 0 ∥ p ( ⋅ ) ] C ∞ p − p + . Then,
∥ w A χ Q ∥ p ( ⋅ ) = λ ≥ ρ ( w A χ Q ) 1 p ∞ C 1 ( n , p ( ⋅ ) ) [ [ w ] A , p ( ⋅ ) ∥ w χ Q 0 ∥ p ( ⋅ ) ] C ∞ p ∞ p + p − . (17) \|w_A\chi_Q\|_{p(\cdot)}=\lambda\ge\frac{\rho(w_A\chi_Q)^{\frac{1}{p_\infty}}}{C_1(n,p(\cdot))\left[[w]_{A,p(\cdot)}\|w\chi_{Q_0}\|_{p(\cdot)}\right]^{\frac{C_\infty}{p_\infty}\frac{p_+}{p_-}}}. \tag*{(17)} ∥ w A χ Q ∥ p ( ⋅ ) = λ ≥ C 1 ( n , p ( ⋅ )) [ [ w ] A , p ( ⋅ ) ∥ w χ Q 0 ∥ p ( ⋅ ) ] p ∞ C ∞ p − p + ρ ( w A χ Q ) p ∞ 1 . (17) Similarly, replacing p ∞ p_\infty p ∞ p ( x ) p(x) p ( x )
1 = ∫ Q ∥ w χ Q ∥ p ( ⋅ ) − p ( x ) w ( x ) p ( x ) d x ≤ e n s C ∞ ∫ Q ∥ w χ Q ∥ p ( ⋅ ) − p ∞ w ( x ) p ( x ) d x + ∫ Q w ( x ) p ( x ) ( e + ∣ x ∣ ) s n p − d x . 1=\int_Q\|w\chi_Q\|_{p(\cdot)}^{-p(x)}w(x)^{p(x)}\,dx\le e^{nsC_\infty}\int_Q\|w\chi_Q\|_{p(\cdot)}^{-p_\infty}w(x)^{p(x)}\,dx+\int_Q\frac{w(x)^{p(x)}}{(e+|x|)^{snp_-}}\,dx. 1 = ∫ Q ∥ w χ Q ∥ p ( ⋅ ) − p ( x ) w ( x ) p ( x ) d x ≤ e n s C ∞ ∫ Q ∥ w χ Q ∥ p ( ⋅ ) − p ∞ w ( x ) p ( x ) d x + ∫ Q ( e + ∣ x ∣ ) s n p − w ( x ) p ( x ) d x . We can split R n \mathbb{R}^n R n A Q k AQ_k A Q k Q k Q_k Q k R n = A Q 0 ∪ ⋃ k = 1 ∞ ( A Q k ∖ A Q k − 1 ) \mathbb{R}^n=AQ_0\cup\bigcup_{k=1}^{\infty}(AQ_k\setminus AQ_{k-1}) R n = A Q 0 ∪ ⋃ k = 1 ∞ ( A Q k ∖ A Q k − 1 )
ρ ( w A χ Q ) = ρ ( ( w χ A Q ) ∘ A ) = ∣ det ( A − 1 ) ∣ ρ ( w χ A Q ) . \rho(w_A\chi_Q)=\rho((w\chi_{AQ})\circ A)=|\det(A^{-1})|\rho(w\chi_{AQ}). ρ ( w A χ Q ) = ρ (( w χ A Q ) ∘ A ) = ∣ det ( A − 1 ) ∣ ρ ( w χ A Q ) . Then for all s > 0 s>0 s > 0
∫ R n w ( x ) p ( x ) ( e + ∣ x ∣ ) s n p − d x = ∫ A Q 0 w ( x ) p ( x ) ( e + ∣ x ∣ ) s n p − d x + ∑ k = 1 ∞ ∫ A Q k ∖ A Q k − 1 w ( x ) p ( x ) ( e + ∣ x ∣ ) s n p − d x ≤ e − s n p − ρ ( w χ A Q 0 ) + c A ∑ k = 1 ∞ e − k n s p − ρ ( w χ A Q k ) ≤ e − s n p − ∣ det ( A ) ∣ ρ ( w A χ Q 0 ) + ∣ det ( A ) ∣ ∑ k = 1 ∞ e − k n s p − ρ ( w A χ Q k ) . \begin{aligned}
\int_{\mathbb{R}^n}\frac{w(x)^{p(x)}}{(e+|x|)^{snp_-}}\,dx
=\int_{AQ_0}\frac{w(x)^{p(x)}}{(e+|x|)^{snp_-}}\,dx+\sum_{k=1}^{\infty}\int_{AQ_k\setminus AQ_{k-1}}\frac{w(x)^{p(x)}}{(e+|x|)^{snp_-}}\,dx \\
&\le e^{-snp_-}\rho(w\chi_{AQ_0})+c_A\sum_{k=1}^{\infty}e^{-knsp_-}\rho(w\chi_{AQ_k}) \\
&\le e^{-snp_-}|\det(A)|\rho(w_A\chi_{Q_0})+|\det(A)|\sum_{k=1}^{\infty}e^{-knsp_-}\rho(w_A\chi_{Q_k}).
\end{aligned} ∫ R n ( e + ∣ x ∣ ) s n p − w ( x ) p ( x ) d x = ∫ A Q 0 ( e + ∣ x ∣ ) s n p − w ( x ) p ( x ) d x + k = 1 ∑ ∞ ∫ A Q k ∖ A Q k − 1 ( e + ∣ x ∣ ) s n p − w ( x ) p ( x ) d x ≤ e − s n p − ρ ( w χ A Q 0 ) + c A k = 1 ∑ ∞ e − k n s p − ρ ( w χ A Q k ) ≤ e − s n p − ∣ det ( A ) ∣ ρ ( w A χ Q 0 ) + ∣ det ( A ) ∣ k = 1 ∑ ∞ e − k n s p − ρ ( w A χ Q k ) . Since ∥ w A χ Q 0 ∥ p ( ⋅ ) = 1 \|w_A\chi_{Q_0}\|_{p(\cdot)}=1 ∥ w A χ Q 0 ∥ p ( ⋅ ) = 1 ∥ w A χ Q k ∥ p ( ⋅ ) ≥ 1 \|w_A\chi_{Q_k}\|_{p(\cdot)}\ge1 ∥ w A χ Q k ∥ p ( ⋅ ) ≥ 1 ρ ( w A χ Q k ) ≤ ∥ w A χ Q k ∥ p ( ⋅ ) p + \rho(w_A\chi_{Q_k})\le\|w_A\chi_{Q_k}\|_{p(\cdot)}^{p_+} ρ ( w A χ Q k ) ≤ ∥ w A χ Q k ∥ p ( ⋅ ) p +
∫ R n w ( x ) p ( x ) ( e + ∣ x ∣ ) s n p − d x ≤ e − s n p − ∣ det ( A ) ∣ + ∣ det ( A ) ∣ ∑ k = 1 ∞ e − k n s p − ∥ w A χ Q k ∥ p ( ⋅ ) p + . \int_{\mathbb{R}^n}\frac{w(x)^{p(x)}}{(e+|x|)^{snp_-}}\,dx\le e^{-snp_-}|\det(A)|+|\det(A)|\sum_{k=1}^{\infty}e^{-knsp_-}\|w_A\chi_{Q_k}\|_{p(\cdot)}^{p_+}. ∫ R n ( e + ∣ x ∣ ) s n p − w ( x ) p ( x ) d x ≤ e − s n p − ∣ det ( A ) ∣ + ∣ det ( A ) ∣ k = 1 ∑ ∞ e − k n s p − ∥ w A χ Q k ∥ p ( ⋅ ) p + . As in the above estimate, we can take s 1 > p + p − s_1>\frac{p_+}{p_-} s 1 > p − p +
∫ R n w ( x ) p ( x ) ( e + ∣ x ∣ ) s 1 n p − d x < 1 2 \int_{\mathbb{R}^n}\frac{w(x)^{p(x)}}{(e+|x|)^{s_1np_-}}\,dx<\frac{1}{2} ∫ R n ( e + ∣ x ∣ ) s 1 n p − w ( x ) p ( x ) d x < 2 1 and
e s 1 n C ∞ = e n C ∞ p + p − − 5 C ∞ p − [ ∣ det ( A ) ∣ K p ( ⋅ ) [ w ] A , p ( ⋅ ) ] C ∞ p + p − . e^{s_1nC_\infty}=e^{nC_\infty\frac{p_+}{p_-}-5\frac{C_\infty}{p_-}}\left[|\det(A)|K_{p(\cdot)}[w]_{A,p(\cdot)}\right]^{C_\infty\frac{p_+}{p_-}}. e s 1 n C ∞ = e n C ∞ p − p + − 5 p − C ∞ [ ∣ det ( A ) ∣ K p ( ⋅ ) [ w ] A , p ( ⋅ ) ] C ∞ p − p + . Then, we have
1 2 ≤ e n s 1 C ∞ ∫ Q ∥ w χ Q ∥ p ( ⋅ ) − p ∞ w ( x ) p ( x ) d x = e n s 1 C ∞ ρ ( w χ Q ) ∥ w χ Q ∥ p ( ⋅ ) p ∞ \frac{1}{2}\le e^{ns_1C_\infty}\int_Q\|w\chi_Q\|_{p(\cdot)}^{-p_\infty}w(x)^{p(x)}\,dx=e^{ns_1C_\infty}\frac{\rho(w\chi_Q)}{\|w\chi_Q\|_{p(\cdot)}^{p_\infty}} 2 1 ≤ e n s 1 C ∞ ∫ Q ∥ w χ Q ∥ p ( ⋅ ) − p ∞ w ( x ) p ( x ) d x = e n s 1 C ∞ ∥ w χ Q ∥ p ( ⋅ ) p ∞ ρ ( w χ Q ) and that imply
∥ w χ Q ∥ p ( ⋅ ) ≤ C 2 ( n , p ( ⋅ ) , A ) [ [ w ] ] A , p ( ⋅ ) C ∞ p ∞ p + p − ρ ( w χ Q ) 1 p ∞ . (18) \|w\chi_Q\|_{p(\cdot)}\le C_2(n,p(\cdot),A)[[w]]_{A,p(\cdot)}^{\frac{C_\infty}{p_\infty}\frac{p_+}{p_-}}\rho(w\chi_Q)^{\frac{1}{p_\infty}}. \tag*{(18)} ∥ w χ Q ∥ p ( ⋅ ) ≤ C 2 ( n , p ( ⋅ ) , A ) [[ w ] ] A , p ( ⋅ ) p ∞ C ∞ p − p + ρ ( w χ Q ) p ∞ 1 . (18) Then, taking account the estimates (5.1) and (5.2) we have
1 = ∣ Q ∣ ∣ Q ∣ ≤ K p ( ⋅ ) [ w ] A A , p ( ⋅ ) ∥ w χ Q ∥ p ( ⋅ ) ∥ w A χ Q ∥ p ( ⋅ ) ≤ C ( n , p ( ⋅ ) , A ) [ [ w ] ] A A , p ( ⋅ ) 2 C ∞ p + p − + 1 ( ρ ( w χ Q ) ρ ( w A χ Q ) ) 1 p ∞ . 1=\frac{|Q|}{|Q|}\leq K_{p(\cdot)}[w]_{A_{A,p(\cdot)}}\frac{\lVert w\chi_Q\rVert_{p(\cdot)}}{\lVert w_A\chi_Q\rVert_{p(\cdot)}}\leq C(n,p(\cdot),A)[[w]]_{A_{A,p(\cdot)}}^{2C_\infty\frac{p_+}{p_-}+1}\left(\frac{\rho(w\chi_Q)}{\rho(w_A\chi_Q)}\right)^{\frac{1}{p_\infty}}. 1 = ∣ Q ∣ ∣ Q ∣ ≤ K p ( ⋅ ) [ w ] A A , p ( ⋅ ) ∥ w A χ Q ∥ p ( ⋅ ) ∥ w χ Q ∥ p ( ⋅ ) ≤ C ( n , p ( ⋅ ) , A ) [[ w ] ] A A , p ( ⋅ ) 2 C ∞ p − p + + 1 ( ρ ( w A χ Q ) ρ ( w χ Q ) ) p ∞ 1 . Summary the four cases, if ∥ w A χ Q 0 ∥ p ( ⋅ ) = 1 \lVert w_A\chi_{Q_0}\rVert_{p(\cdot)}=1 ∥ w A χ Q 0 ∥ p ( ⋅ ) = 1
ρ ( w A χ Q ) ≤ C ( A , n , p ( ⋅ ) ) [ [ w ] ] A A , p ( ⋅ ) max { 2 p + − p − , 2 C ∞ p + p − + p ∞ } ρ ( w χ Q ) . \rho(w_A\chi_Q)\leq C(A,n,p(\cdot))[[w]]_{A_{A,p(\cdot)}}^{\max\{2p_+-p_-,2C_\infty\frac{p_+}{p_-}+p_\infty\}}\rho(w\chi_Q). ρ ( w A χ Q ) ≤ C ( A , n , p ( ⋅ )) [[ w ] ] A A , p ( ⋅ ) m a x { 2 p + − p − , 2 C ∞ p − p + + p ∞ } ρ ( w χ Q ) . where and C ( A , n , p ( ⋅ ) ) C(A,n,p(\cdot)) C ( A , n , p ( ⋅ )) w w w ∥ w A χ Q 0 ∥ p ( ⋅ ) ≠ 1 \lVert w_A\chi_{Q_0}\rVert_{p(\cdot)}\neq1 ∥ w A χ Q 0 ∥ p ( ⋅ ) = 1 u = w ∥ w A χ Q 0 ∥ p ( ⋅ ) u=\frac{w}{\lVert w_A\chi_{Q_0}\rVert_{p(\cdot)}} u = ∥ w A χ Q 0 ∥ p ( ⋅ ) w ∥ w A χ Q 0 ∥ p ( ⋅ ) > 1 \lVert w_A\chi_{Q_0}\rVert_{p(\cdot)}>1 ∥ w A χ Q 0 ∥ p ( ⋅ ) > 1
∥ w A χ Q 0 ∥ p ( ⋅ ) − p − ρ ( w A χ Q ) ≤ ρ ( u A χ Q ) ≤ C ( A , n , p ( ⋅ ) ) [ [ w ] ] A A , p ( ⋅ ) max { 2 p + − p − , 2 C ∞ p + p − + p ∞ } ρ ( u χ Q ) ≤ C ( A , n , p ( ⋅ ) ) [ [ w ] ] A A , p ( ⋅ ) max { 2 p + − p − , 2 C ∞ p + p − + p ∞ } ρ ( w χ Q ) ∥ w A χ Q 0 ∥ p ( ⋅ ) − p + . \begin{aligned}
\lVert w_A\chi_{Q_0}\rVert_{p(\cdot)}^{-p_-}\rho(w_A\chi_Q)\leq\rho(u_A\chi_Q) \\
&\leq C(A,n,p(\cdot))[[w]]_{A_{A,p(\cdot)}}^{\max\{2p_+-p_-,2C_\infty\frac{p_+}{p_-}+p_\infty\}}\rho(u\chi_Q) \\
&\leq C(A,n,p(\cdot))[[w]]_{A_{A,p(\cdot)}}^{\max\{2p_+-p_-,2C_\infty\frac{p_+}{p_-}+p_\infty\}}\rho(w\chi_Q)\lVert w_A\chi_{Q_0}\rVert_{p(\cdot)}^{-p_+}.
\end{aligned} ∥ w A χ Q 0 ∥ p ( ⋅ ) − p − ρ ( w A χ Q ) ≤ ρ ( u A χ Q ) ≤ C ( A , n , p ( ⋅ )) [[ w ] ] A A , p ( ⋅ ) m a x { 2 p + − p − , 2 C ∞ p − p + + p ∞ } ρ ( u χ Q ) ≤ C ( A , n , p ( ⋅ )) [[ w ] ] A A , p ( ⋅ ) m a x { 2 p + − p − , 2 C ∞ p − p + + p ∞ } ρ ( w χ Q ) ∥ w A χ Q 0 ∥ p ( ⋅ ) − p + . Then
ρ ( w A χ Q ) ≤ C ( A , n , p ( ⋅ ) ) [ [ w ] ] A A , p ( ⋅ ) max { 2 p + − p − , 2 C ∞ p + p − + p ∞ } ρ ( w χ Q ) ∥ w A χ Q 0 ∥ p ( ⋅ ) p − − p + . \rho(w_A\chi_Q)\leq C(A,n,p(\cdot))[[w]]_{A_{A,p(\cdot)}}^{\max\{2p_+-p_-,2C_\infty\frac{p_+}{p_-}+p_\infty\}}\rho(w\chi_Q)\lVert w_A\chi_{Q_0}\rVert_{p(\cdot)}^{p_--p_+}. ρ ( w A χ Q ) ≤ C ( A , n , p ( ⋅ )) [[ w ] ] A A , p ( ⋅ ) m a x { 2 p + − p − , 2 C ∞ p − p + + p ∞ } ρ ( w χ Q ) ∥ w A χ Q 0 ∥ p ( ⋅ ) p − − p + . Similarly, if ∥ w A χ Q 0 ∥ p ( ⋅ ) > 1 \lVert w_A\chi_{Q_0}\rVert_{p(\cdot)}>1 ∥ w A χ Q 0 ∥ p ( ⋅ ) > 1
ρ ( w A χ Q ) ≤ C ( A , n , p ( ⋅ ) ) [ [ w ] ] A A , p ( ⋅ ) max { 2 p + − p − , 2 C ∞ p + p − + p ∞ } ρ ( w χ Q ) ∥ w A χ Q 0 ∥ p ( ⋅ ) p + − p − . \rho(w_A\chi_Q)\leq C(A,n,p(\cdot))[[w]]_{A_{A,p(\cdot)}}^{\max\{2p_+-p_-,2C_\infty\frac{p_+}{p_-}+p_\infty\}}\rho(w\chi_Q)\lVert w_A\chi_{Q_0}\rVert_{p(\cdot)}^{p_+-p_-}. ρ ( w A χ Q ) ≤ C ( A , n , p ( ⋅ )) [[ w ] ] A A , p ( ⋅ ) m a x { 2 p + − p − , 2 C ∞ p − p + + p ∞ } ρ ( w χ Q ) ∥ w A χ Q 0 ∥ p ( ⋅ ) p + − p − . Theorem 5.5. Let p ∈ P log p\in\mathcal{P}^{\log} p ∈ P l o g p + < ∞ p_+<\infty p + < ∞ p ( A x ) = p ( x ) p(Ax)=p(x) p ( A x ) = p ( x ) x x x w ∈ A A , p ( ⋅ ) w\in A_{A,p(\cdot)} w ∈ A A , p ( ⋅ )
w A ( x ) ≤ L 3 [ [ w ] ] A A , p ( ⋅ ) 1 p − max { 2 p + − p − , 2 C ∞ p + p − + p ∞ } w ( x ) w_A(x)\leq L_3[[w]]_{A_{A,p(\cdot)}}^{\frac{1}{p_-}\max\{2p_+-p_-,2C_\infty\frac{p_+}{p_-}+p_\infty\}}w(x) w A ( x ) ≤ L 3 [[ w ] ] A A , p ( ⋅ ) p − 1 m a x { 2 p + − p − , 2 C ∞ p − p + + p ∞ } w ( x ) where
L 3 = C ( A , n , p ( ⋅ ) ) max { ∥ w A χ Q 0 ∥ p ( ⋅ ) 1 − p + p − , ∥ w A χ Q 0 ∥ p ( ⋅ ) p + p − − 1 } L_3=C(A,n,p(\cdot))\max\left\{\lVert w_A\chi_{Q_0}\rVert_{p(\cdot)}^{1-\frac{p_+}{p_-}},\lVert w_A\chi_{Q_0}\rVert_{p(\cdot)}^{\frac{p_+}{p_-}-1}\right\} L 3 = C ( A , n , p ( ⋅ )) max { ∥ w A χ Q 0 ∥ p ( ⋅ ) 1 − p − p + , ∥ w A χ Q 0 ∥ p ( ⋅ ) p − p + − 1 } and Q 0 = Q ( 0 , 2 e ) Q_0=Q(0,2e) Q 0 = Q ( 0 , 2 e )
Observe that the exponent max { 2 p + − p − , 2 C ∞ p + p − + p ∞ } 1 p − ≥ 1 \max\left\{2p_+-p_-,2C_\infty\frac{p_+}{p_-}+p_\infty\right\}\frac{1}{p_-}\geq1 max { 2 p + − p − , 2 C ∞ p − p + + p ∞ } p − 1 ≥ 1 p ( ⋅ ) p(\cdot) p ( ⋅ )
Proof. Let x ∈ R n x\in\mathbb{R}^n x ∈ R n Q Q Q x x x ∣ Q ∣ |Q| ∣ Q ∣
ρ ( w A χ Q ) ∣ Q ∣ ≤ L 2 [ [ w ] ] A A , p ( ⋅ ) max { 2 p + − p − , 2 C ∞ p + p − + p ∞ } ρ ( w χ Q ) ∣ Q ∣ . \frac{\rho(w_A\chi_Q)}{|Q|}\leq L_2[[w]]_{A_{A,p(\cdot)}}^{\max\{2p_+-p_-,2C_\infty\frac{p_+}{p_-}+p_\infty\}}\frac{\rho(w\chi_Q)}{|Q|}. ∣ Q ∣ ρ ( w A χ Q ) ≤ L 2 [[ w ] ] A A , p ( ⋅ ) m a x { 2 p + − p − , 2 C ∞ p − p + + p ∞ } ∣ Q ∣ ρ ( w χ Q ) . Since w w w w , w A w,w_A w , w A
w A ( x ) p ( x ) ≤ L 2 [ [ w ] ] A A , p ( ⋅ ) max { 2 p + − p − , 2 C ∞ p + p − + p ∞ } w ( x ) p ( x ) w_A(x)^{p(x)}\leq L_2[[w]]_{A_{A,p(\cdot)}}^{\max\{2p_+-p_-,2C_\infty\frac{p_+}{p_-}+p_\infty\}}w(x)^{p(x)} w A ( x ) p ( x ) ≤ L 2 [[ w ] ] A A , p ( ⋅ ) m a x { 2 p + − p − , 2 C ∞ p − p + + p ∞ } w ( x ) p ( x ) Then,
w A ( x ) ≤ L 2 1 p ( x ) [ [ w ] ] A A , p ( ⋅ ) max { 2 p + − p − , 2 C ∞ p + p − + p + } w ( x ) ≤ L 3 [ [ w ] ] A A , p ( ⋅ ) 1 p − max { 2 p + − p − , 2 C ∞ p + p − + p + } w ( x ) . w_A(x) \le L_2^{\frac{1}{p(x)}} [[w]]_{\mathcal{A}_{A,p(\cdot)}}^{\max\{2p_+ - p_-,2C_\infty\frac{p_+}{p_-+p_+}\}} w(x)
\le L_3 [[w]]_{\mathcal{A}_{A,p(\cdot)}}^{\frac{1}{p_-}} \max\{2p_+ - p_-,2C_\infty\frac{p_+}{p_-+p_+}\} w(x). w A ( x ) ≤ L 2 p ( x ) 1 [[ w ] ] A A , p ( ⋅ ) m a x { 2 p + − p − , 2 C ∞ p − + p + p + } w ( x ) ≤ L 3 [[ w ] ] A A , p ( ⋅ ) p − 1 max { 2 p + − p − , 2 C ∞ p − + p + p + } w ( x ) . Observe that in the case of matrices A A A A k = I A^k=I A k = I k ∈ N k\in\mathbb{N} k ∈ N p ( A x ) = p ( x ) p(Ax)=p(x) p ( A x ) = p ( x ) x x x Corollary 4.4 imply that
A A , p ( ⋅ ) = { w ∈ A p ( ⋅ ) : w ( A x ) ≲ w ( x ) a . e . x ∈ R n } . \mathcal{A}_{A,p(\cdot)}=\{w\in\mathcal{A}_{p(\cdot)}:w(Ax)\lesssim w(x)\ {\rm a.e.}\ x\in\mathbb{R}^n\}. A A , p ( ⋅ ) = { w ∈ A p ( ⋅ ) : w ( A x ) ≲ w ( x ) a.e. x ∈ R n } . In this case, it can be proved that M A − 1 M_{A^{-1}} M A − 1 L p ( ⋅ ) ( w ) L^{p(\cdot)}(w) L p ( ⋅ ) ( w ) Theorem 5.5 and the boundedness on L p ( ⋅ ) ( w ) L^{p(\cdot)}(w) L p ( ⋅ ) ( w ) M M M
Examples
In this final section, we establish a lemma that provides a convenient method for constructing weights in the class A p ( ⋅ ) \mathcal{A}_{p(\cdot)} A p ( ⋅ ) [2 ] . For completeness, we include the proof.
Proposition 6.1. Let p ( ⋅ ) ∈ P log p(\cdot)\in\mathcal{P}^{\log} p ( ⋅ ) ∈ P l o g 1 < p − ≤ p + < ∞ 1<p_-\le p_+<\infty 1 < p − ≤ p + < ∞ p ∞ p_\infty p ∞ α , β ∈ R \alpha,\beta\in\mathbb{R} α , β ∈ R
− n p ( 0 ) < α < n p ′ ( 0 ) , − n p ∞ < β < n p ∞ ′ , -\frac{n}{p(0)}<\alpha<\frac{n}{p'(0)},\qquad-\frac{n}{p_\infty}<\beta<\frac{n}{p'_\infty}, − p ( 0 ) n < α < p ′ ( 0 ) n , − p ∞ n < β < p ∞ ′ n , where p ∞ ′ = p ∞ / ( p ∞ − 1 ) p'_\infty=p_\infty/(p_\infty-1) p ∞ ′ = p ∞ / ( p ∞ − 1 )
w ( x ) = ∣ x ∣ α ( 1 + ∣ x ∣ ) β − α , x ∈ R n ∖ { 0 } , w(x)=|x|^\alpha(1+|x|)^{\beta-\alpha},\qquad x\in\mathbb{R}^n\setminus\{0\}, w ( x ) = ∣ x ∣ α ( 1 + ∣ x ∣ ) β − α , x ∈ R n ∖ { 0 } , belongs to A p ( ⋅ ) \mathcal{A}_{p(\cdot)} A p ( ⋅ ) w w w
Proof. In [2 ] the authors shows that if σ ( x ) = w ( x ) p ( x ) \sigma(x)=w(x)^{p(x)} σ ( x ) = w ( x ) p ( x ) A p ( ⋅ ) ∗ \mathcal{A}^{*}_{p(\cdot)} A p ( ⋅ ) ∗ w ∈ A p ( ⋅ ) w\in\mathcal{A}_{p(\cdot)} w ∈ A p ( ⋅ ) σ ∈ A p ( ⋅ ) ∗ \sigma\in\mathcal{A}^{*}_{p(\cdot)} σ ∈ A p ( ⋅ ) ∗
for every x 0 x_0 x 0 Q x 0 Q_{x_0} Q x 0 x 0 x_0 x 0 σ ∈ A p ( x 0 ) \sigma\in\mathcal{A}_{p(x_0)} σ ∈ A p ( x 0 ) Q x 0 Q_{x_0} Q x 0
σ ∈ A p ∞ \sigma\in\mathcal{A}_{p_\infty} σ ∈ A p ∞ R ∖ Q ( 0 , N ) \mathbb{R}\setminus Q(0,N) R ∖ Q ( 0 , N ) N > 0 N>0 N > 0
Let σ ( x ) = w ( x ) p ( x ) \sigma(x)=w(x)^{p(x)} σ ( x ) = w ( x ) p ( x ) 0 < ∣ x ∣ < 1 2 0<|x|<\frac{1}{2} 0 < ∣ x ∣ < 2 1
σ ( x ) ∣ x ∣ α p ( 0 ) = ∣ x ∣ α ( p ( x ) − p ( 0 ) ) ( 1 + ∣ x ∣ ) ( β − α ) p ( x ) . \frac{\sigma(x)}{|x|^{\alpha p(0)}}=|x|^{\alpha(p(x)-p(0))}(1+|x|)^{(\beta-\alpha)p(x)}. ∣ x ∣ α p ( 0 ) σ ( x ) = ∣ x ∣ α ( p ( x ) − p ( 0 )) ( 1 + ∣ x ∣ ) ( β − α ) p ( x ) . Since p ( ⋅ ) ∈ L H 0 p(\cdot)\in LH_0 p ( ⋅ ) ∈ L H 0
∣ p ( x ) − p ( 0 ) ∣ ∣ log ∣ x ∣ ∣ ≤ C 0 , 0 < ∣ x ∣ < 1 2 . |p(x)-p(0)||\log|x||\le C_0,\qquad0<|x|<\frac{1}{2}. ∣ p ( x ) − p ( 0 ) ∣∣ log ∣ x ∣∣ ≤ C 0 , 0 < ∣ x ∣ < 2 1 . Consequently,
∣ x ∣ α ( p ( x ) − p ( 0 ) ) ≈ 1. |x|^{\alpha(p(x)-p(0))} \approx1. ∣ x ∣ α ( p ( x ) − p ( 0 )) ≈ 1. The second factor is also bounded above and below by positive constants, because 1 ≤ 1 + ∣ x ∣ ≤ 2 1 \le1 + |x| \le2 1 ≤ 1 + ∣ x ∣ ≤ 2 p + < ∞ p_{+} < \infty p + < ∞
σ ( x ) ≈ ∣ x ∣ α p ( 0 ) , 0 < ∣ x ∣ < 1 2 . \sigma(x) \approx|x|^{\alpha p(0)}, \qquad0 < |x| < \frac{1}{2}. σ ( x ) ≈ ∣ x ∣ α p ( 0 ) , 0 < ∣ x ∣ < 2 1 . The assumptions on α \alpha α
− n < α p ( 0 ) < n ( p ( 0 ) − 1 ) . -n < \alpha p(0) < n(p(0)-1). − n < α p ( 0 ) < n ( p ( 0 ) − 1 ) . By the classical characterization of power weights, ∣ x ∣ α p ( 0 ) ∈ A p ( 0 ) |x|^{\alpha p(0)} \in A_{p(0)} ∣ x ∣ α p ( 0 ) ∈ A p ( 0 ) σ \sigma σ A p ( 0 ) A_{p(0)} A p ( 0 )
At every point x 0 ≠ 0 x_{0} \ne0 x 0 = 0 σ \sigma σ x 0 x_{0} x 0 σ \sigma σ A p ( x 0 ) A_{p(x_{0})} A p ( x 0 )
Next, for ∣ x ∣ ≥ 1 |x| \ge1 ∣ x ∣ ≥ 1
σ ( x ) ∣ x ∣ β p ∞ = ∣ x ∣ β ( p ( x ) − p ∞ ) ( 1 + 1 ∣ x ∣ ) ( β − α ) p ( x ) . \frac{\sigma(x)}{|x|^{\beta p_{\infty}}}
= |x|^{\beta(p(x)-p_{\infty})}
\left(1+\frac{1}{|x|}\right)^{(\beta-\alpha)p(x)}. ∣ x ∣ β p ∞ σ ( x ) = ∣ x ∣ β ( p ( x ) − p ∞ ) ( 1 + ∣ x ∣ 1 ) ( β − α ) p ( x ) . The condition p ( ⋅ ) ∈ L H ∞ p(\cdot) \in LH_{\infty} p ( ⋅ ) ∈ L H ∞
∣ p ( x ) − p ∞ ∣ log ( ∣ x ∣ ) ≤ C ∞ log ( ∣ x ∣ ) log ( e + ∣ x ∣ ) ≤ C ∞ , ∣ x ∣ ≥ 1. |p(x)-p_{\infty}|\log(|x|) \le C_{\infty}\frac{\log(|x|)}{\log(e+|x|)} \le C_{\infty}, \qquad|x| \ge1. ∣ p ( x ) − p ∞ ∣ log ( ∣ x ∣ ) ≤ C ∞ log ( e + ∣ x ∣ ) log ( ∣ x ∣ ) ≤ C ∞ , ∣ x ∣ ≥ 1. It follows that
σ ( x ) ≈ ∣ x ∣ β p ∞ , ∣ x ∣ ≥ 1. \sigma(x) \approx|x|^{\beta p_{\infty}}, \qquad|x| \ge1. σ ( x ) ≈ ∣ x ∣ β p ∞ , ∣ x ∣ ≥ 1. Moreover, the assumptions on β \beta β
− n < β p ∞ < n ( p ∞ − 1 ) . -n < \beta p_{\infty} < n(p_{\infty}-1). − n < β p ∞ < n ( p ∞ − 1 ) . Thus ∣ x ∣ β p ∞ ∈ A p ∞ |x|^{\beta p_{\infty}} \in A_{p_{\infty}} ∣ x ∣ β p ∞ ∈ A p ∞ σ \sigma σ A p ∞ A_{p_{\infty}} A p ∞
Hence σ ∈ A p ( ⋅ ) ∗ \sigma\in A^{*}_{p(\cdot)} σ ∈ A p ( ⋅ ) ∗ [2 Theorem 1.2] , the Hardy–Littlewood maximal operator is bounded on the variable Lebesgue space L p ( ⋅ ) ( R n , σ ) L^{p(\cdot)}(\mathbb{R}^{n},\sigma) L p ( ⋅ ) ( R n , σ ) σ \sigma σ L w p ( ⋅ ) L^{p(\cdot)}_{w} L w p ( ⋅ ) [2 Theorem 1.3] , w ∈ A p ( ⋅ ) w \in A_{p(\cdot)} w ∈ A p ( ⋅ )
Example 6.2. Let x ∈ R 2 x \in\mathbb{R}^{2} x ∈ R 2
A = ( 0 − 1 1 0 ) , p ( x ) = 2 + e − ∣ x ∣ 2 , A=\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}, \qquad p(x)=2+e^{-|x|^{2}}, A = ( 0 1 − 1 0 ) , p ( x ) = 2 + e − ∣ x ∣ 2 , and consider the weight
w ( x ) = ∣ x ∣ ( 1 + ∣ x ∣ ) − 3 / 2 , w ( 0 ) = 1. w(x)=|x|(1+|x|)^{-3/2}, \qquad w(0)=1. w ( x ) = ∣ x ∣ ( 1 + ∣ x ∣ ) − 3/2 , w ( 0 ) = 1. The matrix A A A A 4 = I A^{4}=I A 4 = I
p ( A x ) = p ( x ) , w ( A x ) = w ( x ) . p(Ax)=p(x), \qquad w(Ax)=w(x). p ( A x ) = p ( x ) , w ( A x ) = w ( x ) . We first verify the regularity assumptions on the exponent. Notice that
p − = 2 , p + = 3 , p ( 0 ) = 3 , p ∞ = 2. p_{-}=2, \qquad p_{+}=3, \qquad p(0)=3, \qquad p_{\infty}=2. p − = 2 , p + = 3 , p ( 0 ) = 3 , p ∞ = 2. Moreover p ( ⋅ ) p(\cdot) p ( ⋅ )
∣ p ( x ) − p ( y ) ∣ ≤ 2 e ∣ x − y ∣ . |p(x)-p(y)| \le\sqrt{\frac{2}{e}}|x-y|. ∣ p ( x ) − p ( y ) ∣ ≤ e 2 ∣ x − y ∣. Thus p ( ⋅ ) ∈ L H 0 p(\cdot) \in LH_0 p ( ⋅ ) ∈ L H 0 x ∈ R 2 x \in\mathbb{R}^{2} x ∈ R 2
∣ p ( x ) − p ∞ ∣ = e − ∣ x ∣ 2 ≤ C log ( e + ∣ x ∣ ) . |p(x)-p_{\infty}| = e^{-|x|^{2}} \le\frac{C}{\log(e+|x|)}. ∣ p ( x ) − p ∞ ∣ = e − ∣ x ∣ 2 ≤ log ( e + ∣ x ∣ ) C . Therefore, p ∈ L H ∞ p \in LH_{\infty} p ∈ L H ∞
By Proposition 6.1 the radial weight
v α , β ( x ) = ∣ x ∣ α ( 1 + ∣ x ∣ ) β − α v_{\alpha,\beta}(x) = |x|^{\alpha}(1+|x|)^{\beta-\alpha} v α , β ( x ) = ∣ x ∣ α ( 1 + ∣ x ∣ ) β − α belongs to the weight class A p ( ⋅ ) A_{p(\cdot)} A p ( ⋅ )
− n p ( 0 ) < α < n p ′ ( 0 ) , − n p ∞ < β < n p ∞ ′ . -\frac{n}{p(0)} < \alpha< \frac{n}{p'(0)}, \qquad-\frac{n}{p_{\infty}} < \beta< \frac{n}{p'_{\infty}}. − p ( 0 ) n < α < p ′ ( 0 ) n , − p ∞ n < β < p ∞ ′ n . In the present example, n = 2 n=2 n = 2 α = 1 \alpha=1 α = 1 β = − 1 / 2 \beta=-1/2 β = − 1/2
− 2 3 < 1 < 4 3 , − 1 < − 1 2 < 1. -\frac{2}{3} < 1 < \frac{4}{3}, \qquad-1 < -\frac{1}{2} < 1. − 3 2 < 1 < 3 4 , − 1 < − 2 1 < 1. Hence w ∈ A p ( ⋅ ) w \in A_{p(\cdot)} w ∈ A p ( ⋅ )
Finally, since w A = w w_A=w w A = w [ w ] A A , p ( ⋅ ) = [ w ] A p ( ⋅ ) < ∞ [w]_{A_{A,p(\cdot)}}=[w]_{A_{p(\cdot)}}<\infty [ w ] A A , p ( ⋅ ) = [ w ] A p ( ⋅ ) < ∞ w ∈ A A , p ( ⋅ ) w \in A_{A,p(\cdot)} w ∈ A A , p ( ⋅ )
AI Disclosure
ChatGPT was used for language editing and proofreading. All mathematical results, arguments, proofs, and conclusions were developed and proved by the authors.
(G. Ibáñez Firnkorn) Instituto de Matemática (INMABB), Departamento de Matemática, Universidad Nacional del Sur (UNS)-CONICET, Bahía Blanca, Argentina
Email address: gonzalo.ibanez@uns.edu.ar
(L. A. Vallejos) FaMAF, Universidad Nacional de Córdoba, CIEM (CONICET), 5000 Córdoba, Argentina
Email address: lucas.vallejos@unc.edu.ar
