Introduction
We study the extent to which the uncentered Hardy–Littlewood maximal operator decreases derivative seminorms on the real line. Our main result determines the optimal constant, which is characterized by a scalar equation, in the L p L^p L p 1 < p < ∞ 1 < p < \infty 1 < p < ∞ [2 ] with the sharp Lipschitz inequality of Aldaz, Colzani, and Pérez Lázaro [1 ] .
Background and notation
For f ∈ L l o c 1 ( R ) f \in L^1_{\mathrm{loc}}(\mathbb{R}) f ∈ L loc 1 ( R )
M ~ f ( x ) : = sup r , s ≥ 0 r + s > 0 1 r + s ∫ x − r x + s ∣ f ( t ) ∣ d t . (1) \widetilde{M}f(x) := \sup_{\substack{r,s \ge0 \\ r+s>0}} \frac{1}{r+s}\int_{x-r}^{x+s} |f(t)|\,dt. \tag*{(1)} M f ( x ) := r , s ≥ 0 r + s > 0 sup r + s 1 ∫ x − r x + s ∣ f ( t ) ∣ d t . (1) The supremum is over all finite intervals containing x x x R \mathbb{R} R
Kinnunen [12 ] initiated the Sobolev regularity theory of maximal functions by proving boundedness on W 1 , p ( R d ) W^{1,p}(\mathbb{R}^d) W 1 , p ( R d ) 1 < p ≤ ∞ 1 < p \le\infty 1 < p ≤ ∞
∣ ( M ~ f ) ′ ( x ) ∣ ≤ M ~ ( f ′ ) ( x ) for almost every x ∈ R , f ∈ W 1 , p ( R ) . |(\widetilde{M}f)'(x)| \le\widetilde{M}(f')(x) \quad\text{for almost every } x \in\mathbb{R}, \qquad f \in W^{1,p}(\mathbb{R}). ∣ ( M f ) ′ ( x ) ∣ ≤ M ( f ′ ) ( x ) for almost every x ∈ R , f ∈ W 1 , p ( R ) . Here M ~ ( f ′ ) = M ~ ( ∣ f ′ ∣ ) \widetilde{M}(f') = \widetilde{M}(|f'|) M ( f ′ ) = M ( ∣ f ′ ∣ ) L p L^p L p
∥ ( M ~ f ) ′ ∥ p ≤ ∥ M ~ ( f ′ ) ∥ p ≤ C p ∥ f ′ ∥ p . (2) \|(\widetilde{M}f)'\|_p \le\|\widetilde{M}(f')\|_p \le C_p\|f'\|_p. \tag*{(2)} ∥ ( M f ) ′ ∥ p ≤ ∥ M ( f ′ ) ∥ p ≤ C p ∥ f ′ ∥ p . (2) where C p = ∥ M ~ ∥ L p ( R ) → L p ( R ) C_p = \|\widetilde{M}\|_{L^p(\mathbb{R})\to L^p(\mathbb{R})} C p = ∥ M ∥ L p ( R ) → L p ( R ) [10 ] determined this operator norm C p C_p C p
( p − 1 ) x p − p x p − 1 − 1 = 0. (3) (p-1)x^p-px^{p-1}-1=0. \tag*{(3)} ( p − 1 ) x p − p x p − 1 − 1 = 0. (3) In particular, ( p − 1 ) x = p + x 1 − p (p-1)x = p+x^{1-p} ( p − 1 ) x = p + x 1 − p C p > p / ( p − 1 ) C_p > p/(p-1) C p > p / ( p − 1 ) C p → ∞ C_p \to\infty C p → ∞ p ↓ 1 p \downarrow1 p ↓ 1 f ↦ ( M ~ f ) ′ f \mapsto(\widetilde{M}f)' f ↦ ( M f ) ′ W 1 , p ( R ) → L p ( R ) W^{1,p}(\mathbb{R}) \to L^p(\mathbb{R}) W 1 , p ( R ) → L p ( R ) 1 ≤ p < ∞ 1 \le p < \infty 1 ≤ p < ∞ [6 , 14 ] .
2020 Mathematics Subject Classification. Primary 42B25; Secondary 26A45, 46E35.
Key words and phrases. Uncentered maximal operator, sharp derivative inequality, Riesz variation, Sobolev spaces, bounded variation.
Nevertheless, the bound (1.2) provides only an upper bound and does not capture the derivative’s sharp behavior. In particular, as p ↓ 1 p \downarrow1 p ↓ 1 C p → ∞ C_p \to\infty C p → ∞ [2 ] proved that the maximal function of a bounded-variation function is absolutely continuous and has no greater total variation. In particular, for f ∈ W 1 , 1 ( R ) f \in W^{1,1}(\mathbb{R}) f ∈ W 1 , 1 ( R )
∥ ( M ~ f ) ′ ∥ 1 ≤ ∥ f ′ ∥ 1 . (4) \left\|(\widetilde{M}f)'\right\|_1 \le\left\|f'\right\|_1 . \tag*{(4)} ( M f ) ′ 1 ≤ ∥ f ′ ∥ 1 . (4) Thus, the classical L p L^p L p p ↓ 1 p \downarrow1 p ↓ 1 p = 1 p=1 p = 1
Theorem 1 provides a better bound than the C p C_p C p C p > 1 C_p > 1 C p > 1 γ p < 1 \gamma_p < 1 γ p < 1 p > 1 p > 1 p > 1 γ p → 1 \gamma_p \to1 γ p → 1 p ↓ 1 p \downarrow1 p ↓ 1 p = ∞ p=\infty p = ∞ [1 ] proved the sharp Lipschitz inequality with constant 2 − 1 \sqrt{2}-1 2 − 1
Grafakos and Kinnunen [9 Theorem 2.1 and Corollary 2.3] extended the sharp L p L^p L p ( 1 , 1 ) (1,1) ( 1 , 1 ) μ \mu μ R \mathbb{R} R μ \mu μ
As for the discrete setting, for a function f : Z → R f:\mathbb{Z}\to\mathbb{R} f : Z → R
M ~ Z f ( n ) : = sup r , s ≥ 0 1 r + s + 1 ∑ i = − r s ∣ f ( n + i ) ∣ , f ′ ( n ) : = f ( n + 1 ) − f ( n ) . \widetilde{M}_{\mathbb{Z}}f(n) := \sup_{\substack{r,s\ge0}} \frac{1}{r+s+1}\sum_{i=-r}^{s} \lvert f(n+i)\rvert,\qquad f'(n):=f(n+1)-f(n). M Z f ( n ) := r , s ≥ 0 sup r + s + 1 1 i = − r ∑ s ∣ f ( n + i )∣ , f ′ ( n ) := f ( n + 1 ) − f ( n ) . Then, the corresponding derivative map f ↦ ( M ~ Z f ) ′ f \mapsto(\widetilde{M}_{\mathbb{Z}}f)' f ↦ ( M Z f ) ′ W 1 , p ( Z ) W^{1,p}(\mathbb{Z}) W 1 , p ( Z ) ℓ p ( Z ) \ell^p(\mathbb{Z}) ℓ p ( Z ) 1 ≤ p ≤ ∞ 1\le p\le\infty 1 ≤ p ≤ ∞ W 1 , p ( Z ) W^{1,p}(\mathbb{Z}) W 1 , p ( Z ) [3 Theorem 1] obtained the discrete version of the sharp first derivative estimate for the case p = 1 p=1 p = 1
∥ ( M ~ f ) ′ ∥ ℓ 1 ( Z ) ≤ ∥ f ′ ∥ ℓ 1 ( Z ) . (5) \left\|(\widetilde{M}f)'\right\|_{\ell^1(\mathbb{Z})} \le\left\|f'\right\|_{\ell^1(\mathbb{Z})}. \tag*{(5)} ( M f ) ′ ℓ 1 ( Z ) ≤ ∥ f ′ ∥ ℓ 1 ( Z ) . (5) For the case p = ∞ p=\infty p = ∞ [8 Theorem 20] proved the discrete version of the sharp first derivative estimate:
∥ ( M ~ f ) ′ ∥ ℓ ∞ ( Z ) ≤ 1 2 ∥ f ′ ∥ ℓ ∞ ( Z ) . (6) \left\|(\widetilde{M}f)'\right\|_{\ell^\infty(\mathbb{Z})} \le\frac{1}{2}\left\|f'\right\|_{\ell^\infty(\mathbb{Z})}. \tag*{(6)} ( M f ) ′ ℓ ∞ ( Z ) ≤ 2 1 ∥ f ′ ∥ ℓ ∞ ( Z ) . (6) For 1 < p < ∞ 1<p<\infty 1 < p < ∞ [13 Theorems 2.3 and 2.4] determined the discrete version of the sharp first-difference constant for characteristic functions and showed that the constant for general functions is strictly larger.
It is noteworthy that for p = 1 p=1 p = 1 (5) and the continuous setting (4) . However, on the other endpoint, for p = ∞ p=\infty p = ∞ (16) is 2 − 1 \sqrt{2}-1 2 − 1 1 2 \frac{1}{2} 2 1 (6) . Nevertheless, given a compact interval I ⊆ R I\subseteq\mathbb{R} I ⊆ R
M ~ I f ( x ) : = sup x ∈ J ⊂ I 0 < ∣ J ∣ < ∞ 1 ∣ J ∣ ∫ J ∣ f ∣ , \widetilde{M}_I f(x) := \sup_{\substack{x\in J\subset I\\0<|J|<\infty}} \frac{1}{|J|}\int_J |f|, M I f ( x ) := x ∈ J ⊂ I 0 < ∣ J ∣ < ∞ sup ∣ J ∣ 1 ∫ J ∣ f ∣ , which will share the same optimal constant with the discrete setting at p = ∞ p=\infty p = ∞ [1 Corollary 4.5] . Therefore, the corresponding estimate between the integer and the real number is not always straightforward.
Besides the domains R \mathbb{R} R Z \mathbb{Z} Z R n \mathbb{R}^n R n Z n \mathbb{Z}^n Z n R n \mathbb{R}^n R n [4 , 5 , 11 ] . Additionally, beyond the uncentered maximal operator, we may study a similar question for the centered maximal operator, non-tangential maximal operator, and the maximal operator of convolution type [4 , 7 , 11 , 16 ] .
The arguments and results below all concern derivative seminorms for the continuous uncentered maximal operator.
For 1 ≤ p < ∞ 1 \le p < \infty 1 ≤ p < ∞ p p p
Var p ( f ) : = sup x 0 < ⋯ < x m ( ∑ j = 1 m ∣ f ( x j ) − f ( x j − 1 ) ∣ p ∣ x j − x j − 1 ∣ p − 1 ) 1 / p . (7) \operatorname{Var}_p(f) := \sup_{x_0<\cdots<x_m}\left(\sum_{j=1}^{m}\frac{|f(x_j)-f(x_{j-1})|^p}{|x_j-x_{j-1}|^{p-1}}\right)^{1/p}. \tag*{(7)} Var p ( f ) := x 0 < ⋯ < x m sup ( j = 1 ∑ m ∣ x j − x j − 1 ∣ p − 1 ∣ f ( x j ) − f ( x j − 1 ) ∣ p ) 1/ p . (7) We write Var ( f ) = Var 1 ( f ) \operatorname{Var}(f)=\operatorname{Var}_1(f) Var ( f ) = Var 1 ( f ) f f f f ′ ∈ L p f' \in L^p f ′ ∈ L p Var p ( f ) = ∥ f ′ ∥ p \operatorname{Var}_p(f)=\|f'\|_p Var p ( f ) = ∥ f ′ ∥ p p p p p > 1 p>1 p > 1 [15 Chapter IX §4, Theorem 7] . For a bounded-variation function f f f D f Df D f ∣ D f ∣ ( R ) |Df|(\mathbb{R}) ∣ D f ∣ ( R ) B V BV B V f f f B V BV B V f ∈ L l o c 1 ( R ) f \in L^1_{\mathrm{loc}}(\mathbb{R}) f ∈ L loc 1 ( R ) ∣ D f ∣ ( R ) < ∞ |Df|(\mathbb{R})<\infty ∣ D f ∣ ( R ) < ∞ f ∈ L 1 ( R ) f \in L^1(\mathbb{R}) f ∈ L 1 ( R ) ∣ D f ∣ ( R ) |Df|(\mathbb{R}) ∣ D f ∣ ( R ) Var ( f ) \operatorname{Var}(f) Var ( f ) χ { 0 } \chi_{\{0\}} χ { 0 } x ↦ f ( x + ) x\mapsto f(x+) x ↦ f ( x + ) [2 Lemma 3.7] for the corresponding identification using a canonical representative.
The sharp derivative and Riesz variation inequality
For 1 < p < ∞ 1<p<\infty 1 < p < ∞ γ p ∈ [ 0 , 1 ] \gamma_p\in[0,1] γ p ∈ [ 0 , 1 ]
( 2 p − 1 ) x + ( p − 1 ) x ( 2 p − 1 ) / ( p − 1 ) = p for x ∈ [ 0 , 1 ] . (8) (2p-1)x+(p-1)x^{(2p-1)/(p-1)}=p \qquad\text{for }x\in[0,1]. \tag*{(8)} ( 2 p − 1 ) x + ( p − 1 ) x ( 2 p − 1 ) / ( p − 1 ) = p for x ∈ [ 0 , 1 ] . (8) The left side is strictly increasing in x x x γ p \gamma_p γ p A C l o c ( R ) AC_{\mathrm{loc}}(\mathbb{R}) A C loc ( R ) R \mathbb{R} R
Theorem 1. Let 1 < p < ∞ 1<p<\infty 1 < p < ∞ Suppose that f ∈ A C l o c ( R ) f\in AC_{\mathrm{loc}}(\mathbb{R}) f ∈ A C loc ( R ) f ′ ∈ L p ( R ) f'\in L^p(\mathbb{R}) f ′ ∈ L p ( R ) M ~ f \widetilde{M}f M f M ~ f \widetilde{M}f M f
∥ ( M ~ f ) ′ ∥ p = Var p ( M ~ f ) ≤ γ p ∥ f ′ ∥ p = γ p Var p ( f ) . (9) \|(\widetilde{M}f)'\|_p=\operatorname{Var}_p(\widetilde{M}f)\le\gamma_p\|f'\|_p=\gamma_p\operatorname{Var}_p(f). \tag*{(9)} ∥ ( M f ) ′ ∥ p = Var p ( M f ) ≤ γ p ∥ f ′ ∥ p = γ p Var p ( f ) . (9) In particular, the inequality holds for every f ∈ W 1 , p ( R ) f\in W^{1,p}(\mathbb{R}) f ∈ W 1 , p ( R ) M ~ f ∈ W 1 , p ( R ) \widetilde{M}f\in W^{1,p}(\mathbb{R}) M f ∈ W 1 , p ( R ) γ p \gamma_p γ p f n f_n f n [ − 1 , 1 ] [-1,1] [ − 1 , 1 ] [ 0 , 1 ) [0,1) [ 0 , 1 )
lim n → ∞ ∥ ( M ~ f n ) ′ ∥ p ∥ f n ′ ∥ p = γ p . \lim_{n\to\infty}\frac{\|(\widetilde{M}f_n)'\|_p}{\|f_n'\|_p}=\gamma_p. n → ∞ lim ∥ f n ′ ∥ p ∥ ( M f n ) ′ ∥ p = γ p . Remark 2. Specifically, for p = 2 p=2 p = 2
γ 2 = 2 + 1 3 − 2 − 1 3 ≈ 0.5961 , γ 2 3 + 3 γ 2 = 2. \gamma_2=\sqrt[3]{\sqrt{2}+1}-\sqrt[3]{\sqrt{2}-1}\approx0.5961,\qquad\gamma_2^3+3\gamma_2=2. γ 2 = 3 2 + 1 − 3 2 − 1 ≈ 0.5961 , γ 2 3 + 3 γ 2 = 2. Additionally, in order to obtain a rough upper bound of γ p \gamma_p γ p ( p − 1 ) x ( 2 p − 1 ) / ( p − 1 ) (p-1)x^{(2p-1)/(p-1)} ( p − 1 ) x ( 2 p − 1 ) / ( p − 1 ) (8) and obtain the estimate
γ p ≤ p 2 p − 1 . \gamma_p\le\frac{p}{2p-1}. γ p ≤ 2 p − 1 p . Compared to the argument in (1.2), which gives a bound C 2 = 2 + 1 ≈ 2.4142 C_2=\sqrt{2}+1\approx2.4142 C 2 = 2 + 1 ≈ 2.4142 p = 2 p=2 p = 2
Note that every W 1 , p ( R ) W^{1,p}(\mathbb{R}) W 1 , p ( R ) L p L^p L p M ~ f ∈ L p \widetilde{M}f\in L^p M f ∈ L p f ( x ) = log ( 2 + ∣ x ∣ ) f(x)=\log(2+|x|) f ( x ) = log ( 2 + ∣ x ∣ ) f ∈ A C l o c ( R ) f\in AC_{\mathrm{loc}}(\mathbb{R}) f ∈ A C loc ( R ) f ′ ∈ L p f'\in L^p f ′ ∈ L p p > 1 p>1 p > 1 M ~ f = ∞ \widetilde{M}f=\infty M f = ∞
The optimal constant and the classical endpoints
The constant in Theorem 1 has an exact relation to the sharp Hölder constants of Aldaz, Colzani, and Pérez Lázaro [1 Remark 3.3, formula (3.3.1)] . For 0 < α ≤ 1 0 < \alpha\le1 0 < α ≤ 1
H α : = min 0 ≤ t ≤ 1 1 + t 1 + α ( 1 + α ) ( 1 + t ) . (10) H_\alpha:= \min_{0 \le t \le1} \frac{1+t^{1+\alpha}}{(1+\alpha)(1+t)}. \tag*{(10)} H α := 0 ≤ t ≤ 1 min ( 1 + α ) ( 1 + t ) 1 + t 1 + α . (10) Their theorem identifies H α H_\alpha H α
[ M ~ f ] C 0 , α ≤ H α [ f ] C 0 , α , (11) [\widetilde{M}f]_{C^{0,\alpha}} \le H_\alpha[f]_{C^{0,\alpha}}, \tag*{(11)} [ M f ] C 0 , α ≤ H α [ f ] C 0 , α , (11) for all globally α \alpha α f f f
[ f ] C 0 , α : = sup x ≠ y ∣ f ( x ) − f ( y ) ∣ ∣ x − y ∣ α , [f]_{C^{0,\alpha}} := \sup_{x \ne y} \frac{|f(x)-f(y)|}{|x-y|^\alpha}, [ f ] C 0 , α := x = y sup ∣ x − y ∣ α ∣ f ( x ) − f ( y ) ∣ , especially Lip ( f ) = [ f ] C 0 , 1 \operatorname{Lip}(f) = [f]_{C^{0,1}} Lip ( f ) = [ f ] C 0 , 1
The following proposition demonstrates the relation between H α H_\alpha H α γ p \gamma_p γ p
Proposition 3. For every 1 < p < ∞ 1 < p < \infty 1 < p < ∞
γ p = H 1 − 1 / p . (12) \gamma_p = H_{1-1/p}. \tag*{(12)} γ p = H 1 − 1/ p . (12) The function p ↦ γ p p \mapsto\gamma_p p ↦ γ p
lim p ↓ 1 γ p = 1 , lim p → ∞ γ p = 2 − 1. (13) \lim_{p \downarrow1} \gamma_p = 1,\qquad\lim_{p \to\infty} \gamma_p = \sqrt{2}-1. \tag*{(13)} p ↓ 1 lim γ p = 1 , p → ∞ lim γ p = 2 − 1. (13) If r = 2 − 1 r = \sqrt{2}-1 r = 2 − 1
γ p = r + 1 − r − r 2 log r 2 ( 1 + r ) 1 p + O ( p − 2 ) . (14) \gamma_p = r + \frac{1-r-r^2\log r}{2(1+r)}\frac{1}{p} + O(p^{-2}). \tag*{(14)} γ p = r + 2 ( 1 + r ) 1 − r − r 2 log r p 1 + O ( p − 2 ) . (14) The scalar minimum H α H_\alpha H α (11) are due to [1 ] . Our new assertion is that this same number, at α = 1 − 1 / p \alpha= 1-1/p α = 1 − 1/ p L p L^p L p
For every f ∈ L l o c 1 ( R ) f \in L^1_{\mathrm{loc}}(\mathbb{R}) f ∈ L loc 1 ( R ) ∣ D f ∣ ( R ) < ∞ |Df|(\mathbb{R}) < \infty ∣ D f ∣ ( R ) < ∞ f f f [2 Theorem 2.5] proved that M ~ f \widetilde{M}f M f
∥ ( M ~ f ) ′ ∥ 1 = Var ( M ~ f ) ≤ ∣ D f ∣ ( R ) . (15) \|(\widetilde{M}f)'\|_1 = \operatorname{Var}(\widetilde{M}f) \le|Df|(\mathbb{R}). \tag*{(15)} ∥ ( M f ) ′ ∥ 1 = Var ( M f ) ≤ ∣ D f ∣ ( R ) . (15) In particular, the right side is ∥ f ′ ∥ 1 \|f'\|_1 ∥ f ′ ∥ 1 f ∈ W 1 , 1 ( R ) f \in W^{1,1}(\mathbb{R}) f ∈ W 1 , 1 ( R ) [1 Corollary 3.6] established the sharp whole-line Lipschitz bound
∥ ( M ~ f ) ′ ∥ ∞ ≤ ( 2 − 1 ) ∥ f ′ ∥ ∞ (16) \|(\widetilde{M}f)'\|_\infty\le(\sqrt{2}-1)\|f'\|_\infty \tag*{(16)} ∥ ( M f ) ′ ∥ ∞ ≤ ( 2 − 1 ) ∥ f ′ ∥ ∞ (16) for globally Lipschitz inputs with finite maximal function. Both results are classical and sharp. Section 3 explains how their quantitative inequalities follow from Theorem 1 by limiting arguments and approximation.
Proof strategy
We first prove Theorem 1 for nonnegative, continuous, compactly supported, and piecewise-affine functions. On the smooth intervals where ( M ~ f ) ′ (\widetilde{M}f)' ( M f ) ′ x x x a ( x ) a(x) a ( x ) [18 Lemma 2.2] , relate ( M ~ f ) ′ (\widetilde{M}f)' ( M f ) ′ f f f a ( x ) a(x) a ( x )
The key observation is that the change of variables x ↦ a ( x ) x \mapsto a(x) x ↦ a ( x ) R \mathbb{R} R ∥ ( M ~ f ) ′ ∥ p \|(\widetilde{M}f)'\|_p ∥ ( M f ) ′ ∥ p ∥ f ′ ∥ p \|f'\|_p ∥ f ′ ∥ p (8) .
Sections 2.1 and 2.2 establish the local formulas, the desired partition, and the injectivity of the endpoint map. Section 2.3 proves the derivative estimate for piecewise-affine inputs. Section 2.4 extends it by approximation to the full class of inputs in Theorem 1, and Section 2.5 establishes sharpness through an explicit family of functions. Finally, Section 3 proves Proposition 3 and derives the quantitative endpoint bounds.
Proof of Theorem 1
We first assume that f ≥ 0 f \ge0 f ≥ 0
Maximizing intervals
By sublinearity, we have
∣ M ~ f ( x + h ) − M ~ f ( x ) ∣ ≤ Lip ( f ) ∣ h ∣ . (17) \left|\widetilde{M}f(x+h)-\widetilde{M}f(x)\right| \le\operatorname{Lip}(f)|h|. \tag*{(17)} M f ( x + h ) − M f ( x ) ≤ Lip ( f ) ∣ h ∣. (17) This Lipschitz bound will provide the regularity needed below.
On the open set
Ω : = { x ∈ R : M ~ f ( x ) > f ( x ) } , \Omega:= \{x \in\mathbb{R} : \widetilde{M}f(x) > f(x)\}, Ω := { x ∈ R : M f ( x ) > f ( x )} , the supremum defining M ~ f ( x ) \widetilde{M}f(x) M f ( x ) I I I 1 ∣ I ∣ ∫ I f \frac{1}{|I|}\int_I f ∣ I ∣ 1 ∫ I f f ( x ) f(x) f ( x ) ℓ \ell ℓ ∥ f ∥ 1 / ℓ \lVert f\rVert_1/\ell ∥ f ∥ 1 / ℓ
At almost every point of the contact set { x ∈ R : M ~ f ( x ) = f ( x ) } \{x \in\mathbb{R} : \widetilde{M}f(x) = f(x)\} { x ∈ R : M f ( x ) = f ( x )}
( M ~ f ) ′ ( x ) = f ′ ( x ) = 0. (18) (\widetilde{M}f)'(x) = f'(x) = 0. \tag*{(18)} ( M f ) ′ ( x ) = f ′ ( x ) = 0. (18) Indeed, since both M ~ f \widetilde{M}f M f f f f ( M ~ f ) ′ = f ′ (\widetilde{M}f)' = f' ( M f ) ′ = f ′ f f f x x x f ( x ) f(x) f ( x )
M ~ f ( x ) = max { 1 h ∫ [ x − h , x ] f , 1 h ∫ [ x , x + h ] f } > f ( x ) (19) \widetilde{M}f(x) = \max\left\{\frac{1}{h}\int_{[x-h,x]} f,\frac{1}{h}\int_{[x,x+h]} f\right\} > f(x) \tag*{(19)} M f ( x ) = max { h 1 ∫ [ x − h , x ] f , h 1 ∫ [ x , x + h ] f } > f ( x ) (19) for some sufficiently small h > 0 h>0 h > 0 M ~ f ( x ) = f ( x ) \widetilde{M}f(x)=f(x) M f ( x ) = f ( x )
The local formulas in the next lemma are the differentiation identities in Weigt [18 Lemma 2.2, equations (2.3)–(2.5)] , proved there under a unimodality and two-sided slope hypothesis. We include the details and adapt his proof for completeness in the present piecewise-affine setting.
Lemma 4. Suppose that x ∈ Ω x \in\Omega x ∈ Ω M ~ f \widetilde{M}f M f C 1 C^1 C 1 x x x ( M ~ f ) ′ ( x ) < 0 (\widetilde{M}f)'(x)<0 ( M f ) ′ ( x ) < 0 [ a ( x ) , x ] [a(x),x] [ a ( x ) , x ]
f ( a ( x ) ) = M ~ f ( x ) , ( M ~ f ) ′ ( x ) = f ( x ) − M ~ f ( x ) x − a ( x ) . (20) f(a(x))=\widetilde{M}f(x), \qquad(\widetilde{M}f)'(x)=\frac{f(x)-\widetilde{M}f(x)}{x-a(x)}. \tag*{(20)} f ( a ( x )) = M f ( x ) , ( M f ) ′ ( x ) = x − a ( x ) f ( x ) − M f ( x ) . (20) Additionally, if M ~ f \widetilde{M}f M f x x x f ′ ( a ( x ) ) ≠ 0 f'(a(x))\ne0 f ′ ( a ( x )) = 0
a ′ ( x ) = ( M ~ f ) ′ ( x ) f ′ ( a ( x ) ) , a'(x)=\frac{(\widetilde{M}f)'(x)}{f'(a(x))}, a ′ ( x ) = f ′ ( a ( x )) ( M f ) ′ ( x ) , ( M ~ f ) ′ ′ ( x ) = f ′ ( x ) − ( 2 − a ′ ( x ) ) ( M ~ f ) ′ ( x ) x − a ( x ) . (21) (\widetilde{M}f)''(x)=\frac{f'(x)-(2-a'(x))(\widetilde{M}f)'(x)}{x-a(x)}. \tag*{(21)} ( M f ) ′′ ( x ) = x − a ( x ) f ′ ( x ) − ( 2 − a ′ ( x )) ( M f ) ′ ( x ) . (21) When ( M ~ f ) ′ ( x ) > 0 (\widetilde{M}f)'(x)>0 ( M f ) ′ ( x ) > 0 [ x , a ( x ) ] [x,a(x)] [ x , a ( x )] x − a ( x ) < 0 x-a(x)<0 x − a ( x ) < 0
Proof. Suppose first that ( M ~ f ) ′ ( x ) < 0 (\widetilde{M}f)'(x)<0 ( M f ) ′ ( x ) < 0 [ a , b ] [a,b] [ a , b ] x x x
1 b − a ∫ a b f ≤ max { 1 x − a ∫ a x f , 1 b − x ∫ x b f } , \frac{1}{b-a}\int_a^b f \le\max\left\{\frac{1}{x-a}\int_a^x f,\frac{1}{b-x}\int_x^b f\right\}, b − a 1 ∫ a b f ≤ max { x − a 1 ∫ a x f , b − x 1 ∫ x b f } , it suffices to consider those intervals such that x x x M ~ f \widetilde{M}f M f [ x , b ] [x,b] [ x , b ] M ~ f ( x + h ) ≥ M ~ f ( x ) \widetilde{M}f(x+h)\ge\widetilde{M}f(x) M f ( x + h ) ≥ M f ( x ) h > 0 h>0 h > 0 ( M ~ f ) ′ ( x ) < 0 (\widetilde{M}f)'(x)<0 ( M f ) ′ ( x ) < 0 [ a , x ] [a,x] [ a , x ]
Fix such an interval [ a , x ] [a,x] [ a , x ] f f f
0 = d d u ( 1 x − u ∫ u x f ( t ) d t ) ∣ u = a = M ~ f ( x ) − f ( a ) x − a . 0=\left.\frac{d}{du}\left(\frac{1}{x-u}\int_u^x f(t)\,dt\right)\right|_{u=a}=\frac{\widetilde{M}f(x)-f(a)}{x-a}. 0 = d u d ( x − u 1 ∫ u x f ( t ) d t ) u = a = x − a M f ( x ) − f ( a ) . Hence f ( a ) = M ~ f ( x ) f(a)=\widetilde{M}f(x) f ( a ) = M f ( x )
t ↦ 1 t − a ∫ a t f ( u ) d u t\mapsto\frac{1}{t-a}\int_a^t f(u)\,du t ↦ t − a 1 ∫ a t f ( u ) d u touches the graph of M ~ f \widetilde{M}f M f t = x t=x t = x M ~ f \widetilde{M}f M f t = x t=x t = x
( M ~ f ) ′ ( x ) = d d t ( 1 t − a ∫ a t f ( u ) d u ) ∣ t = x = f ( x ) − M ~ f ( x ) x − a . (\widetilde{M}f)'(x)=\left.\frac{d}{dt}\left(\frac{1}{t-a}\int_a^t f(u)\,du\right)\right|_{t=x}=\frac{f(x)-\widetilde{M}f(x)}{x-a}. ( M f ) ′ ( x ) = d t d ( t − a 1 ∫ a t f ( u ) d u ) t = x = x − a f ( x ) − M f ( x ) . In particular,
a = x − f ( x ) − M ~ f ( x ) ( M ~ f ) ′ ( x ) . (22) a=x-\frac{f(x)-\widetilde{M}f(x)}{(\widetilde{M}f)'(x)}. \tag*{(22)} a = x − ( M f ) ′ ( x ) f ( x ) − M f ( x ) . (22) so the maximizing interval is unique. Denote its left endpoint by a ( x ) a(x) a ( x ) a ( x ) a(x) a ( x ) M ~ f \widetilde{M}f M f
Moreover, consider a point x x x f f f C 2 C^2 C 2 x x x f ′ ( a ( x ) ) ≠ 0 f'(a(x))\ne0 f ′ ( a ( x )) = 0 f ( a ( x ) ) = M ~ f ( x ) f(a(x))=\widetilde{M}f(x) f ( a ( x )) = M f ( x )
f ′ ( a ( x ) ) a ′ ( x ) = ( M ~ f ) ′ ( x ) ⇒ a ′ ( x ) = ( M ~ f ) ′ ( x ) f ′ ( a ( x ) ) . f'(a(x))a'(x)=(\widetilde{M}f)'(x)\Rightarrow a'(x)=\frac{(\widetilde{M}f)'(x)}{f'(a(x))}. f ′ ( a ( x )) a ′ ( x ) = ( M f ) ′ ( x ) ⇒ a ′ ( x ) = f ′ ( a ( x )) ( M f ) ′ ( x ) . Finally, differentiating (2.4) gives
( M ~ f ) ′ ′ ( x ) = f ′ ( x ) − ( M ~ f ) ′ ( x ) x − a ( x ) − ( f ( x ) − M ~ f ( x ) ) ( 1 − a ′ ( x ) ) ( x − a ( x ) ) 2 = f ′ ( x ) − ( 2 − a ′ ( x ) ) ( M ~ f ) ′ ( x ) x − a ( x ) , \begin{aligned}
(\widetilde{M}f)''(x)=\frac{f'(x)-(\widetilde{M}f)'(x)}{x-a(x)}-\frac{(f(x)-\widetilde{M}f(x))(1-a'(x))}{(x-a(x))^2} \\
&=\frac{f'(x)-(2-a'(x))(\widetilde{M}f)'(x)}{x-a(x)},
\end{aligned} ( M f ) ′′ ( x ) = x − a ( x ) f ′ ( x ) − ( M f ) ′ ( x ) − ( x − a ( x ) ) 2 ( f ( x ) − M f ( x )) ( 1 − a ′ ( x )) = x − a ( x ) f ′ ( x ) − ( 2 − a ′ ( x )) ( M f ) ′ ( x ) , which proves (2.5). Applying the argument to t ↦ f ( − t ) t\mapsto f(-t) t ↦ f ( − t ) ( M ~ f ) ′ ( x ) > 0 (\widetilde{M}f)'(x)>0 ( M f ) ′ ( x ) > 0 [ x , a ( x ) ] [x,a(x)] [ x , a ( x )] x − a ( x ) < 0 x-a(x)<0 x − a ( x ) < 0
Smooth pieces and endpoint injectivity
Under the assumption that f ≥ 0 f\ge0 f ≥ 0 R \mathbb{R} R
Lemma 5 (Finite smooth partition). There is a finite partition of R \mathbb{R} R J J J f ∣ J f|_J f ∣ J M ~ f ∣ J \widetilde{M}f|_J M f ∣ J
( M ~ f ) ′ ∣ J > 0 , ( M ~ f ) ′ ∣ J < 0 , or ( M ~ f ) ′ ∣ J ≡ 0. (\widetilde{M}f)'|_J>0,\qquad(\widetilde{M}f)'|_J<0,\qquad\text{or}\qquad(\widetilde{M}f)'|_J\equiv0. ( M f ) ′ ∣ J > 0 , ( M f ) ′ ∣ J < 0 , or ( M f ) ′ ∣ J ≡ 0. The partition can also be chosen so that ( M ~ f ) ′ (\widetilde{M}f)' ( M f ) ′ J J J
Proof. Write f ( t ) = m i t + b i f(t)=m_i t+b_i f ( t ) = m i t + b i I i I_i I i P ( t ) = ∫ − ∞ t f ( y ) d y P(t)=\int_{-\infty}^{t} f(y)\,dy P ( t ) = ∫ − ∞ t f ( y ) d y P i = P ∣ I i P_i=P|_{I_i} P i = P ∣ I i I i I_i I i
P i ( t ) : = 1 2 m i t 2 + b i t + c i P_i(t):=\frac{1}{2}m_i t^2+b_i t+c_i P i ( t ) := 2 1 m i t 2 + b i t + c i where c i c_i c i P ( t ) P(t) P ( t ) R \mathbb{R} R x ∈ I i x\in I_i x ∈ I i
A ( u , x ) : = { P ( x ) − P ( u ) x − u x ≠ u , f ( x ) x = u , A(u,x):=\begin{cases}
\dfrac{P(x)-P(u)}{x-u} & x\ne u,\\
f(x) & x=u,
\end{cases} A ( u , x ) := ⎩ ⎨ ⎧ x − u P ( x ) − P ( u ) f ( x ) x = u , x = u , which is the average of f f f u u u x x x (19) ,
M ~ f ( x ) = sup u ≠ x A ( u , x ) . \widetilde{M}f(x)=\sup_{u\ne x} A(u,x). M f ( x ) = u = x sup A ( u , x ) . If a maximizing endpoint u = u 0 u=u_0 u = u 0 I j I_j I j
0 = ∂ u A ( u 0 , x ) = − f ( u ) x − u + P ( x ) − P ( u ) ( x − u ) 2 = A ( u , x ) − f ( u ) x − u , 0=\partial_u A(u_0,x)=-\frac{f(u)}{x-u}+\frac{P(x)-P(u)}{(x-u)^2}=\frac{A(u,x)-f(u)}{x-u}, 0 = ∂ u A ( u 0 , x ) = − x − u f ( u ) + ( x − u ) 2 P ( x ) − P ( u ) = x − u A ( u , x ) − f ( u ) , ⇒ 0 = 1 2 m j u 2 − m j x u + P i ( x ) − b j x − c j . \Rightarrow0=\frac{1}{2}m_j u^2-m_jxu+P_i(x)-b_jx-c_j. ⇒ 0 = 2 1 m j u 2 − m j xu + P i ( x ) − b j x − c j . For m j ≠ 0 m_j\ne0 m j = 0
D i j ( x ) : = x 2 − 2 m j ( P i ( x ) − b j x − c j ) , u i j ± ( x ) : = x ± D i j ( x ) , D_{ij}(x):=x^2-\frac{2}{m_j}\bigl(P_i(x)-b_jx-c_j\bigr),\qquad u_{ij}^{\pm}(x):=x\pm\sqrt{D_{ij}(x)}, D ij ( x ) := x 2 − m j 2 ( P i ( x ) − b j x − c j ) , u ij ± ( x ) := x ± D ij ( x ) , where u i j ± ( x ) u_{ij}^{\pm}(x) u ij ± ( x ) a ( x ) a(x) a ( x ) M ~ f ( x ) \widetilde{M}f(x) M f ( x ) x ∈ I i x\in I_i x ∈ I i Table 1 :
Table 1. Candidates for M ~ f ( x ) \widetilde{M}f(x) M f ( x )
In the last row, one may choose any u ∈ I j ∖ { x } u\in I_j\setminus\{x\} u ∈ I j ∖ { x } a ( x ) a(x) a ( x ) x ∈ I i x\in I_i x ∈ I i C i ( x ) C_i(x) C i ( x )
M ~ f ( x ) = max C i ( x ) . (23) \widetilde{M}f(x)=\max C_i(x). \tag*{(23)} M f ( x ) = max C i ( x ) . (23) Based on Table 1 , first cut at the endpoints of I i I_i I i
In Table 1 , while the first type candidates always exist, the second type candidates need to satisfy two conditions: D i j ( x ) ≥ 0 D_{ij}(x)\ge0 D ij ( x ) ≥ 0 u i j ± ( x ) ∈ I j ‾ u_{ij}^{\pm}(x)\in\overline{I_j} u ij ± ( x ) ∈ I j Table 2 , respectively: The first one is D i j ( x ) D_{ij}(x) D ij ( x ) u i j ± ( x ) u_{ij}^{\pm}(x) u ij ± ( x ) D i j ( x ) − ( v − x ) 2 D_{ij}(x)-(v-x)^2 D ij ( x ) − ( v − x ) 2 v v v I j I_j I j v = x ± D i j ( x ) = u i j ( x ) v=x\pm\sqrt{D_{ij}(x)}=u_{ij}(x) v = x ± D ij ( x ) = u ij ( x ) u i j ± ( x ) ∈ I j ‾ u_{ij}^{\pm}(x)\in\overline{I_j} u ij ± ( x ) ∈ I j P i ( x ) − b j ( x ) − c j = 0 P_i(x)-b_j(x)-c_j=0 P i ( x ) − b j ( x ) − c j = 0 m j = 0 m_j=0 m j = 0 Table 2 suffices to determine its existence. Hence, with these finitely many polynomials mentioned in Table 2 , we can use their zeros and endpoints of intervals I j I_j I j J J J f ∣ J f|_J f ∣ J M ~ f \widetilde{M}f M f J J J J J J
Table 2. Polynomials needed for smooth partition
Figure 1. M ~ f ( x ) \widetilde{M}f(x) M f ( x )
Next, we aim to show that the intersection of these candidate curves is either finite or a subinterval. Note that every candidate is both algebraic and analytic. Let k 1 ( x ) , k 2 ( x ) k_1(x), k_2(x) k 1 ( x ) , k 2 ( x ) Q ( x , y ) ∈ R [ x , y ] Q(x,y)\in\mathbb{R}[x,y] Q ( x , y ) ∈ R [ x , y ] Q ( x , k 1 ( x ) − k 2 ( x ) ) = 0 Q(x,k_1(x)-k_2(x))=0 Q ( x , k 1 ( x ) − k 2 ( x )) = 0 x x x k 1 ( x ) = k 2 ( x ) k_1(x)=k_2(x) k 1 ( x ) = k 2 ( x ) Q ( x , 0 ) = 0 Q(x,0)=0 Q ( x , 0 ) = 0 Q ∈ R [ x , y ] Q\in\mathbb{R}[x,y] Q ∈ R [ x , y ] y ∣ Q ( x , y ) y\mid Q(x,y) y ∣ Q ( x , y ) Q ( x , y ) = c y Q(x,y)=cy Q ( x , y ) = cy c ∈ R c\in\mathbb{R} c ∈ R k 1 ( x ) ≡ k 2 ( x ) k_1(x)\equiv k_2(x) k 1 ( x ) ≡ k 2 ( x )
Since the intersection is either finite or a subinterval, there exists a finite refinement of the previous partition such that, on each interval J J J f ∣ J f|_J f ∣ J M ~ f ∣ J \widetilde{M}f|_J M f ∣ J M ~ f \widetilde{M}f M f Table 1 and thus M ~ f \widetilde{M}f M f J J J
Since M ~ f ∣ J \widetilde{M}f|_J M f ∣ J J J J ( M ~ f ) ′ (\widetilde{M}f)' ( M f ) ′
( M ~ f ) ′ ∣ J > 0 , ( M ~ f ) ′ ∣ J < 0 , or ( M ~ f ) ′ ∣ J = 0. (\widetilde{M}f)'|_J>0,\qquad(\widetilde{M}f)'|_J<0,\qquad\text{or}\qquad(\widetilde{M}f)'|_J=0. ( M f ) ′ ∣ J > 0 , ( M f ) ′ ∣ J < 0 , or ( M f ) ′ ∣ J = 0. Lemma 6. Let J J J ( M ~ f ) ′ ∣ J (\widetilde{M}f)'|_J ( M f ) ′ ∣ J
H : = { f ( v ) ∣ v is a finite endpoint of an affine interval } ∪ { b j ∣ m j = 0 } , \mathcal{H}:=\{f(v)\mid v\text{ is a finite endpoint of an affine interval}\}\cup\{b_j\mid m_j=0\}, H := { f ( v ) ∣ v is a finite endpoint of an affine interval } ∪ { b j ∣ m j = 0 } , and P : = { x ∈ R ∣ M ~ f ( x ) ∈ H , ( M ~ f ) ′ ( x ) ≠ 0 } \mathcal{P}:=\{x\in\mathbb{R}\mid\widetilde{M}f(x)\in\mathcal{H},(\widetilde{M}f)'(x)\ne0\} P := { x ∈ R ∣ M f ( x ) ∈ H , ( M f ) ′ ( x ) = 0 } P \mathcal{P} P J ∖ P J\setminus\mathcal{P} J ∖ P
{ f ′ ( a ( x ) ) > 0 , a ′ ( x ) < 0 if ( M ~ f ) ′ ( x ) < 0 , f ′ ( a ( x ) ) < 0 , a ′ ( x ) < 0 if ( M ~ f ) ′ ( x ) > 0. (24) \begin{cases}
f'(a(x))>0,\ a'(x)<0 & \text{if }(\widetilde{M}f)'(x)<0,\\
f'(a(x))<0,\ a'(x)<0 & \text{if }(\widetilde{M}f)'(x)>0. \tag*{(24)}
\end{cases} { f ′ ( a ( x )) > 0 , a ′ ( x ) < 0 f ′ ( a ( x )) < 0 , a ′ ( x ) < 0 if ( M f ) ′ ( x ) < 0 , if ( M f ) ′ ( x ) > 0. (24) At every finite partition point x 0 x_0 x 0 ( M ~ f ) ′ ( x 0 ± ) (\widetilde{M}f)'(x_0\pm) ( M f ) ′ ( x 0 ± ) x 0 ∈ Ω x_0\in\Omega x 0 ∈ Ω
( M ~ f ) ′ ( x 0 − ) ≤ ( M ~ f ) ′ ( x 0 + ) . (25) (\widetilde{M}f)'(x_0-)\leq(\widetilde{M}f)'(x_0+). \tag*{(25)} ( M f ) ′ ( x 0 − ) ≤ ( M f ) ′ ( x 0 + ) . (25) Proof. For each h ∈ H h\in\mathcal{H} h ∈ H M ~ f ( x ) = h \widetilde{M}f(x)=h M f ( x ) = h J J J h ∈ H h\in\mathcal{H} h ∈ H M ~ f \widetilde{M}f M f J J J P \mathcal{P} P J J J a ( x ) a(x) a ( x ) f ( a ( x ) ) = M ~ f ( x ) ∉ H f(a(x))=\widetilde{M}f(x)\notin\mathcal{H} f ( a ( x )) = M f ( x ) ∈ / H a ( x ) a(x) a ( x ) I j I_j I j m j ≠ 0 m_j\ne0 m j = 0
0 ≥ ∂ u u A ( a ( x ) , x ) = A ( a ( x ) , x ) − f ( a ( x ) ) x − a ( x ) − f ′ ( a ( x ) ) x − a ( x ) + A ( a ( x ) , x ) − f ( a ( x ) ) ( x − a ( x ) ) 2 = − f ′ ( a ( x ) ) x − a ( x ) (26) 0\geq\partial_{uu}A(a(x),x)=\frac{\dfrac{A(a(x),x)-f(a(x))}{x-a(x)}-f'(a(x))}{x-a(x)}+\frac{A(a(x),x)-f(a(x))}{(x-a(x))^2}=-\frac{f'(a(x))}{x-a(x)} \tag*{(26)} 0 ≥ ∂ uu A ( a ( x ) , x ) = x − a ( x ) x − a ( x ) A ( a ( x ) , x ) − f ( a ( x )) − f ′ ( a ( x )) + ( x − a ( x ) ) 2 A ( a ( x ) , x ) − f ( a ( x )) = − x − a ( x ) f ′ ( a ( x )) (26) because A ( a ( x ) , x ) = M ~ f ( x ) = f ( a ( x ) ) A(a(x),x)=\widetilde{M}f(x)=f(a(x)) A ( a ( x ) , x ) = M f ( x ) = f ( a ( x )) J J J ( M ~ f ) ′ ≠ 0 (\widetilde{M}f)'\ne0 ( M f ) ′ = 0
a ( x ) = x − f ( x ) − M ~ f ( x ) ( M ~ f ) ′ ( x ) , f ( a ( x ) ) = M ~ f ( x ) . a(x)=x-\frac{f(x)-\widetilde{M}f(x)}{(\widetilde{M}f)'(x)},\qquad f(a(x))=\widetilde{M}f(x). a ( x ) = x − ( M f ) ′ ( x ) f ( x ) − M f ( x ) , f ( a ( x )) = M f ( x ) . In particular, a ( x ) < x a(x)<x a ( x ) < x ( M ~ f ) ′ < 0 (\widetilde{M}f)'<0 ( M f ) ′ < 0 a ( x ) > x a(x)>x a ( x ) > x ( M ~ f ) ′ > 0 (\widetilde{M}f)'>0 ( M f ) ′ > 0 f ′ ( a ( x ) ) > 0 f'(a(x))>0 f ′ ( a ( x )) > 0 ( M ~ f ) ′ ( x ) < 0 (\widetilde{M}f)'(x)<0 ( M f ) ′ ( x ) < 0 f ′ ( a ( x ) ) < 0 f'(a(x))<0 f ′ ( a ( x )) < 0 ( M ~ f ) ′ ( x ) > 0 (\widetilde{M}f)'(x)>0 ( M f ) ′ ( x ) > 0 f ( a ( x ) ) = M ~ f ( x ) f(a(x))=\widetilde{M}f(x) f ( a ( x )) = M f ( x )
a ′ ( x ) = ( M ~ f ) ′ ( x ) f ′ ( a ( x ) ) < 0 a'(x)=\frac{(\widetilde{M}f)'(x)}{f'(a(x))}<0 a ′ ( x ) = f ′ ( a ( x )) ( M f ) ′ ( x ) < 0 in both cases. Figure 2 illustrates the endpoint identity.
Figure 2. An example: a ( 1 2 ) = ( 1 − 2 ) / 2 a\left(\frac{1}{2}\right)=(1-\sqrt{2})/2 a ( 2 1 ) = ( 1 − 2 ) /2 f ( a ) = M ~ f ( 1 2 ) = ( 3 − 2 ) / 2 f(a)=\widetilde{M}f\left(\frac{1}{2}\right)=(3-\sqrt{2})/2 f ( a ) = M f ( 2 1 ) = ( 3 − 2 ) /2
In Figure 2 , the shaded area equals the area of the rectangle of width 1 2 − a \frac{1}{2}-a 2 1 − a f ( a ) f(a) f ( a ) f f f 1 2 \frac{1}{2} 2 1 M ~ f \widetilde{M}f M f a ( 1 2 ) a\left(\frac{1}{2}\right) a ( 2 1 )
f ( a ) = 1 1 2 − a ∫ a 1 2 f . f(a)=\frac{1}{\frac{1}{2}-a}\int_a^{\frac{1}{2}}f. f ( a ) = 2 1 − a 1 ∫ a 2 1 f . The derivative ( M ~ f ) ′ (\widetilde{M}f)' ( M f ) ′ ∣ ( M ~ f ) ′ ∣ ≤ Lip ( f ) \lvert(\widetilde{M}f)'\rvert\leq\operatorname{Lip}(f) ∣( M f ) ′ ∣ ≤ Lip ( f ) (17) ; hence its one-sided limits at finite endpoints exist and are finite. For h > 0 h>0 h > 0
M ~ f ( x 0 + h ) − M ~ f ( x 0 ) h = ( M ~ f ) ′ ( ξ h ) , x 0 < ξ h < x 0 + h . \frac{\widetilde{M}f(x_0+h)-\widetilde{M}f(x_0)}{h}=(\widetilde{M}f)'(\xi_h),\qquad x_0<\xi_h<x_0+h. h M f ( x 0 + h ) − M f ( x 0 ) = ( M f ) ′ ( ξ h ) , x 0 < ξ h < x 0 + h . Letting h ↓ 0 h\downarrow0 h ↓ 0
For x 0 ∈ Ω x_0\in\Omega x 0 ∈ Ω [ u , v ] [u,v] [ u , v ] x 0 x_0 x 0 t t t x 0 x_0 x 0
Q ( t ) : = { 1 v − u ∫ u v f , u < x 0 < v , 1 t − u ∫ u t f , x 0 = v , 1 v − t ∫ t v f , x 0 = u . Q(t):=
\begin{cases}
\frac{1}{v-u}\int_u^v f, & u<x_0<v,\\[6pt]
\frac{1}{t-u}\int_u^t f, & x_0=v,\\[6pt]
\frac{1}{v-t}\int_t^v f, & x_0=u.
\end{cases} Q ( t ) := ⎩ ⎨ ⎧ v − u 1 ∫ u v f , t − u 1 ∫ u t f , v − t 1 ∫ t v f , u < x 0 < v , x 0 = v , x 0 = u . Then Q ∈ C 1 Q\in C^1 Q ∈ C 1 x 0 x_0 x 0 Q ≤ M ~ f Q\leq\widetilde{M}f Q ≤ M f Q ( x 0 ) = M ~ f ( x 0 ) Q(x_0)=\widetilde{M}f(x_0) Q ( x 0 ) = M f ( x 0 ) h > 0 h>0 h > 0
M ~ f ( x 0 − h ) − M ~ f ( x 0 ) − h ≤ Q ( x 0 − h ) − Q ( x 0 ) − h , Q ( x 0 + h ) − Q ( x 0 ) h ≤ M ~ f ( x 0 + h ) − M ~ f ( x 0 ) h . \frac{\widetilde{M}f(x_0-h)-\widetilde{M}f(x_0)}{-h}
\leq
\frac{Q(x_0-h)-Q(x_0)}{-h},
\qquad
\frac{Q(x_0+h)-Q(x_0)}{h}
\leq
\frac{\widetilde{M}f(x_0+h)-\widetilde{M}f(x_0)}{h}. − h M f ( x 0 − h ) − M f ( x 0 ) ≤ − h Q ( x 0 − h ) − Q ( x 0 ) , h Q ( x 0 + h ) − Q ( x 0 ) ≤ h M f ( x 0 + h ) − M f ( x 0 ) . Thus ( M ~ f ) ′ ( x 0 − ) ≤ Q ′ ( x 0 ) ≤ ( M ~ f ) ′ ( x 0 + ) (\widetilde{M}f)'(x_0-)\leq Q'(x_0)\leq(\widetilde{M}f)'(x_0+) ( M f ) ′ ( x 0 − ) ≤ Q ′ ( x 0 ) ≤ ( M f ) ′ ( x 0 + ) (25) ; see Figure 3 for a visual illustration.
Figure 3. Schematic of the touching inequality Q ≤ M ~ f Q\leq\widetilde{M}f Q ≤ M f Q ( x 0 ) = M ~ f ( x 0 ) Q(x_0)=\widetilde{M}f(x_0) Q ( x 0 ) = M f ( x 0 ) M ~ f \widetilde{M}f M f Q ′ ( x 0 ) Q'(x_0) Q ′ ( x 0 )
Let P ′ \mathcal{P}' P ′ P ∗ : = P ∪ P ′ \mathcal{P}^*:=\mathcal{P}\cup\mathcal{P}' P ∗ := P ∪ P ′
E : = { x ∈ R ∣ x ∉ P , x ∉ P ′ , ( M ~ f ) ′ ( x ) ≠ 0 } . E:=\{x\in\mathbb{R}\mid x\notin\mathcal{P},\ x\notin\mathcal{P}',\ (\widetilde{M}f)'(x)\neq0\}. E := { x ∈ R ∣ x ∈ / P , x ∈ / P ′ , ( M f ) ′ ( x ) = 0 } . Write E + = { x ∈ E ∣ ( M ~ f ) ′ ( x ) > 0 } E_+=\{x\in E\mid(\widetilde{M}f)'(x)>0\} E + = { x ∈ E ∣ ( M f ) ′ ( x ) > 0 } E − = { x ∈ E ∣ ( M ~ f ) ′ ( x ) < 0 } E_-=\{x\in E\mid(\widetilde{M}f)'(x)<0\} E − = { x ∈ E ∣ ( M f ) ′ ( x ) < 0 } a ( x ) a(x) a ( x ) x x x
Lemma 7. The function x ↦ a ( x ) x\mapsto a(x) x ↦ a ( x ) E − E_- E − E + E_+ E +
Proof. Prove by contradiction. Suppose that x 1 < x 2 x_1<x_2 x 1 < x 2 x 1 , x 2 ∈ E − x_1,x_2\in E_- x 1 , x 2 ∈ E − a ( x 1 ) = a ( x 2 ) a(x_1)=a(x_2) a ( x 1 ) = a ( x 2 )
M ~ f ( x 1 ) = f ( a ( x 1 ) ) = f ( a ( x 2 ) ) = M ~ f ( x 2 ) . \widetilde{M}f(x_1)=f(a(x_1))=f(a(x_2))=\widetilde{M}f(x_2). M f ( x 1 ) = f ( a ( x 1 )) = f ( a ( x 2 )) = M f ( x 2 ) . By (18) , we know that ( M ~ f ) ′ ( x i ) < 0 (\widetilde{M}f)'(x_i)<0 ( M f ) ′ ( x i ) < 0 f ( x i ) < M ~ f ( x i ) f(x_i)<\widetilde{M}f(x_i) f ( x i ) < M f ( x i ) a ( x i ) < x i a(x_i)<x_i a ( x i ) < x i i = 1 , 2 i=1,2 i = 1 , 2 x 1 x_1 x 1 x 2 x_2 x 2 a ( x 1 ) a(x_1) a ( x 1 ) f f f f ( x 2 ) < M ~ f ( x 2 ) f(x_2) < \widetilde{M}f(x_2) f ( x 2 ) < M f ( x 2 ) [ a ( x 2 ) , x 2 ] [a(x_2),x_2] [ a ( x 2 ) , x 2 ] [ a ( x 1 ) , b ] [a(x_1),b] [ a ( x 1 ) , b ] x 1 < b x_1 < b x 1 < b x 1 ∈ [ a ( x 1 ) , b ] x_1 \in[a(x_1),b] x 1 ∈ [ a ( x 1 ) , b ]
Figure 4. The position of x 1 x_1 x 1 x 2 x_2 x 2 a : = a ( x 1 ) = a ( x 2 ) a := a(x_1) = a(x_2) a := a ( x 1 ) = a ( x 2 )
1 b − a ( x 1 ) ∫ a ( x 1 ) b f > 1 x 2 − a ( x 2 ) ∫ a ( x 2 ) x 2 f = M ~ f ( x 2 ) = M ~ f ( x 1 ) , \frac{1}{b-a(x_1)}\int_{a(x_1)}^b f > \frac{1}{x_2-a(x_2)}\int_{a(x_2)}^{x_2}f = \widetilde{M}f(x_2) = \widetilde{M}f(x_1), b − a ( x 1 ) 1 ∫ a ( x 1 ) b f > x 2 − a ( x 2 ) 1 ∫ a ( x 2 ) x 2 f = M f ( x 2 ) = M f ( x 1 ) , which contradicts the definition of the maximal operator. By symmetry, x ↦ a ( x ) x \mapsto a(x) x ↦ a ( x ) E + E_+ E +
Additionally, by (24) ,
{ a ( x ) : x ∈ E − } ⊂ { t : f ′ ( t ) > 0 } , { a ( x ) : x ∈ E + } ⊂ { t : f ′ ( t ) < 0 } . \{a(x):x\in E_-\}\subset\{t:f'(t)>0\},\qquad\{a(x):x\in E_+\}\subset\{t:f'(t)<0\}. { a ( x ) : x ∈ E − } ⊂ { t : f ′ ( t ) > 0 } , { a ( x ) : x ∈ E + } ⊂ { t : f ′ ( t ) < 0 } . Thus, the images are disjoint. □
The derivative estimate
In this section, we will use the refined partition P ∗ \mathcal{P}^* P ∗ R > 0 R>0 R > 0 supp f ⊂ [ − R , R ] \operatorname{supp} f\subset[-R,R] supp f ⊂ [ − R , R ] x > R x>R x > R M ~ f \widetilde{M}f M f f ( x ) = 0 < M ~ f ( x ) f(x)=0<\widetilde{M}f(x) f ( x ) = 0 < M f ( x ) x x x [ a ( x ) , x ] [a(x),x] [ a ( x ) , x ] − R ≤ a ( x ) < R -R\le a(x)<R − R ≤ a ( x ) < R h > 0 h>0 h > 0 ∫ x − h x f ( y ) d y = 0 \int_{x-h}^x f(y)\,\mathrm{d}y=0 ∫ x − h x f ( y ) d y = 0
M ~ f ( x − h ) − M ~ f ( x ) ≥ 1 ( x − h ) − a ( x ) ∫ a ( x ) x − h f ( y ) d y − 1 x − a ( x ) ∫ a ( x ) x f ( y ) d y = h ( x − h − a ( x ) ) ( x − a ( x ) ) ∫ a ( x ) x f ( y ) d y . \begin{aligned}
\widetilde{M}f(x-h)-\widetilde{M}f(x)\ge\frac{1}{(x-h)-a(x)}\int_{a(x)}^{x-h}f(y)\,\mathrm{d}y-\frac{1}{x-a(x)}\int_{a(x)}^x f(y)\,\mathrm{d}y \\
&=\frac{h}{(x-h-a(x))(x-a(x))}\int_{a(x)}^x f(y)\,\mathrm{d}y.
\end{aligned} M f ( x − h ) − M f ( x ) ≥ ( x − h ) − a ( x ) 1 ∫ a ( x ) x − h f ( y ) d y − x − a ( x ) 1 ∫ a ( x ) x f ( y ) d y = ( x − h − a ( x )) ( x − a ( x )) h ∫ a ( x ) x f ( y ) d y . i.e. ( M ~ f ) ′ ( x ) < 0 (\widetilde{M}f)'(x)<0 ( M f ) ′ ( x ) < 0 x > R x>R x > R
M ~ f ( x ) ≤ ∥ f ∥ 1 x − a ( x ) ≤ ∥ f ∥ 1 x − R , ∣ ( M ~ f ) ′ ( x ) ∣ ≤ ∣ M ~ f ( x ) ∣ ∣ x − a ( x ) ∣ ≤ ∥ f ∥ 1 ( x − R ) 2 . \widetilde{M}f(x)\le\frac{\lVert f\rVert_1}{x-a(x)}\le\frac{\lVert f\rVert_1}{x-R},\qquad|(\widetilde{M}f)'(x)|\le\frac{|\widetilde{M}f(x)|}{|x-a(x)|}\le\frac{\lVert f\rVert_1}{(x-R)^2}. M f ( x ) ≤ x − a ( x ) ∥ f ∥ 1 ≤ x − R ∥ f ∥ 1 , ∣ ( M f ) ′ ( x ) ∣ ≤ ∣ x − a ( x ) ∣ ∣ M f ( x ) ∣ ≤ ( x − R ) 2 ∥ f ∥ 1 . Reflection gives the similar estimates for the case x < − R x<-R x < − R
M ~ f ( x ) ≤ ∥ f ∥ 1 ∣ x ∣ − R , ∣ ( M ~ f ) ′ ( x ) ∣ ≤ ∥ f ∥ 1 ( ∣ x ∣ − R ) 2 for all ∣ x ∣ > R . (27) \widetilde{M}f(x)\le\frac{\lVert f\rVert_1}{|x|-R},\qquad|(\widetilde{M}f)'(x)|\le\frac{\lVert f\rVert_1}{(|x|-R)^2}\quad\text{for all }|x|>R. \tag*{(27)} M f ( x ) ≤ ∣ x ∣ − R ∥ f ∥ 1 , ∣ ( M f ) ′ ( x ) ∣ ≤ ( ∣ x ∣ − R ) 2 ∥ f ∥ 1 for all ∣ x ∣ > R . (27) On [ − R − 1 , R + 1 ] [-R-1,R+1] [ − R − 1 , R + 1 ] ∣ ( M ~ f ) ′ ( x ) ∣ ≤ Lip ( f ) |(\widetilde{M}f)'(x)|\le\operatorname{Lip}(f) ∣ ( M f ) ′ ( x ) ∣ ≤ Lip ( f ) (17) ; outside [ − R − 1 , R + 1 ] [-R-1,R+1] [ − R − 1 , R + 1 ] (27) applies. Thus ∫ R ∣ ( M ~ f ) ′ ( x ) ∣ r d x < ∞ \int_{\mathbb{R}}|(\widetilde{M}f)'(x)|^r\,\mathrm{d}x<\infty ∫ R ∣ ( M f ) ′ ( x ) ∣ r d x < ∞ r > 1 / 2 r>1/2 r > 1/2 f f f f ′ ( a ( x ) ) f'(a(x)) f ′ ( a ( x )) E E E ∣ f ′ ( a ( x ) ) ∣ |f'(a(x))| ∣ f ′ ( a ( x )) ∣ x ∈ E x\in E x ∈ E
Our goal is the following proposition:
Proposition 8. Let 1 < p < ∞ 1 < p < \infty 1 < p < ∞ f f f
∫ R f ′ ( x ) ( M ~ f ) ′ ( x ) ∣ ( M ~ f ) ′ ( x ) ∣ p − 2 d x ≥ ( 2 − 1 p ) ∫ R ∣ ( M ~ f ) ′ ( x ) ∣ p d x + p − 1 p ∫ E ∣ ( M ~ f ) ′ ( x ) ∣ p + 1 ∣ f ′ ( a ( x ) ) ∣ d x , (28) \int_{\mathbb{R}} f'(x)(\widetilde{M}f)'(x)\lvert(\widetilde{M}f)'(x)\rvert^{p-2}\,dx
\ge\left(2-\frac{1}{p}\right)\int_{\mathbb{R}}\lvert(\widetilde{M}f)'(x)\rvert^p\,dx+\frac{p-1}{p}\int_E\frac{\lvert(\widetilde{M}f)'(x)\rvert^{p+1}}{\lvert f'(a(x))\rvert}\,dx, \tag*{(28)} ∫ R f ′ ( x ) ( M f ) ′ ( x ) ∣( M f ) ′ ( x ) ∣ p − 2 d x ≥ ( 2 − p 1 ) ∫ R ∣( M f ) ′ ( x ) ∣ p d x + p p − 1 ∫ E ∣ f ′ ( a ( x ))∣ ∣( M f ) ′ ( x ) ∣ p + 1 d x , (28) ∫ E ∣ f ′ ( a ( x ) ) ∣ p − 1 ∣ ( M ~ f ) ′ ( x ) ∣ d x ≤ ∫ R ∣ f ′ ( x ) ∣ p d x . (29) \int_E\lvert f'(a(x))\rvert^{p-1}\lvert(\widetilde{M}f)'(x)\rvert\,dx\le\int_{\mathbb{R}}\lvert f'(x)\rvert^p\,dx. \tag*{(29)} ∫ E ∣ f ′ ( a ( x )) ∣ p − 1 ∣( M f ) ′ ( x )∣ d x ≤ ∫ R ∣ f ′ ( x ) ∣ p d x . (29) To prove (28) , we use a signed identity that treats increasing and decreasing intervals together. Choose the refined partition P ∗ \mathcal{P}^* P ∗
− ∞ = t 0 < t 1 < ⋯ < t m < t m + 1 = + ∞ . -\infty=t_0<t_1<\cdots<t_m<t_{m+1}=+\infty. − ∞ = t 0 < t 1 < ⋯ < t m < t m + 1 = + ∞. On each open interval ( t j , t j + 1 ) (t_j,t_{j+1}) ( t j , t j + 1 ) ( M ~ f ) ′ ( x ) (\widetilde{M}f)'(x) ( M f ) ′ ( x ) E − E_- E − E + E_+ E + (21) , reads
f ′ ( x ) = ( 2 − a ′ ( x ) ) ( M ~ f ) ′ ( x ) + ( x − a ( x ) ) ( M ~ f ) ′ ′ ( x ) . f'(x)=(2-a'(x))(\widetilde{M}f)'(x)+(x-a(x))(\widetilde{M}f)''(x). f ′ ( x ) = ( 2 − a ′ ( x )) ( M f ) ′ ( x ) + ( x − a ( x )) ( M f ) ′′ ( x ) . It holds for either sign of ( M ~ f ) ′ ( x ) (\widetilde{M}f)'(x) ( M f ) ′ ( x ) x − a ( x ) > 0 x-a(x)>0 x − a ( x ) > 0 x − a ( x ) < 0 x-a(x)<0 x − a ( x ) < 0 ( M ~ f ) ′ ( x ) ∣ ( M ~ f ) ′ ( x ) ∣ p − 2 (\widetilde{M}f)'(x)\lvert(\widetilde{M}f)'(x)\rvert^{p-2} ( M f ) ′ ( x ) ∣( M f ) ′ ( x ) ∣ p − 2 [ u , v ] [u,v] [ u , v ]
∫ u v f ′ ( x ) ( M ~ f ) ′ ( x ) ∣ ( M ~ f ) ′ ( x ) ∣ p − 2 d x = ∫ u v ( 2 − a ′ ( x ) ) ∣ ( M ~ f ) ′ ( x ) ∣ p d x + ∫ u v ( x − a ( x ) ) ( M ~ f ) ′ ′ ( x ) ( M ~ f ) ′ ( x ) ∣ ( M ~ f ) ′ ( x ) ∣ p − 2 d x . (30) \int_u^v f'(x)(\widetilde{M}f)'(x)\lvert(\widetilde{M}f)'(x)\rvert^{p-2}\,dx
=\int_u^v(2-a'(x))\lvert(\widetilde{M}f)'(x)\rvert^p\,dx+\int_u^v(x-a(x))(\widetilde{M}f)''(x)(\widetilde{M}f)'(x)\lvert(\widetilde{M}f)'(x)\rvert^{p-2}\,dx. \tag*{(30)} ∫ u v f ′ ( x ) ( M f ) ′ ( x ) ∣( M f ) ′ ( x ) ∣ p − 2 d x = ∫ u v ( 2 − a ′ ( x )) ∣( M f ) ′ ( x ) ∣ p d x + ∫ u v ( x − a ( x )) ( M f ) ′′ ( x ) ( M f ) ′ ( x ) ∣( M f ) ′ ( x ) ∣ p − 2 d x . (30) Observe that
d d x ∣ ( M ~ f ) ′ ( x ) ∣ p = p ( M ~ f ) ′ ( x ) ∣ ( M ~ f ) ′ ( x ) ∣ p − 2 ( M ~ f ) ′ ′ ( x ) , \frac{d}{dx}\lvert(\widetilde{M}f)'(x)\rvert^p=p(\widetilde{M}f)'(x)\lvert(\widetilde{M}f)'(x)\rvert^{p-2}(\widetilde{M}f)''(x), d x d ∣( M f ) ′ ( x ) ∣ p = p ( M f ) ′ ( x ) ∣( M f ) ′ ( x ) ∣ p − 2 ( M f ) ′′ ( x ) , which could be related to the second term of the right-hand side of equation (30) . Thus, by integration by parts,
∫ u v f ′ ( x ) ( M ~ f ) ′ ( x ) ∣ ( M ~ f ) ′ ( x ) ∣ p − 2 d x = ∫ u v ( 2 − a ′ ( x ) ) ∣ ( M ~ f ) ′ ( x ) ∣ p d x + ∫ u v x − a ( x ) p d d x ∣ ( M ~ f ) ′ ( x ) ∣ p d x = ( 2 − 1 p ) ∫ u v ∣ ( M ~ f ) ′ ( x ) ∣ p d x + 1 − p p ∫ u v a ′ ( x ) ∣ ( M ~ f ) ′ ( x ) ∣ p d x − [ ( M ~ f ( x ) − f ( x ) ) ( M ~ f ) ′ ( x ) ∣ ( M ~ f ) ′ ( x ) ∣ p − 2 p ] x = u x = v . (31) \begin{aligned}
\int_u^v f'(x)(\widetilde{M}f)'(x)\lvert(\widetilde{M}f)'(x)\rvert^{p-2}\,dx
={}&\int_u^v(2-a'(x))\lvert(\widetilde{M}f)'(x)\rvert^p\,dx+\int_u^v\frac{x-a(x)}{p}\frac{d}{dx}\lvert(\widetilde{M}f)'(x)\rvert^p\,dx\\
={}&\left(2-\frac{1}{p}\right)\int_u^v\lvert(\widetilde{M}f)'(x)\rvert^p\,dx+\frac{1-p}{p}\int_u^v a'(x)\lvert(\widetilde{M}f)'(x)\rvert^p\,dx\\
&-\left[\frac{(\widetilde{M}f(x)-f(x))(\widetilde{M}f)'(x)\lvert(\widetilde{M}f)'(x)\rvert^{p-2}}{p}\right]_{x=u}^{x=v}. \tag*{(31)}
\end{aligned} ∫ u v f ′ ( x ) ( M f ) ′ ( x ) ∣( M f ) ′ ( x ) ∣ p − 2 d x = = ∫ u v ( 2 − a ′ ( x )) ∣( M f ) ′ ( x ) ∣ p d x + ∫ u v p x − a ( x ) d x d ∣( M f ) ′ ( x ) ∣ p d x ( 2 − p 1 ) ∫ u v ∣( M f ) ′ ( x ) ∣ p d x + p 1 − p ∫ u v a ′ ( x ) ∣( M f ) ′ ( x ) ∣ p d x − [ p ( M f ( x ) − f ( x )) ( M f ) ′ ( x ) ∣( M f ) ′ ( x ) ∣ p − 2 ] x = u x = v . (31) Here we used
( x − a ( x ) ) ( M ~ f ) ′ ( x ) = f ( x ) − M ~ f ( x ) (x-a(x))(\widetilde{M}f)'(x)=f(x)-\widetilde{M}f(x) ( x − a ( x )) ( M f ) ′ ( x ) = f ( x ) − M f ( x ) by (20) . Also, by equations (21) and (24) , for every x ∈ E x\in E x ∈ E
− a ′ ( x ) ∣ ( M ~ f ) ′ ( x ) ∣ p = ∣ ( M ~ f ) ′ ( x ) ∣ p + 1 ∣ f ′ ( a ( x ) ) ∣ ( x ∈ E ) . (32) -a'(x)\lvert(\widetilde{M}f)'(x)\rvert^p=\frac{\lvert(\widetilde{M}f)'(x)\rvert^{p+1}}{\lvert f'(a(x))\rvert}\qquad(x\in E). \tag*{(32)} − a ′ ( x ) ∣( M f ) ′ ( x ) ∣ p = ∣ f ′ ( a ( x ))∣ ∣( M f ) ′ ( x ) ∣ p + 1 ( x ∈ E ) . (32) Passing to the endpoints of a smooth interval. We first let u ↓ t j u\downarrow t_j u ↓ t j v ↑ t j + 1 v\uparrow t_{j+1} v ↑ t j + 1 (31) are integrable, by the bounds at the beginning of this subsection and (2.16). Thus the bulk integrals converge by dominated convergence theorem. Next, we will examine the contribution of a finite/infinite partition point t j t_j t j (31) below.
Consider the integral in (31) and let u ↓ t j u \downarrow t_j u ↓ t j v ↑ t j + 1 v \uparrow t_{j+1} v ↑ t j + 1 t j t_j t j P ∗ \mathcal{P}^* P ∗ t j t_j t j
R j ± : = ± M ~ f ( t j ) − f ( t j ) p ( M ~ f ) ′ ( t j ± ) ∣ ( M ~ f ) ′ ( t j ± ) ∣ p − 2 , 1 ≤ j ≤ m . R_j^{\pm} := \pm\frac{\widetilde{M}f(t_j)-f(t_j)}{p}(\widetilde{M}f)'(t_j\pm)|(\widetilde{M}f)'(t_j\pm)|^{p-2}, \qquad1 \leq j \leq m. R j ± := ± p M f ( t j ) − f ( t j ) ( M f ) ′ ( t j ± ) ∣ ( M f ) ′ ( t j ± ) ∣ p − 2 , 1 ≤ j ≤ m . where r ∣ r ∣ p − 2 r|r|^{p-2} r ∣ r ∣ p − 2 0 0 0 r = 0 r=0 r = 0
R 0 + : = lim x → − ∞ M ~ f ( x ) − f ( x ) p ( M ~ f ) ′ ( x ) ∣ ( M ~ f ) ′ ( x ) ∣ p − 2 , R_0^+ := \lim_{x\to-\infty}\frac{\widetilde{M}f(x)-f(x)}{p}(\widetilde{M}f)'(x)|(\widetilde{M}f)'(x)|^{p-2}, R 0 + := x → − ∞ lim p M f ( x ) − f ( x ) ( M f ) ′ ( x ) ∣ ( M f ) ′ ( x ) ∣ p − 2 , R m + 1 − : = − lim x → + ∞ M ~ f ( x ) − f ( x ) p ( M ~ f ) ′ ( x ) ∣ ( M ~ f ) ′ ( x ) ∣ p − 2 . R_{m+1}^- := -\lim_{x\to+\infty}\frac{\widetilde{M}f(x)-f(x)}{p}(\widetilde{M}f)'(x)|(\widetilde{M}f)'(x)|^{p-2}. R m + 1 − := − x → + ∞ lim p M f ( x ) − f ( x ) ( M f ) ′ ( x ) ∣ ( M f ) ′ ( x ) ∣ p − 2 . Lemma 9. The boundary term in (31) equals R j + + R j + 1 − R_j^+ + R_{j+1}^- R j + + R j + 1 − 1 ≤ j ≤ m 1 \leq j \leq m 1 ≤ j ≤ m
R j + + R j − ≥ 0 , R 0 + = R m + 1 − = 0. R_j^+ + R_j^- \geq0, \qquad R_0^+ = R_{m+1}^- = 0. R j + + R j − ≥ 0 , R 0 + = R m + 1 − = 0. Consequently, the total boundary contribution is nonnegative:
∑ j = 0 m ( R j + + R j + 1 − ) = ∑ j = 1 m ( R j + + R j − ) ≥ 0. \sum_{j=0}^{m}(R_j^+ + R_{j+1}^-) = \sum_{j=1}^{m}(R_j^+ + R_j^-) \geq0. j = 0 ∑ m ( R j + + R j + 1 − ) = j = 1 ∑ m ( R j + + R j − ) ≥ 0. Proof. The derivative ( M ~ f ) ′ (\widetilde{M}f)' ( M f ) ′ M ~ f − f \widetilde{M}f-f M f − f (31) satisfies
− [ M ~ f ( x ) − f ( x ) p ( M ~ f ) ′ ( x ) ∣ ( M ~ f ) ′ ( x ) ∣ p − 2 ] x = u x = v ⟶ R j + + R j + 1 − . -\left[\frac{\widetilde{M}f(x)-f(x)}{p}(\widetilde{M}f)'(x)|(\widetilde{M}f)'(x)|^{p-2}\right]_{x=u}^{x=v} \longrightarrow R_j^+ + R_{j+1}^-. − [ p M f ( x ) − f ( x ) ( M f ) ′ ( x ) ∣ ( M f ) ′ ( x ) ∣ p − 2 ] x = u x = v ⟶ R j + + R j + 1 − . Fix a finite partition point t j t_j t j M ~ f ( t j ) = f ( t j ) \widetilde{M}f(t_j)=f(t_j) M f ( t j ) = f ( t j ) R j + = R j − = 0 R_j^+=R_j^-=0 R j + = R j − = 0 t j ∈ Ω t_j\in\Omega t j ∈ Ω
( M ~ f ) ′ ( t j − ) ≤ ( M ~ f ) ′ ( t j + ) . (\widetilde{M}f)'(t_j-) \leq(\widetilde{M}f)'(t_j+). ( M f ) ′ ( t j − ) ≤ ( M f ) ′ ( t j + ) . Since r ↦ r ∣ r ∣ p − 2 r\mapsto r|r|^{p-2} r ↦ r ∣ r ∣ p − 2 p > 1 p>1 p > 1
R j + + R j − = M ~ f ( t j ) − f ( t j ) p ( ( M ~ f ) ′ ( t j + ) ∣ ( M ~ f ) ′ ( t j + ) ∣ p − 2 − ( M ~ f ) ′ ( t j − ) ∣ ( M ~ f ) ′ ( t j − ) ∣ p − 2 ) ≥ 0. R_j^+ + R_j^- = \frac{\widetilde{M}f(t_j)-f(t_j)}{p}\left((\widetilde{M}f)'(t_j+)|(\widetilde{M}f)'(t_j+)|^{p-2}-(\widetilde{M}f)'(t_j-)|(\widetilde{M}f)'(t_j-)|^{p-2}\right)\geq0. R j + + R j − = p M f ( t j ) − f ( t j ) ( ( M f ) ′ ( t j + ) ∣ ( M f ) ′ ( t j + ) ∣ p − 2 − ( M f ) ′ ( t j − ) ∣ ( M f ) ′ ( t j − ) ∣ p − 2 ) ≥ 0. Finally, choose R > 0 R>0 R > 0 supp f ⊂ [ − R , R ] \operatorname{supp} f\subset[-R,R] supp f ⊂ [ − R , R ] ∣ x ∣ > R |x|>R ∣ x ∣ > R
∣ M ~ f ( x ) − f ( x ) p ( M ~ f ) ′ ( x ) ∣ ( M ~ f ) ′ ( x ) ∣ p − 2 ∣ ≤ ∥ f ∥ 1 p p ( ∣ x ∣ − R ) 2 p − 1 → 0. \left|\frac{\widetilde{M}f(x)-f(x)}{p}(\widetilde{M}f)'(x)|(\widetilde{M}f)'(x)|^{p-2}\right|\leq\frac{\lVert f\rVert_1^p}{p(|x|-R)^{2p-1}}\to0. p M f ( x ) − f ( x ) ( M f ) ′ ( x ) ∣ ( M f ) ′ ( x ) ∣ p − 2 ≤ p ( ∣ x ∣ − R ) 2 p − 1 ∥ f ∥ 1 p → 0. Thus R 0 + = R m + 1 − = 0 R_0^+=R_{m+1}^-=0 R 0 + = R m + 1 − = 0
Now, we are ready to prove Proposition 8.
Proof of Proposition 8. Now sum (2.15) over the finitely many nonconstant intervals ( t j , t j + 1 ) (t_j,t_{j+1}) ( t j , t j + 1 )
M ~ f ( t j ) − f ( t j ) p ( ( M ~ f ) ′ ( t j + ) ∣ ( M ~ f ) ′ ( t j + ) ∣ p − 2 − ( M ~ f ) ′ ( t j − ) ∣ ( M ~ f ) ′ ( t j − ) ∣ p − 2 ) , (33) \frac{\widetilde{M}f(t_j)-f(t_j)}{p}\left((\widetilde{M}f)'(t_j+)|(\widetilde{M}f)'(t_j+)|^{p-2}-(\widetilde{M}f)'(t_j-)|(\widetilde{M}f)'(t_j-)|^{p-2}\right), \tag*{(33)} p M f ( t j ) − f ( t j ) ( ( M f ) ′ ( t j + ) ∣ ( M f ) ′ ( t j + ) ∣ p − 2 − ( M f ) ′ ( t j − ) ∣ ( M f ) ′ ( t j − ) ∣ p − 2 ) , (33) and all those contributions are nonnegative. There is no contribution from infinity. Outside E E E ( M ~ f ) ′ ( x ) (\widetilde{M}f)'(x) ( M f ) ′ ( x ) (31) and (2.16, we conclude that
∫ R f ′ ( x ) ( M ~ f ) ′ ( x ) ∣ ( M ~ f ) ′ ( x ) ∣ p − 2 d x ≥ ( 2 − 1 p ) ∫ R ∣ ( M ~ f ) ′ ( x ) ∣ p d x + p − 1 p ∫ E ∣ ( M ~ f ) ′ ( x ) ∣ p + 1 ∣ f ′ ( a ( x ) ) ∣ d x . \int_{\mathbb{R}} f'(x)(\widetilde{M}f)'(x)\lvert(\widetilde{M}f)'(x)\rvert^{p-2}\,dx
\geq\left(2-\frac{1}{p}\right)\int_{\mathbb{R}}\lvert(\widetilde{M}f)'(x)\rvert^p\,dx+\frac{p-1}{p}\int_E\frac{\lvert(\widetilde{M}f)'(x)\rvert^{p+1}}{\lvert f'(a(x))\rvert}\,dx. ∫ R f ′ ( x ) ( M f ) ′ ( x ) ∣( M f ) ′ ( x ) ∣ p − 2 d x ≥ ( 2 − p 1 ) ∫ R ∣( M f ) ′ ( x ) ∣ p d x + p p − 1 ∫ E ∣ f ′ ( a ( x ))∣ ∣( M f ) ′ ( x ) ∣ p + 1 d x . This completes the proof of (28) .
For (29) , use the first formula in (21) and change variables on each smooth interval J ⊂ E J\subset E J ⊂ E
∫ J ∣ f ′ ( a ( x ) ) ∣ p − 1 ∣ ( M ~ f ) ′ ( x ) ∣ d x = ∫ J ∣ f ′ ( a ( x ) ) ∣ p ∣ a ′ ( x ) ∣ d x = ∫ { a ( x ) : x ∈ J } ∣ f ′ ( t ) ∣ p d t . \begin{aligned}
\int_J\lvert f'(a(x))\rvert^{p-1}\lvert(\widetilde{M}f)'(x)\rvert\,dx
=\int_J\lvert f'(a(x))\rvert^p\lvert a'(x)\rvert\,dx \\
&=\int_{\{a(x):x\in J\}}\lvert f'(t)\rvert^p\,dt.
\end{aligned} ∫ J ∣ f ′ ( a ( x )) ∣ p − 1 ∣( M f ) ′ ( x )∣ d x = ∫ J ∣ f ′ ( a ( x )) ∣ p ∣ a ′ ( x )∣ d x = ∫ { a ( x ) : x ∈ J } ∣ f ′ ( t ) ∣ p d t . Lemma 7 makes these image sets pairwise disjoint, also between the two signs of ( M ~ f ) ′ (\widetilde{M}f)' ( M f ) ′ ∫ R ∣ f ′ ( t ) ∣ p d t \int_{\mathbb{R}}\lvert f'(t)\rvert^p\,dt ∫ R ∣ f ′ ( t ) ∣ p d t (29) .
Next, we will first prove Theorem 1 for those functions f ≥ 0 f\geq0 f ≥ 0 Theorem 1 is valid for general function f f f Theorem 1 .
Proof of (9) in Theorem 1 for piecewise-affine f f f Hölder’s inequality and (29) give
∥ ( M ~ f ) ′ ∥ p p = ∫ E ( ∣ ( M ~ f ) ′ ( x ) ∣ p + 1 ∣ f ′ ( a ( x ) ) ∣ ) ( p − 1 ) / p ( ∣ f ′ ( a ( x ) ) ∣ p − 1 ∣ ( M ~ f ) ′ ( x ) ∣ ) 1 / p d x ≤ ( ∫ E ∣ ( M ~ f ) ′ ( x ) ∣ p + 1 ∣ f ′ ( a ( x ) ) ∣ d x ) ( p − 1 ) / p ∥ f ′ ∥ p . \begin{aligned}
\lVert(\widetilde{M}f)'\rVert_p^p
=\int_E\left(\frac{\lvert(\widetilde{M}f)'(x)\rvert^{p+1}}{\lvert f'(a(x))\rvert}\right)^{(p-1)/p}\left(\lvert f'(a(x))\rvert^{p-1}\lvert(\widetilde{M}f)'(x)\rvert\right)^{1/p}\,dx \\
&\leq\left(\int_E\frac{\lvert(\widetilde{M}f)'(x)\rvert^{p+1}}{\lvert f'(a(x))\rvert}\,dx\right)^{(p-1)/p}\lVert f'\rVert_p.
\end{aligned} ∥( M f ) ′ ∥ p p = ∫ E ( ∣ f ′ ( a ( x ))∣ ∣( M f ) ′ ( x ) ∣ p + 1 ) ( p − 1 ) / p ( ∣ f ′ ( a ( x )) ∣ p − 1 ∣( M f ) ′ ( x )∣ ) 1/ p d x ≤ ( ∫ E ∣ f ′ ( a ( x ))∣ ∣( M f ) ′ ( x ) ∣ p + 1 d x ) ( p − 1 ) / p ∥ f ′ ∥ p . If ∥ ( M ~ f ) ′ ∥ p = 0 \lVert(\widetilde{M}f)'\rVert_p=0 ∥( M f ) ′ ∥ p = 0 ∥ f ′ ∥ p > 0 \lVert f'\rVert_p>0 ∥ f ′ ∥ p > 0
∫ E ∣ ( M ~ f ) ′ ( x ) ∣ p + 1 ∣ f ′ ( a ( x ) ) ∣ d x ≥ ∥ ( M ~ f ) ′ ∥ p p 2 / ( p − 1 ) ∥ f ′ ∥ p p / ( p − 1 ) . \int_E\frac{\lvert(\widetilde{M}f)'(x)\rvert^{p+1}}{\lvert f'(a(x))\rvert}\,dx\geq\frac{\lVert(\widetilde{M}f)'\rVert_p^{p^2/(p-1)}}{\lVert f'\rVert_p^{p/(p-1)}}. ∫ E ∣ f ′ ( a ( x ))∣ ∣( M f ) ′ ( x ) ∣ p + 1 d x ≥ ∥ f ′ ∥ p p / ( p − 1 ) ∥( M f ) ′ ∥ p p 2 / ( p − 1 ) . A second application of Hölder to the left side of (28) yields
∥ f ′ ∥ p ∥ ( M ~ f ) ′ ∥ p p − 1 ≥ ( 2 − 1 p ) ∥ ( M ~ f ) ′ ∥ p p + p − 1 p ∥ ( M ~ f ) ′ ∥ p p 2 / ( p − 1 ) ∥ f ′ ∥ p p / ( p − 1 ) . \lVert f'\rVert_p\lVert(\widetilde{M}f)'\rVert_p^{p-1}
\geq\left(2-\frac{1}{p}\right)\lVert(\widetilde{M}f)'\rVert_p^p+\frac{p-1}{p}\frac{\lVert(\widetilde{M}f)'\rVert_p^{p^2/(p-1)}}{\lVert f'\rVert_p^{p/(p-1)}}. ∥ f ′ ∥ p ∥( M f ) ′ ∥ p p − 1 ≥ ( 2 − p 1 ) ∥( M f ) ′ ∥ p p + p p − 1 ∥ f ′ ∥ p p / ( p − 1 ) ∥( M f ) ′ ∥ p p 2 / ( p − 1 ) . After division and multiplication by p p p
( 2 p − 1 ) ∥ ( M ~ f ) ′ ∥ p ∥ f ′ ∥ p + ( p − 1 ) ( ∥ ( M ~ f ) ′ ∥ p ∥ f ′ ∥ p ) ( 2 p − 1 ) / ( p − 1 ) ≤ p . (2p-1)\frac{\lVert(\widetilde{M}f)'\rVert_p}{\lVert f'\rVert_p}+(p-1)\left(\frac{\lVert(\widetilde{M}f)'\rVert_p}{\lVert f'\rVert_p}\right)^{(2p-1)/(p-1)}\leq p. ( 2 p − 1 ) ∥ f ′ ∥ p ∥( M f ) ′ ∥ p + ( p − 1 ) ( ∥ f ′ ∥ p ∥( M f ) ′ ∥ p ) ( 2 p − 1 ) / ( p − 1 ) ≤ p . This gives (8) . Additionally, since ( 2 p − 1 ) x + ( p − 1 ) x ( 2 p − 1 ) / ( p − 1 ) (2p-1)x+(p-1)x^{(2p-1)/(p-1)} ( 2 p − 1 ) x + ( p − 1 ) x ( 2 p − 1 ) / ( p − 1 ) (8) has a unique solution γ p \gamma_p γ p [ 0 , 1 ] [0,1] [ 0 , 1 ] (9) for piecewise-affine inputs. ▫
Approximation
We will use lower semicontinuity of the Riesz variation under pointwise convergence: Assume u j ( x ) → u ( x ) ∈ R u_j(x) \to u(x) \in\mathbb{R} u j ( x ) → u ( x ) ∈ R x x x 1 ≤ p < ∞ 1 \le p < \infty 1 ≤ p < ∞ P = { x 0 < ⋯ < x m } P = \{x_0 < \cdots< x_m\} P = { x 0 < ⋯ < x m }
S P ( u ) : = ( ∑ j = 1 m ∣ u ( x j ) − u ( x j − 1 ) ∣ p ∣ x j − x j − 1 ∣ p − 1 ) 1 p . S_P(u) := \left(\sum_{j=1}^{m} \frac{\lvert u(x_j)-u(x_{j-1})\rvert^p}{\lvert x_j-x_{j-1}\rvert^{p-1}}\right)^{\frac{1}{p}}. S P ( u ) := ( j = 1 ∑ m ∣ x j − x j − 1 ∣ p − 1 ∣ u ( x j ) − u ( x j − 1 ) ∣ p ) p 1 . Then, by the convergence and the definition, for all P P P
S P ( u ) = lim inf n → ∞ S P ( u n ) ≤ lim inf n → ∞ Var p ( u n ) ⇒ Var p ( u ) ≤ lim inf n → ∞ Var p ( u n ) . (34) S_P(u) = \liminf_{n\to\infty} S_P(u_n) \le\liminf_{n\to\infty} \operatorname{Var}_p(u_n) \Rightarrow\operatorname{Var}_p(u) \le\liminf_{n\to\infty} \operatorname{Var}_p(u_n). \tag*{(34)} S P ( u ) = n → ∞ lim inf S P ( u n ) ≤ n → ∞ lim inf Var p ( u n ) ⇒ Var p ( u ) ≤ n → ∞ lim inf Var p ( u n ) . (34) Indeed, pass to the limit along each fixed finite partition in (1.7), and then take the supremum over partitions. For p > 1 p>1 p > 1 Var p ( u ) = ∥ u ′ ∥ p \operatorname{Var}_p(u)=\lVert u'\rVert_p Var p ( u ) = ∥ u ′ ∥ p [15 Chapter IX §4, Theorem 7] .
Proof of (1.9) in Theorem 1. Assume that f ≥ 0 f\ge0 f ≥ 0 W 1 , p ( R ) W^{1,p}(\mathbb{R}) W 1 , p ( R ) [ A , B ] [A,B] [ A , B ] supp f \operatorname{supp} f supp f f j f_j f j [ A , B ] [A,B] [ A , B ] [ A , B ] [A,B] [ A , B ] f j → f f_j\to f f j → f I I I
f j ′ ∣ I = 1 ∣ I ∣ ∫ I f ′ ( t ) d t . f'_j\big|_I = \frac{1}{\lvert I\rvert}\int_I f'(t)\,\mathrm{d}t. f j ′ I = ∣ I ∣ 1 ∫ I f ′ ( t ) d t . The corresponding cell-average operators are contractions on L p ( [ A , B ] ) L^p([A,B]) L p ([ A , B ]) L p ( [ A , B ] ) L^p([A,B]) L p ([ A , B ]) [17 Theorem 3.14] , it follows that f j ′ → f ′ f'_j\to f' f j ′ → f ′ L p ( R ) L^p(\mathbb{R}) L p ( R ) f f f
∥ M ~ f − M ~ g ∥ ∞ ≤ ∥ f − g ∥ ∞ (35) \lVert\widetilde{M}f-\widetilde{M}g\rVert_\infty\le\lVert f-g\rVert_\infty \tag*{(35)} ∥ M f − M g ∥ ∞ ≤ ∥ f − g ∥ ∞ (35) gives uniform convergence M ~ f j → M ~ f \widetilde{M}f_j\to\widetilde{M}f M f j → M f f j f_j f j f j f_j f j
Var p ( M ~ f ) ≤ lim inf j → ∞ Var p ( M ~ f j ) ≤ lim inf j → ∞ γ p ∥ f j ′ ∥ p = γ p ∥ f ′ ∥ p . (36) \operatorname{Var}_p(\widetilde{M}f)\le\liminf_{j\to\infty}\operatorname{Var}_p(\widetilde{M}f_j)\le\liminf_{j\to\infty}\gamma_p\lVert f'_j\rVert_p=\gamma_p\lVert f'\rVert_p. \tag*{(36)} Var p ( M f ) ≤ j → ∞ lim inf Var p ( M f j ) ≤ j → ∞ lim inf γ p ∥ f j ′ ∥ p = γ p ∥ f ′ ∥ p . (36) In particular, since p > 1 p>1 p > 1 Var p ( M ~ f ) < ∞ \operatorname{Var}_p(\widetilde{M}f)<\infty Var p ( M f ) < ∞ M ~ f ∈ A C l o c ( R ) \widetilde{M}f\in AC_{\mathrm{loc}}(\mathbb{R}) M f ∈ A C loc ( R ) ( M ~ f ) ′ ∈ L p ( R ) (\widetilde{M}f)'\in L^p(\mathbb{R}) ( M f ) ′ ∈ L p ( R ) ∥ ( M ~ f ) ′ ∥ p = Var p ( M ~ f ) \lVert(\widetilde{M}f)'\rVert_p=\operatorname{Var}_p(\widetilde{M}f) ∥( M f ) ′ ∥ p = Var p ( M f ) f ∈ W 1 , p f\in W^{1,p} f ∈ W 1 , p
Next, suppose that f ≥ 0 f\ge0 f ≥ 0 f ′ ∈ L p f'\in L^p f ′ ∈ L p η R ( x ) \eta_R(x) η R ( x ) [ − R , R ] [-R,R] [ − R , R ] [ − 2 R , 2 R ] [-2R,2R] [ − 2 R , 2 R ]
∥ ( η R f ) ′ ∥ p ≤ ∥ f ′ ∥ p + ∥ f ∥ ∞ ( 2 R 1 − p ) 1 / p . \lVert(\eta_Rf)'\rVert_p\le\lVert f'\rVert_p+\lVert f\rVert_\infty(2R^{1-p})^{1/p}. ∥( η R f ) ′ ∥ p ≤ ∥ f ′ ∥ p + ∥ f ∥ ∞ ( 2 R 1 − p ) 1/ p . The error term ∥ f ∥ ∞ ( 2 R 1 − p ) 1 / p \lVert f\rVert_\infty(2R^{1-p})^{1/p} ∥ f ∥ ∞ ( 2 R 1 − p ) 1/ p R → ∞ R\to\infty R → ∞ 0 ≤ η R ( x ) f ( x ) ≤ f ( x ) 0\le\eta_R(x)f(x)\le f(x) 0 ≤ η R ( x ) f ( x ) ≤ f ( x ) f f f
M ~ ( η R f ) ( x ) ⟶ M ~ f ( x ) . (37) \widetilde{M}(\eta_Rf)(x)\longrightarrow\widetilde{M}f(x). \tag*{(37)} M ( η R f ) ( x ) ⟶ M f ( x ) . (37) All these maximal functions are finite, as they are bounded by ∥ f ∥ ∞ \lVert f\rVert_\infty ∥ f ∥ ∞ η R f \eta_Rf η R f
Var p ( M ~ f ) ≤ lim inf R → ∞ Var p ( M ~ η R f ) ≤ lim inf R → ∞ γ p ∥ ( η R f ) ′ ∥ p = γ p ∥ f ′ ∥ p . \operatorname{Var}_p(\widetilde{M}f)\le\liminf_{R\to\infty}\operatorname{Var}_p(\widetilde{M}\eta_Rf)\le\liminf_{R\to\infty}\gamma_p\lVert(\eta_Rf)'\rVert_p=\gamma_p\lVert f'\rVert_p. Var p ( M f ) ≤ R → ∞ lim inf Var p ( M η R f ) ≤ R → ∞ lim inf γ p ∥( η R f ) ′ ∥ p = γ p ∥ f ′ ∥ p . Similarly, by the boundedness of Var p ( M ~ f ) \operatorname{Var}_p(\widetilde{M}f) Var p ( M f ) ∥ ( M ~ f ) ′ ∥ p = Var p ( M ~ f ) \lVert(\widetilde{M}f)'\rVert_p=\operatorname{Var}_p(\widetilde{M}f) ∥( M f ) ′ ∥ p = Var p ( M f ) f f f
For a general nonnegative function f f f M ~ f ( x 0 ) < ∞ \widetilde{M}f(x_0)<\infty M f ( x 0 ) < ∞ [ a , b ] [a,b] [ a , b ] x 0 x_0 x 0 h ∈ R h\in\mathbb{R} h ∈ R
1 b − a ∫ a + h b + h f − 1 b − a ∫ a b f = 1 b − a ∫ a b ∫ t t + h f ′ ( u ) d u d t ≤ 1 b − a ∫ a b ∥ f ′ ∥ p ∣ h ∣ 1 − 1 / p d t = ∥ f ′ ∥ p ∣ h ∣ 1 − 1 / p . \begin{aligned}
\frac{1}{b-a}\int_{a+h}^{b+h}f-\frac{1}{b-a}\int_a^b f=\frac{1}{b-a}\int_a^b\int_t^{t+h}f'(u)\,\mathrm{d}u\mathrm{d}t \\
&\leq\frac{1}{b-a}\int_a^b\lVert f'\rVert_p|h|^{1-1/p}\,\mathrm{d}t=\lVert f'\rVert_p|h|^{1-1/p}.
\end{aligned} b − a 1 ∫ a + h b + h f − b − a 1 ∫ a b f = b − a 1 ∫ a b ∫ t t + h f ′ ( u ) d u d t ≤ b − a 1 ∫ a b ∥ f ′ ∥ p ∣ h ∣ 1 − 1/ p d t = ∥ f ′ ∥ p ∣ h ∣ 1 − 1/ p . Taking supremum over all intervals [ a , b ] [a,b] [ a , b ] x 0 x_0 x 0
M ~ f ( x 0 + h ) = sup x 0 ∈ [ a , b ] 1 b − a ∫ a + h b + h f ≤ M ~ f ( x 0 ) + ∥ f ′ ∥ p ∣ h ∣ 1 − 1 / p . \widetilde{M}f(x_0+h)=\sup_{x_0\in[a,b]}\frac{1}{b-a}\int_{a+h}^{b+h}f\leq\widetilde{M}f(x_0)+\lVert f'\rVert_p|h|^{1-1/p}. M f ( x 0 + h ) = x 0 ∈ [ a , b ] sup b − a 1 ∫ a + h b + h f ≤ M f ( x 0 ) + ∥ f ′ ∥ p ∣ h ∣ 1 − 1/ p . Additionally, replacing h h h − h -h − h x 0 x_0 x 0 ( x 0 + h ) (x_0+h) ( x 0 + h )
M ~ f ( x 0 ) = M ~ f ( ( x 0 + h ) − h ) ≤ M ~ f ( x 0 + h ) + ∥ f ′ ∥ p ∣ h ∣ 1 − 1 / p . \widetilde{M}f(x_0)=\widetilde{M}f((x_0+h)-h)\leq\widetilde{M}f(x_0+h)+\lVert f'\rVert_p|h|^{1-1/p}. M f ( x 0 ) = M f (( x 0 + h ) − h ) ≤ M f ( x 0 + h ) + ∥ f ′ ∥ p ∣ h ∣ 1 − 1/ p . To sum up,
∣ M ~ f ( x 0 + h ) − M ~ f ( x 0 ) ∣ ≤ ∥ f ′ ∥ p ∣ h ∣ 1 − 1 / p . \left|\widetilde{M}f(x_0+h)-\widetilde{M}f(x_0)\right|\leq\lVert f'\rVert_p|h|^{1-1/p}. M f ( x 0 + h ) − M f ( x 0 ) ≤ ∥ f ′ ∥ p ∣ h ∣ 1 − 1/ p . Thus, once M ~ f \widetilde{M}f M f f K ( x ) = min { f ( x ) , K } f_K(x)=\min\{f(x),K\} f K ( x ) = min { f ( x ) , K } ∣ f K ′ ( x ) ∣ ≤ ∣ f ′ ( x ) ∣ |f'_K(x)|\leq|f'(x)| ∣ f K ′ ( x ) ∣ ≤ ∣ f ′ ( x ) ∣ M ~ f K ( x ) ↑ M ~ f ( x ) \widetilde{M}f_K(x)\mathbin{\uparrow}\widetilde{M}f(x) M f K ( x ) ↑ M f ( x ) f K f_K f K
Var p ( M ~ f ) ≤ lim inf K → ∞ Var p ( M ~ f K ) ≤ lim inf K → ∞ γ p ∥ f K ′ ∥ p ≤ γ p ∥ f ′ ∥ p . \operatorname{Var}_p(\widetilde{M}f)\leq\liminf_{K\to\infty}\operatorname{Var}_p(\widetilde{M}f_K)\leq\liminf_{K\to\infty}\gamma_p\lVert f'_K\rVert_p\leq\gamma_p\lVert f'\rVert_p. Var p ( M f ) ≤ K → ∞ lim inf Var p ( M f K ) ≤ K → ∞ lim inf γ p ∥ f K ′ ∥ p ≤ γ p ∥ f ′ ∥ p . Similarly, by the boundedness of Var p ( M ~ f ) \operatorname{Var}_p(\widetilde{M}f) Var p ( M f ) ∥ ( M ~ f ) ′ ∥ p = Var p ( M ~ f ) \lVert(\widetilde{M}f)'\rVert_p=\operatorname{Var}_p(\widetilde{M}f) ∥( M f ) ′ ∥ p = Var p ( M f ) f f f
Replacing a signed function f f f ∣ f ∣ |f| ∣ f ∣ M ~ ∣ f ∣ = M ~ f \widetilde{M}|f|=\widetilde{M}f M ∣ f ∣ = M f ∣ ( ∣ f ∣ ) ′ ( x ) ∣ ≤ ∣ f ′ ( x ) ∣ \left|(|f|)'(x)\right|\leq|f'(x)| ∣ ( ∣ f ∣ ) ′ ( x ) ∣ ≤ ∣ f ′ ( x ) ∣
Sharpness
For 0 < α ≤ 1 0<\alpha\leq1 0 < α ≤ 1
( 1 + α ) t α + α t 1 + α − 1 = 0. (1+\alpha)t^\alpha+\alpha t^{1+\alpha}-1=0. ( 1 + α ) t α + α t 1 + α − 1 = 0. The left-hand side is strictly increasing, negative at zero, and positive at one. Hence, there is a unique minimizer t α ∈ ( 0 , 1 ) t_\alpha\in(0,1) t α ∈ ( 0 , 1 )
( 1 + α ) t α α + α t α 1 + α = 1 , H α = 1 + t α 1 + α ( 1 + α ) ( 1 + t α ) = t α α . (1+\alpha)t_\alpha^\alpha+\alpha t_\alpha^{1+\alpha}=1,\qquad H_\alpha=\frac{1+t_\alpha^{1+\alpha}}{(1+\alpha)(1+t_\alpha)}=t_\alpha^\alpha. ( 1 + α ) t α α + α t α 1 + α = 1 , H α = ( 1 + α ) ( 1 + t α ) 1 + t α 1 + α = t α α . In particular,
( 1 + α ) H α + α H α 1 + 1 / α = 1. (38) (1+\alpha)H_\alpha+\alpha H_\alpha^{1+1/\alpha}=1. \tag*{(38)} ( 1 + α ) H α + α H α 1 + 1/ α = 1. (38) Taking α = 1 − 1 / p \alpha=1-1/p α = 1 − 1/ p H 1 − 1 / p = γ p H_{1-1/p}=\gamma_p H 1 − 1/ p = γ p
Proof of the sharpness of Theorem 1. Fix 1 − 1 / p < θ ≤ 1 1-1/p<\theta\leq1 1 − 1/ p < θ ≤ 1
f θ ( x ) = max { 1 − ∣ x ∣ θ , 0 } . f_\theta(x)=\max\{1-|x|^\theta,0\}. f θ ( x ) = max { 1 − ∣ x ∣ θ , 0 } . Then f θ ∈ W 1 , p ( R ) f_\theta\in W^{1,p}(\mathbb{R}) f θ ∈ W 1 , p ( R )
∥ f θ ′ ∥ p p = 2 θ p ∫ 0 1 x p ( θ − 1 ) d x = 2 θ p p θ − p + 1 . \lVert f'_\theta\rVert_p^p=2\theta^p\int_0^1x^{p(\theta-1)}\,\mathrm{d}x=\frac{2\theta^p}{p\theta-p+1}. ∥ f θ ′ ∥ p p = 2 θ p ∫ 0 1 x p ( θ − 1 ) d x = pθ − p + 1 2 θ p . For a symmetrically decreasing function supported on [ − 1 , 1 ] [-1,1] [ − 1 , 1 ] x > 0 x>0 x > 0 [ − t x , x ] [-tx,x] [ − t x , x ] 0 ≤ t ≤ 1 0\leq t\leq1 0 ≤ t ≤ 1 x x x f θ f_\theta f θ [ a , x ] [a,x] [ a , x ] f θ f_\theta f θ − x ≤ a ≤ 0 -x\leq a\leq0 − x ≤ a ≤ 0 [ − t x , x ] [-tx,x] [ − t x , x ] 0 ≤ t ≤ 1 0\leq t\leq1 0 ≤ t ≤ 1 0 < x < 1 0 < x < 1 0 < x < 1
M ~ f θ ( x ) = max 0 ≤ t ≤ 1 1 ( t + 1 ) x ∫ − t x x f θ ( y ) d y = 1 − x θ min 0 ≤ t ≤ 1 1 + t 1 + θ ( 1 + θ ) ( 1 + t ) = 1 − H θ x θ ⇒ ( M ~ f θ ) ′ ( x ) = − θ H θ x θ − 1 \begin{aligned}
\widetilde{M}f_\theta(x) = \max_{0 \le t \le1} \frac{1}{(t+1)x}\int_{-tx}^{x} f_\theta(y)\,dy = 1 - x^\theta\min_{0 \le t \le1} \frac{1+t^{1+\theta}}{(1+\theta)(1+t)} = 1 - H_\theta x^\theta\\
&\Rightarrow(\widetilde{M}f_\theta)'(x) = -\theta H_\theta x^{\theta-1}
\end{aligned} M f θ ( x ) = 0 ≤ t ≤ 1 max ( t + 1 ) x 1 ∫ − t x x f θ ( y ) d y = 1 − x θ 0 ≤ t ≤ 1 min ( 1 + θ ) ( 1 + t ) 1 + t 1 + θ = 1 − H θ x θ ⇒ ( M f θ ) ′ ( x ) = − θ H θ x θ − 1 Thus, by a straightforward estimate and (9) ,
∥ ( M ~ f θ ) ′ ∥ p p ≥ 2 ∫ 0 1 ( θ H θ x θ − 1 ) p d x = 2 θ p H θ p p ( θ − 1 ) + 1 = H θ p ∥ f θ ′ ∥ p p ⇒ H θ ≤ ∥ ( M ~ f θ ) ′ ∥ p ∥ f θ ′ ∥ p ≤ γ p \begin{aligned}
\left\|(\widetilde{M}f_\theta)'\right\|_p^p \ge2\int_0^1 \left(\theta H_\theta x^{\theta-1}\right)^p dx = \frac{2\theta^p H_\theta^p}{p(\theta-1)+1} = H_\theta^p\left\|f_\theta'\right\|_p^p \\
&\Rightarrow H_\theta\le\frac{\left\|(\widetilde{M}f_\theta)'\right\|_p}{\left\|f_\theta'\right\|_p} \le\gamma_p
\end{aligned} ( M f θ ) ′ p p ≥ 2 ∫ 0 1 ( θ H θ x θ − 1 ) p d x = p ( θ − 1 ) + 1 2 θ p H θ p = H θ p ∥ f θ ′ ∥ p p ⇒ H θ ≤ ∥ f θ ′ ∥ p ( M f θ ) ′ p ≤ γ p As θ ↓ 1 − 1 / p \theta\downarrow1 - 1/p θ ↓ 1 − 1/ p H θ → γ p H_\theta\to\gamma_p H θ → γ p
lim θ ↓ 1 − 1 / p ∥ ( M ~ f θ ) ′ ∥ p ∥ f θ ′ ∥ p = γ p . \lim_{\theta\downarrow1-1/p} \frac{\left\|(\widetilde{M}f_\theta)'\right\|_p}{\left\|f_\theta'\right\|_p} = \gamma_p. θ ↓ 1 − 1/ p lim ∥ f θ ′ ∥ p ( M f θ ) ′ p = γ p . Together with the proof in Section 2.4, we complete the proof of Theorem 1. □
The optimal constant and its endpoint limits
We first prove the monotonicity, limits, and asymptotic expansion of the optimal constant. We then show how the quantitative bounds of the classical endpoint theorems arise as limits of the finite-p p p
The optimal constant γ p \gamma_p γ p
Now, we are ready to prove Proposition 3.
Proof of Proposition 3. The identity γ p = H 1 − 1 / p \gamma_p = H_{1-1/p} γ p = H 1 − 1/ p (38) . To see monotonicity, write
H α = min 0 ≤ t ≤ 1 1 + t 1 + α ( 1 + t ) ( 1 + α ) = min 0 ≤ t ≤ 1 1 1 + t ∫ − t 1 ∣ u ∣ α d u . H_\alpha= \min_{0 \le t \le1} \frac{1+t^{1+\alpha}}{(1+t)(1+\alpha)} = \min_{0 \le t \le1} \frac{1}{1+t}\int_{-t}^{1} |u|^\alpha\,du. H α = 0 ≤ t ≤ 1 min ( 1 + t ) ( 1 + α ) 1 + t 1 + α = 0 ≤ t ≤ 1 min 1 + t 1 ∫ − t 1 ∣ u ∣ α d u . Write t α ∈ [ 0 , 1 ] t_\alpha\in[0,1] t α ∈ [ 0 , 1 ] g α ( t ) : = 1 1 + t ∫ − t 1 ∣ u ∣ α d u g_\alpha(t) := \frac{1}{1+t}\int_{-t}^{1}|u|^\alpha\,du g α ( t ) := 1 + t 1 ∫ − t 1 ∣ u ∣ α d u H α = g α ( t α ) H_\alpha= g_\alpha(t_\alpha) H α = g α ( t α ) t t t g α ( t ) g_\alpha(t) g α ( t ) α \alpha α α < α ′ \alpha< \alpha' α < α ′
H α = g α ( t α ) > g α ′ ( t α ) ≥ g α ′ ( t α ′ ) = H α ′ H_\alpha= g_\alpha(t_\alpha) > g_{\alpha'}(t_\alpha) \ge g_{\alpha'}(t_{\alpha'}) = H_{\alpha'} H α = g α ( t α ) > g α ′ ( t α ) ≥ g α ′ ( t α ′ ) = H α ′ Therefore, H α H_\alpha H α γ p \gamma_p γ p
The expression in (10) extends continuously to α = 0 \alpha= 0 α = 0 t ∈ [ 0 , 1 ] t \in[0,1] t ∈ [ 0 , 1 ] H α → 1 H_\alpha\to1 H α → 1 α ↓ 0 \alpha\downarrow0 α ↓ 0 α = 1 \alpha= 1 α = 1 (38) gives H 1 2 + 2 H 1 = 1 H_1^2 + 2H_1 = 1 H 1 2 + 2 H 1 = 1 H 1 = 2 − 1 H_1 = \sqrt{2} - 1 H 1 = 2 − 1 (13) .
For the first-order expansion, differentiate (38) at α = 1 \alpha= 1 α = 1 r = 2 − 1 r = \sqrt{2} - 1 r = 2 − 1
d H α d α ∣ α = 1 = − r + r 2 − r 2 log r 2 ( 1 + r ) = − 1 − r − r 2 log r 2 ( 1 + r ) . \left.\frac{dH_\alpha}{d\alpha}\right|_{\alpha=1} = -\frac{r+r^2-r^2\log r}{2(1+r)} = -\frac{1-r-r^2\log r}{2(1+r)}. d α d H α α = 1 = − 2 ( 1 + r ) r + r 2 − r 2 log r = − 2 ( 1 + r ) 1 − r − r 2 log r . Taylor expansion at α = 1 \alpha= 1 α = 1 α = 1 − 1 / p \alpha= 1 - 1/p α = 1 − 1/ p (14) and completes Proposition 3. □
Endpoint behavior of γ p \gamma_p γ p
By a limit argument, one may define γ 1 = 1 , γ ∞ = 2 − 1 \gamma_1=1,\gamma_\infty=\sqrt{2}-1 γ 1 = 1 , γ ∞ = 2 − 1 p = 1 p=1 p = 1 p = ∞ p=\infty p = ∞ [1 , 2 ] . Therefore, the quantitative endpoint result
∥ ( M ~ f ) ′ ∥ 1 ≤ ∥ f ′ ∥ 1 , ∥ ( M ~ f ) ′ ∥ ∞ ≤ ( 2 − 1 ) ∥ f ′ ∥ ∞ (39) \left\|(\widetilde{M}f)'\right\|_1\leq\left\|f'\right\|_1,\qquad\left\|(\widetilde{M}f)'\right\|_\infty\leq(\sqrt{2}-1)\left\|f'\right\|_\infty \tag*{(39)} ( M f ) ′ 1 ≤ ∥ f ′ ∥ 1 , ( M f ) ′ ∞ ≤ ( 2 − 1 ) ∥ f ′ ∥ ∞ (39) in [1 , 2 ] can be recovered by a limit argument as follows:
Proof of (3.1). First, consider the side p → ∞ p\to\infty p → ∞
Let f f f f ′ ∈ L p f'\in L^p f ′ ∈ L p p > 1 p>1 p > 1 (17) makes ( M ~ f ) ′ (\widetilde{M}f)' ( M f ) ′ Theorem 1 shows that ( M ~ f ) ′ (\widetilde{M}f)' ( M f ) ′ L p L^p L p
∥ ( M ~ f ) ′ ∥ ∞ = lim p → ∞ ∥ ( M ~ f ) ′ ∥ p ≤ ( lim p → ∞ γ p ) ( lim p → ∞ ∥ f ′ ∥ p ) = ( 2 − 1 ) ∥ f ′ ∥ ∞ . \left\|(\widetilde{M}f)'\right\|_\infty=\lim_{p\to\infty}\left\|(\widetilde{M}f)'\right\|_p
\leq\left(\lim_{p\to\infty}\gamma_p\right)\left(\lim_{p\to\infty}\left\|f'\right\|_p\right)=(\sqrt{2}-1)\left\|f'\right\|_\infty. ( M f ) ′ ∞ = p → ∞ lim ( M f ) ′ p ≤ ( p → ∞ lim γ p ) ( p → ∞ lim ∥ f ′ ∥ p ) = ( 2 − 1 ) ∥ f ′ ∥ ∞ . For Lipschitz functions, these derivative norms equal the Lipschitz seminorms. This proves (16) for compact support.
Next, apply this result to bounded Lipschitz functions by using the function η R \eta_R η R (37) .
Finally, considering f K ( x ) = min { ∣ f ( x ) ∣ , K } f_K(x)=\min\{|f(x)|,K\} f K ( x ) = min { ∣ f ( x ) ∣ , K } K → ∞ K\to\infty K → ∞ § 2.4 , one can generalize the above result to all function f ∈ A C l o c ( R ) f\in AC_{\mathrm{loc}}(\mathbb{R}) f ∈ A C loc ( R ) f ′ ∈ L ∞ ( R ) f'\in L^\infty(\mathbb{R}) f ′ ∈ L ∞ ( R ) M ~ f \widetilde{M}f M f (16) in [1 ] as a limit of Theorem 1 .
Second, let p → 1 p\to1 p → 1 f f f f ′ ∈ L 1 ∩ L ∞ f'\in L^1\cap L^\infty f ′ ∈ L 1 ∩ L ∞ Theorem 1 , applied at p = 2 p=2 p = 2 M ~ f \widetilde{M}f M f J J J p > 1 p>1 p > 1
∫ J ∣ ( M ~ f ) ′ ( x ) ∣ d x ≤ ∣ J ∣ 1 − 1 / p ∥ ( M ~ f ) ′ ∥ p ≤ ∣ J ∣ 1 − 1 / p γ p ∥ f ′ ∥ p . \int_J\left|(\widetilde{M}f)'(x)\right|\,dx\leq|J|^{1-1/p}\left\|(\widetilde{M}f)'\right\|_p\leq|J|^{1-1/p}\gamma_p\left\|f'\right\|_p. ∫ J ( M f ) ′ ( x ) d x ≤ ∣ J ∣ 1 − 1/ p ( M f ) ′ p ≤ ∣ J ∣ 1 − 1/ p γ p ∥ f ′ ∥ p . As p → 1 p\to1 p → 1 ∫ J ∣ ( M ~ f ) ′ ( x ) ∣ d x ≤ γ 1 ∥ f ′ ∥ 1 \int_J\left|(\widetilde{M}f)'(x)\right|\,dx\leq\gamma_1\left\|f'\right\|_1 ∫ J ( M f ) ′ ( x ) d x ≤ γ 1 ∥ f ′ ∥ 1 J J J J = [ − R , R ] J=[-R,R] J = [ − R , R ] R → ∞ R\to\infty R → ∞
∥ ( M ~ f ) ′ ∥ 1 = Var ( M ~ f ) ≤ ∥ f ′ ∥ 1 . (40) \left\|(\widetilde{M}f)'\right\|_1=\operatorname{Var}(\widetilde{M}f)\leq\left\|f'\right\|_1. \tag*{(40)} ( M f ) ′ 1 = Var ( M f ) ≤ ∥ f ′ ∥ 1 . (40) The variation identity follows from local absolute continuity and integrability of the derivative.
We next pass to B V BV B V f f f ∣ f ∣ |f| ∣ f ∣ f ≥ 0 f\geq0 f ≥ 0 ∣ D f ∣ ( R ) < ∞ |Df|(\mathbb{R})<\infty ∣ D f ∣ ( R ) < ∞ ρ ε \rho_\varepsilon ρ ε f ε = f ∗ ρ ε f_\varepsilon=f*\rho_\varepsilon f ε = f ∗ ρ ε B V BV B V
f ε ′ = D f ∗ ρ ε ∈ L 1 ∩ L ∞ , ∥ f ε ′ ∥ 1 ≤ ∣ D f ∣ ( R ) . f_\varepsilon'=Df*\rho_\varepsilon\in L^1\cap L^\infty,\qquad\left\|f_\varepsilon'\right\|_1\leq|Df|(\mathbb{R}). f ε ′ = D f ∗ ρ ε ∈ L 1 ∩ L ∞ , ∥ f ε ′ ∥ 1 ≤ ∣ D f ∣ ( R ) . Thus (3.2) gives Var ( M ~ f ε ) ≤ ∣ D f ∣ ( R ) \operatorname{Var}(\widetilde{M}f_\varepsilon)\leq|Df|(\mathbb{R}) Var ( M f ε ) ≤ ∣ D f ∣ ( R ) M ~ f \widetilde{M}f M f [2 Lemmas 3.3–3.4] . Fix x ∈ R x\in\mathbb{R} x ∈ R I I I x x x I I I
1 ∣ I ∣ ∫ I f ε ( t ) d t = ∫ R ρ ε ( y ) ( 1 ∣ I ∣ ∫ I f ( t − y ) d t ) d y ≤ ∫ R ρ ε ( y ) M ~ f ( x − y ) d y ⇒ M ~ f ε ( x ) ≤ ( ( M ~ f ) ∗ ρ ε ) ( x ) ⟶ ε → 0 M ~ f ( x ) ⇒ lim sup ε → 0 M ~ f ε ( x ) ≤ M ~ f ( x ) . (41) \begin{aligned}
\frac{1}{|I|}\int_I f_\varepsilon(t)\,dt=\int_{\mathbb{R}}\rho_\varepsilon(y)\left(\frac{1}{|I|}\int_I f(t-y)\,dt\right)\,dy\leq\int_{\mathbb{R}}\rho_\varepsilon(y)\widetilde{M}f(x-y)\,dy \\
&\Rightarrow\widetilde{M}f_\varepsilon(x)\leq((\widetilde{M}f)*\rho_\varepsilon)(x)
\underset{\varepsilon\to0}{\longrightarrow}\widetilde{M}f(x) \\
&\Rightarrow\limsup_{\varepsilon\to0}\widetilde{M}f_\varepsilon(x)\leq\widetilde{M}f(x). \tag*{(41)}
\end{aligned} ∣ I ∣ 1 ∫ I f ε ( t ) d t = ∫ R ρ ε ( y ) ( ∣ I ∣ 1 ∫ I f ( t − y ) d t ) d y ≤ ∫ R ρ ε ( y ) M f ( x − y ) d y ⇒ M f ε ( x ) ≤ (( M f ) ∗ ρ ε ) ( x ) ε → 0 ⟶ M f ( x ) ⇒ ε → 0 lim sup M f ε ( x ) ≤ M f ( x ) . (41) For the reverse lower-limit inequality, fix any finite averaging interval I I I x x x f ε → f f_\varepsilon\to f f ε → f L 1 L^1 L 1
1 ∣ I ∣ ∫ I f ( t ) d t = lim inf ε → 0 1 ∣ I ∣ ∫ I f ε ( t ) d t ≤ lim inf ε → 0 M ~ f ε ( x ) ⇒ M ~ f ( x ) ≤ lim inf ε → 0 M ~ f ε ( x ) . \frac{1}{|I|}\int_I f(t)\,dt=\liminf_{\varepsilon\to0}\frac{1}{|I|}\int_I f_\varepsilon(t)\,dt\leq\liminf_{\varepsilon\to0}\widetilde{M}f_\varepsilon(x)\Rightarrow\widetilde{M}f(x)\leq\liminf_{\varepsilon\to0}\widetilde{M}f_\varepsilon(x). ∣ I ∣ 1 ∫ I f ( t ) d t = ε → 0 lim inf ∣ I ∣ 1 ∫ I f ε ( t ) d t ≤ ε → 0 lim inf M f ε ( x ) ⇒ M f ( x ) ≤ ε → 0 lim inf M f ε ( x ) . Together with (3.3), we get M ~ f ε → M ~ f \widetilde{M}f_{\varepsilon} \to\widetilde{M}f M f ε → M f
By (2.18) and (3.2),
Var ( M ~ f ) ≤ lim inf ε ↓ 0 Var ( M ~ f ε ) ≤ ∣ D f ∣ ( R ) . \operatorname{Var}(\widetilde{M}f) \le\liminf_{\varepsilon\downarrow0}\operatorname{Var}(\widetilde{M}f_{\varepsilon}) \le|Df|(\mathbb{R}). Var ( M f ) ≤ ε ↓ 0 lim inf Var ( M f ε ) ≤ ∣ D f ∣ ( R ) . This is the total-variation bound in (1.15), obtained from Theorem 1 by p ↓ 1 p \downarrow1 p ↓ 1 M ~ f \widetilde{M}f M f ∥ ( M ~ f ) ′ ∥ 1 \lVert(\widetilde{M}f)'\rVert_1 ∥( M f ) ′ ∥ 1 [2 Theorem 2.5] . We use that regularity result here without reproducing its proof. □
Acknowledgements
J.M. was partially supported by the Simons Foundation Grant #453576.
Statement on AI use
The initial ideas and overall structure of this paper are from the authors. Then, in collaboration with AI models (including GPT 6 Astra, Claude Opus 5.5), we developed several technical lemmas, visual illustrations, and concise arguments to complete the proof. All mathematical proofs, derivations, and final conclusions were verified by the authors, who take full responsibility for them.
References
Department of Mathematics, Virginia Polytechnic Institute and State University, 225 Stanger Street, Blacksburg, VA 24061-1026, USA Email address: sungyi@vt.edu
Department of Mathematics, Virginia Polytechnic Institute and State University, 225 Stanger Street, Blacksburg, VA 24061-1026, USA Email address: josemadrid@vt.edu
Department of Mathematics, Virginia Polytechnic Institute and State University, 225 Stanger Street, Blacksburg, VA 24061-1026, USA Email address: palsson@vt.edu
