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A Note on Fractional Sobolev inequalities revisited

Published Nov 02, 2026Fresh — added to the Hub within the last 30 daysv14 min read

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Abstract

In this paper, we extend the analysis of homogeneous fractional Sobolev inequalities in the paper by A.D. Nguyen, J. I. Dıaz, and Q.-H. Nguyen (2020). We combine their subcritical proof with the first author's adaptation to the supercritical case to provide a short interpolation-based proof for the critical case.

Introduction

In [3], Nguyen et al. used a novel application of ordinary and sharp Hardy-Littlewood maximal functions to prove that for $s_1-\frac{N}{p_1}=s_2-\frac{N}{p_2}$ , $s_2<s_1$ the embedding

\dot{W}^{s_1,p_1}(\mathbb{R}^{N}) \hookrightarrow\dot{W}^{s_2,p_2}(\mathbb{R}^{N})

of homogeneous Sobolev spaces holds. This result was derived by applying the inequality

\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\frac{\lvert u(x+h)-u(x)\rvert^{q}}{\lVert h\rVert^{N+\sigma q}}\,dh\,dx \lesssim

\int_{\mathbb{R}^{N}}(M(u)(x))^{(1-\theta)q}\left(\sup_{r>0}\frac{1}{r^{N}}\int_{B(0,r)}\frac{\lvert u(x+z)-u(x)\rvert}{\lVert z\rVert^{\sigma}}\,dz\right)^{\theta q}\,dx,

where $1\leq q<\infty$ and $0<\theta,\sigma<1$ , in combination with the Sobolev-Gagliardo-Nirenberg embedding. This method relies on the intrinsic Gagliardo semi-norm,

\lvert u\rvert_{W^{s,p}(\mathbb{R}^{N})}=\left(\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\frac{\lvert u(x+h)-u(x)\rvert^{p}}{\lVert h\rVert^{N+sp}}\,dh\,dx\right)^{\frac{1}{p}},

where $1\leq p,q<\infty$ and $0<s<1$ . The method offers a simpler alternative to classical proofs, including those from [6] and [4]. The former is presented as Theorem 17.85 of [1] and uses the Littlewood-Paley decomposition to prove the critical and supercritical cases, that is, when $s_1-\frac{N}{p_1}=0$ and $s_1-\frac{N}{p_1}>0$ respectively. The latter is presented as Theorem 7.35 in [2] and uses anisotropic $L^p$ spaces to prove the general case in Besov spaces.

Although the proof from Nguyen et al. is simpler, their analysis cannot cover the critical and supercritical cases when $s_1<1$ and $s_2<1$ . In these settings, standard embeddings are not sufficiently compatible with maximal functions for an identical approach.

The first author extended the result of Nguyen et al. in Theorem 7.34 of [2] by modifying the Hardy–Littlewood maximal function. He proved the inequality

\int_{\mathbb{R}^{N}}\left(\frac{1}{\lVert h\rVert^{N}}\int_{B(0,2\lVert h\rVert)}\lvert u(x+z)-u(x)\rvert\,\mathrm{d}z\right)^{p_{2}}\frac{\mathrm{d}h}{\lVert h\rVert^{N+s_{2}p_{2}}}\lesssim\lvert u\rvert_{\mathcal{C}^{p_{2}-p_{1}}_{0,\alpha}(\mathbb{R}^{N})}\left(\sup_{r>0}\frac{1}{r^{N}}\int_{B(0,r)}\lvert u(x+z)-u(x)\rvert\left(\frac{\mathrm{d}z}{\lVert z\rVert^{s_{1}}}\right)^{p_{1}}\right)

where $0<s_{2}<s_{1}<1$ , $1\leq p_{1},p_{2}\leq\infty$ , $s_{1}-\frac{N}{p_{1}}=s_{2}-\frac{N}{p_{2}}=\alpha>0$ . After an application of Morrey’s embedding theorem, this handles the supercritical case. Nevertheless, the first author resorted to Solonnikov’s long and classical method from [4] to prove the critical case in Theorem 7.35. An alternative proof akin to Nguyen et al.’s result may be available through the space $\operatorname{BMO}(\mathbb{R}^{N})$ , although no such proof is forthcoming. Instead, in this paper, we succinctly prove the critical case using standard results of interpolation and the short theorems taken from both [2] and [3]. Our result is as follows.

Theorem 1.1. For $0<s_{2}<s_{1}<1$ , $1\leq p_{1},p_{2}<\infty$ where $s_{1}-\frac{N}{p_{1}}=s_{2}-\frac{N}{p_{2}}=0$ ,

\dot{W}^{s_{1},p_{1}}(\mathbb{R}^{N})\hookrightarrow\dot{W}^{s_{2},p_{2}}(\mathbb{R}^{N}). \tag*{(1)}

Preliminaries

In this paper, we write $A(x)\lesssim B(x)$ if for some constant $c>0$ independent of $x$ , $A(x)\leq cB(x)$ for all $x\in\operatorname{dom}(A)$ .

Combining the aforementioned results of [2] and [3], the statement is proven in the noncritical case.

Theorem 2.1. For $0<s_{2}<s_{1}<1$ , $1\leq p_{1},p_{2}<\infty$ where $s_{1}-\frac{N}{p_{1}}=s_{2}-\frac{N}{p_{2}}\ne0$ ,

\dot{W}^{s_{1},p_{1}}(\mathbb{R}^{N})\hookrightarrow\dot{W}^{s_{2},p_{2}}(\mathbb{R}^{N}). \tag*{(2)}

We make use of several classical results on interpolation, which are extended to interpolation on semi-normed spaces. A more comprehensive discussion of the subject can be found in [1] and [5], although we present a brief overview of the relevant terms and theorems in this paper.

Definition 2.1. Two semi-normed spaces $(X_{0},\lvert\cdot\rvert_{X_{0}})$ and $(X_{1},\lvert\cdot\rvert_{X_{1}})$ are compatible if both embed into the same larger vector space $X$ .

Both the intersection and the sum of compatible spaces inherit natural semi-norm structures. The real interpolation spaces defined below are particular examples of intermediate spaces between $X_{0}\cap X_{1}$ and $X_{0}+X_{1}$ .

Definition 2.2. For compatible semi-normed spaces $(X_{0},\lvert\cdot\rvert_{X_{0}})$ and $(X_{1},\lvert\cdot\rvert_{X_{1}})$ and any $x\in X_{0}+X_{1}$ , define the semi-norm $K(x,t;X_{0},X_{1})=\inf(\lvert x_{0}\rvert_{X_{0}}+t\lvert x_{1}\rvert_{X_{1}})$ , where the infimum is taken over all $x_{0}\in X_{0}$ , $x_{1}\in X_{1}$ with $x_{0}+x_{1}=x$ . Where $X_{0}$ and $X_{1}$ are apparent, we write $K(x,t)$ instead.

Definition 2.3 (K Method of Interpolation). For compatible semi-normed spaces $(X_{0},\lvert\cdot\rvert_{X_{0}})$ and $(X_{1},\lvert\cdot\rvert_{X_{1}})$ , $0<\theta<1$ , and $1\leq p\leq\infty$ , define

\lvert x\rvert_{(X_{0},X_{1})_{\theta,p}}=\left(\int_{0}^{\infty}\frac{K(x,t)^{p}}{t^{1+\theta p}}\,\mathrm{d}t\right)^{\frac{1}{p}} \tag*{(3)}

if $p < \infty$ , and

|x|_{(X_0,X_1)_{\theta,p}}=\sup_{t>0}t^{-\theta}K(x,t)

otherwise. $(X_0,X_1)_{\theta,p}$ is the set of all $x\in X_0+X_1$ such that this value is finite.

Spaces generated by the K-method have several useful properties, the most important of which is the following.

Theorem 2.2 (Reiteration). Let $(X_0,|\cdot|_{X_0})$ and $(X_1,|\cdot|_{X_1})$ be compatible complete semi-normed spaces with complete intersection. For $0\le s_0<s_1\le1$ , $0<\theta<1$ , $s=\theta s_1+(1-\theta)s_0$ , $1\le q_0,q_1,q_2\le\infty$ ,

(X_{s_0,q_0},X_{s_1,q_1})_{\theta,q_2}=(X_0,X_1)_{s,q_2} \tag*{(4)}

with equivalence of seminorms, where

X_{s_i,q_i} = \begin{cases} X_{s_i} & s_i\in\{0,1\},\\ (X_0,X_1)_{s_i,q_i} & s_i\notin\{0,1\}. \end{cases} \tag*{(5)}

The reiteration theorem is proved in [5]. Although it requires a few tricks, the method is largely long algebraic manipulation. We also use a few standard inclusions of interpolation spaces.

Theorem 2.3. Let $(X_0,|\cdot|_{X_0}),(X_1,|\cdot|_{X_1})$ be an admissible pair of semi-normed spaces, $0<\theta<1$ , $1\le p\le q<\infty$ . In this setting,

(X_0,X_1)_{\theta,p}\hookrightarrow(X_0,X_1)_{\theta,q}\hookrightarrow(X_0,X_1)_{\theta,\infty}. \tag*{(6)}

Proof. We present a proof taken from [1]. As $K(x,\cdot)$ is increasing, for $t>0$ , by (3),

t^{-\theta}K(x,t)\lesssim\left(\int_t^\infty(K(x,t))^q\frac{dz}{z^{1+\theta q}}\right)^{1/q} \le\left(\int_t^\infty(K(x,z))^q\frac{dz}{z^{1+\theta q}}\right)^{1/q} \le|x|_{(X_0,X_1)_{\theta,q}}.

Taking the supremum of the left side yields the second embedding. Further,

|x|_{(X_0,X_1)_{\theta,q}}^q =\int_0^\infty(K(x,t))^q\frac{dt}{t^{1+\theta q}} =\int_0^\infty\left(t^{-\theta}K(x,t)\right)^{q-p}(K(x,t))^p\frac{dt}{t^{1+\theta p}} \lesssim|x|_{(X_0,X_1)_{\theta,\infty}}^{q-p}\int_0^\infty(K(x,t))^p\frac{dt}{t^{1+\theta p}} \lesssim|x|_{(X_0,X_1)_{\theta,\infty}}^{q-p}|x|_{(X_0,X_1)_{\theta,p}}^p,

where the final inequality uses the second embedding applied to $(X_0,X_1)_{\theta,p}$ . □

Theorem 2.4. Let $(X_0,|\cdot|_{X_0}),(X_1,|\cdot|_{X_1})$ and $(Y_0,|\cdot|_{Y_0}),(Y_1,|\cdot|_{Y_1})$ be two pairs of compatible semi-normed spaces, and let $T:X_0+X_1\to Y_0+Y_1$ be linear. Suppose $T:X_i\to Y_i$ and $|T(x_i)|_{Y_i}\lesssim|x_i|_{X_i}$ for all $x_i\in X_i$ , $i=0,1$ . For all $0<\theta<1$ , $1\le p\le\infty$ , $|T(x)|_{(Y_0,Y_1)_{\theta,p}}\lesssim|x|_{(X_0,X_1)_{\theta,p}}$ for all $x\in(X_0,X_1)_{\theta,p}$ .

Proof. We present an adapted proof from [1]. Let $x_0 \in X_0$ , $x_1 \in X_1$ be such that $x_0+x_1=x$ . By linearity,

K(T(x),t;Y_0,Y_1)\leq|T(x_0)|_{Y_0}+t|T(x_1)|_{Y_1}\lesssim|x_0|_{X_0}+t|x_1|_{X_1}

Taking the infimum yields $K(T(x),t;Y_0,Y_1)\lesssim K(x,t;X_0,X_1)$ . The result follows. $\square$

We conclude our preliminaries with a specific application of the K-method to Sobolev spaces.

Theorem 2.5. For $1\leq p<\infty$ , $0<\theta<1$ ,

\left(L^p(\mathbb{R}^N),\dot{W}^{1,p}(\mathbb{R}^N)\right)_{\theta,p}=\dot{W}^{\theta,p}(\mathbb{R}^N). \tag*{(7)}

This result is proven in [2] using mollification to decompose an arbitrary function. Note that $\dot{W}^{1,p}(\mathbb{R}^N)\cap L^p(\mathbb{R}^N)=W^{1,p}(\mathbb{R}^N)$ is a Banach space. Thus, we may apply the reiteration theorem to Sobolev spaces.

Proof

Proof of Theorem 1.1. Observe that $1<p_1=\frac{N}{s_1}<p_2=\frac{N}{s_2}<\infty$ . Let $0<d_1<s_1<t_1<1$ and let $0<\theta<1$ such that $\theta t_1+(1-\theta)d_1=s_1$ . Define $d_2=d_1+\frac{N}{p_2}-\frac{N}{p_1}$ and $t_2=t_1+\frac{N}{p_2}-\frac{N}{p_1}$ . Because

d_1-\frac{N}{p_1}=d_2-\frac{N}{p_2}=d_1-s_1<0,\quad t_1-\frac{N}{p_1}=t_2-\frac{N}{p_2}=t_1-s_1>0,

we may apply Theorems 2.2 and 2.5 to write

\dot{W}^{s_1,p_1}(\mathbb{R}^N)=\left(\dot{W}^{d_1,p_1}(\mathbb{R}^N),\dot{W}^{t_1,p_1}(\mathbb{R}^N)\right)_{\theta,p_1}.

By taking $d_1,t_1$ close to $s_1$ , we ensure $0<d_2,t_2<1$ , since $0<s_2=s_1+\frac{N}{p_2}-\frac{N}{p_1}<1$ . By Theorems 2.1 and 2.4,

\left(\dot{W}^{d_1,p_1}(\mathbb{R}^N),\dot{W}^{t_1,p_1}(\mathbb{R}^N)\right)_{\theta,p_1}\hookrightarrow\left(\dot{W}^{d_2,p_2}(\mathbb{R}^N),\dot{W}^{t_2,p_2}(\mathbb{R}^N)\right)_{\theta,p_1},

where we use the identity map as our linear transformation. By Theorem 2.3

\left(\dot{W}^{d_2,p_2}(\mathbb{R}^N),\dot{W}^{t_2,p_2}(\mathbb{R}^N)\right)_{\theta,p_1}\hookrightarrow\left(\dot{W}^{d_2,p_2}(\mathbb{R}^N),\dot{W}^{t_2,p_2}(\mathbb{R}^N)\right)_{\theta,p_2}.

A final application of Theorems 2.2 and 2.5 completes the proof, since

s_2=s_1+\frac{N}{p_2}-\frac{N}{p_1}=\theta t_2+(1-\theta)d_2.

$\square$

Acknowledgments

Asher Wheaton was supported by the summer research support of NSF grant DMS #2508400 This paper was written without using AI.

Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh PA 15213-3890, USA

Email address: [email protected]

Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh PA 15213-3890, USA

Email address: [email protected]

References

[1]G. Leoni. A first course in Sobolev spaces, volume 181 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, second edition, 2017.
[2]G. Leoni. A first course in fractional Sobolev spaces, volume 229 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, [2023] ©2023.arxiv.org/abs/2303.05940
[3]A. D. Nguyen, J. I. Díaz, and Q.-H. Nguyen. Fractional Sobolev inequalities revisited: the maximal function approach. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 31(1):225–236, 2020.DOI
[4]V. A. Solonnikov. Inequalities for functions of the classes W̃ρ(rⁿ). J Math Sci, 3(4):549–564, Apr. 1975.
[5]N. Stevenson and I. Tice. A truncated real interpolation method and characterizations of screened Sobolev spaces. Commun. Pure Appl. Anal., 19(12):5509–5566, 2020.arxiv.org/abs/2003.12518
[6]H. Triebel. Interpolation theory, function spaces, differential operators. Johann Ambrosius Barth, Heidelberg, second edition, 1995.

A Note on Fractional Sobolev inequalities revisited

Abstract

Introduction

Preliminaries

Proof

Acknowledgments

References

Publication

Taxonomy

Authors

AI contributions

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