A weak inequality in fractional homogeneous Sobolev spaces

TL;DR

This work develops a unified framework connecting fractional Sobolev spaces and generalized Littlewood-Paley constructions. It proves a weak-type bound for the fractional difference quotient $\mathcal{D}_{s,q}$ in terms of the homogeneous Sobolev norm $\|f\|_{\dot{L}^p_s}$ under the parameter constraint $1<p<q$, $2\le q<\infty$, $0<s=n(\frac{1}{p}-\frac{1}{q})<1$, and establishes $L^p$ bounds for the Littlewood-Paley-Poisson function $\mathfrak{g}_{s,q}$ on homogeneous Triebel-Lizorkin spaces $\dot{F}^s_{p,q}$. It further proves weak-type $(p,p)$ bounds for the generalized Littlewood-Paley functions $\mathcal{G}_{\lambda,q}$ and $\mathcal{R}_{s,q}$, expanding classical results for square-function type operators to broader kernel families and parameter regimes. The methods combine Whitney decomposition, Poisson/harmonic-analytic techniques (including subharmonicity and Hörmander-type estimates) with Triebel-Lizorkin and Hardy-Littlewood-Sobolev machinery to relate smoothness, difference quotients, and Littlewood-Paley theory in a robust, quantitative way. These results enhance the understanding of fractional regularity and provide tools for analyzing fractional Sobolev and Triebel-Lizorkin spaces via generalized square-function constructs.

Abstract

In this paper, we prove the following inequality \begin{equation*} \|\big(\int_{\mathbb{R}^n}\frac{|f(\cdot+y)-f(\cdot)|^q}{|y|^{n+sq}}dy\big)^{\frac{1}{q}}\|_{L^{p,\infty}(\mathbb{R}^n)}\lesssim\|f\|_{\dot{L}^p_s(\mathbb{R}^n)}, \end{equation*} where $\|\cdot\|_{L^{p,\infty}(\mathbb{R}^n)}$ is the weak $L^p$ quasinorm and $\|\cdot\|_{\dot{L}^p_s(\mathbb{R}^n)}$ is the homogeneous Sobolev norm, and parameters satisfy the condition that $1<p<q$, $2\leq q<\infty$, and $0<s=n(\frac{1}{p}-\frac{1}{q})<1$. Furthermore, we prove the estimate $\|\mathfrak{g}_{s,q}(f)\|_{L^p(\mathbb{R}^n)}\lesssim\|f\|_{\dot{F}^s_{p,q}(\mathbb{R}^n)}$ when $0<p,q<\infty$, $-1<s<1$, $\|\cdot\|_{\dot{F}^s_{p,q}(\mathbb{R}^n)}$ denotes the homogeneous Triebel-Lizorkin quasinorm and the Littlewood-Paley-Poisson function $\mathfrak{g}_{s,q}(f)(\cdot)$ is a generalization of the classical Littlewood-Paley $g$-function. Moreover, we prove the weak type $(p,p)$ boundedness of the $\mathcal{G}_{λ,q}$-function and the $\mathcal{R}_{s,q}$-function, where the $\mathcal{G}_{λ,q}$-function is a generalization of the well-known classical Littlewood-Paley $g_λ^*$-function.

Figures (2)

• Figure 1: Lemma \ref{['lemma2']} (1). Point $y$ is an arbitrary point in the cube $Q_m$, and point $z$ is an arbitrary point in the cube $Q_j$.
• Figure 2: Lemma \ref{['lemma2']} (2). If cubes $Q_j$ and $Q_m$ have disjoint interiors, then the two $n$-dimensional balls inscribed in $Q_j$ and $Q_m$ are disjoint balls, and thus the sum of radii of these two balls is less than or equal to the distance between the centers of these two balls.

Theorems & Definitions (43)

• Proposition 1
• proof : Proof of Proposition \ref{['proposition1']}
• Definition 2
• Theorem 3: cf. Corollary 1.7 (i) of Wang2023
• Definition 4: cf. section 1.3.3 of 14modern
• Remark 5
• Theorem 6
• Theorem 7
• Corollary 8
• proof : Proof of Corollary \ref{['corollary1']}