A weak inequality in fractional homogeneous Sobolev spaces

Abstract

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