A weak inequality in fractional homogeneous Sobolev spaces (2312.14662v6)

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 $\mathcal{G}{\lambda,q}$-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 $g_{\lambda}^*$-function.