A weak inequality in fractional homogeneous Sobolev spaces

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 $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{L^{p,\mathrm{\infty }}({\mathbb{R}}^n)}$ is the weak $L^p$ quasinorm and $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{{\dot{L}}_s^p({\mathbb{R}}^n)}$ is the homogeneous Sobolev norm, and parameters satisfy the condition that $1<p<q$, $2\le q<\mathrm{\infty }$, and $0<s=n(\frac{1}{p}-\frac{1}{q})<1$. Furthermore, we prove the estimate $\mathrm{\parallel }{\mathfrak{g}}_{s,q}(f){\mathrm{\parallel }}_{L^p({\mathbb{R}}^n)}\lesssim \mathrm{\parallel }f{\mathrm{\parallel }}_{{\dot{F}}_{p,q}^s({\mathbb{R}}^n)}$ when $0<p,q<\mathrm{\infty }$, $-1<s<1$, $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{{\dot{F}}_{p,q}^s({\mathbb{R}}^n)}$ denotes the homogeneous Triebel-Lizorkin quasinorm and the Littlewood-Paley-Poisson function ${\mathfrak{g}}_{s,q}(f)(\cdot )$ is a generalization of the classical 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 $g_{\lambda }^{\ast }$-function. We also prove that when $0<p,q<\mathrm{\infty }$ and $-\mathrm{\infty }<s\le \max \{0,n(\frac{1}{p}-\frac{1}{q})\}$, we have \begin{equation*} |\big(\int_{$\mathbb{R}$^n}\frac{|f(\cdot+y)-f(\cdot)|^q}{|y|^{n+sq}}dy\big)^{\frac{1}{q}}|_{L^{p}($\mathbb{R}$^n)}=\infty. \end{equation*}