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 $\|\cdot\|_{L^{p,\infty}(\mathbb{R}^n)}$ 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. We also prove that when $0<p,q<\infty$ and $-\infty<s\leq\max\{0,n(\frac{1}{p}-\frac{1}{q})\}$, 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*}