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Sobolev trace theorem on Morrey-type spaces on $\beta$-Hausdorff dimensional surfaces

Research paper by Marcelo F. de Almeida, Lidiane S. M. Lima

Indexed on: 07 May '21Published on: 03 Nov '19Published in: arXiv - Mathematics - Analysis of PDEs



Abstract

In this note we strengthen to Morrey-Lorentz spaces the Sobolev-trace principle discovered by D. R. Adams and extended to many functions spaces by Adams, Xiao and Liu. More precisely, we show that Riesz potential $I_{\alpha}$ mapping \begin{equation} I_{\alpha}:\mathcal{M}_{pl}^{\lambda}(\mathbb{R}^n,d\nu)\longrightarrow\mathcal{M}_{qs}^{\lambda_{\star}}(M,\,d\mu),\nonumber \end{equation} continuously if, and only if the Radon measure $\mu$, supported on a $\beta-$dimensional surface $M\subset\mathbb{R}^n$ satisfies $\mu(B_r(x))\leq C r^{\beta}$ for every $x\in M$ and $r>0$, provided $ n-\alpha p<\beta\leq n,\; \alpha=\frac{n}{\lambda}-\frac{\beta}{\lambda_\ast}\; \text{ and }\;\frac{\lambda_\ast}{q}\leq \frac{\lambda}{p}\nonumber.\,$ In particular, we obtain Sobolev trace theorem on Lipschitz domain and we show that distributional solutions of fractional Laplace equation $(-\Delta_x)^{\frac{\delta}{2}}v=f$ satisfies $v\in \mathcal{M}_{qs}^{\lambda_{\star}}(M,\,d\mu)$ if provided $f\in \mathcal{M}_{pl}^{\lambda}(\mathbb{R}^n,d\nu)$, where \begin{equation} \Vert f \Vert_{\mathcal{M}^{\lambda_\star}_{qs}(M,d\mu)}=\sup_{x\,\in\, \text{supp}(\mu),\, r>0}r^{-\beta \left(\frac{1}{q}-\frac{1}{\lambda_\star}\right)} \Vert f\Vert_{L^{qs}(\mu\lfloor_{M}(B_r) )}.\nonumber \end{equation}