Adams' trace principle on Morrey-Lorentz spaces over $\beta$-Hausdorff dimensional surfaces

Abstract

In this paper we strengthen to Morrey-Lorentz spaces the famous trace principle introduced by Adams. More precisely, we show that Riesz potential $I_{\alpha }$ is continuous \begin{equation} \Vert I_αf\Vert_{\mathcal{M}_{q, \infty}^{λ_{\ast}}(dμ)}\lesssim \Arrowvertμ\Arrowvert_β^{{1}/{q}}\,\Vert f\Vert_{\mathcal{M}_{p, \infty}^λ(dν)}\nonumber\\[0.02in] \end{equation} if and only if the Radon measure $d\mu$ supported in $\Omega \subset {\mathbb{R}}^n$ is controlled by $$\Arrowvertμ\Arrowvert_β=\sup_{x\in\mathbb{R}^n,\,r>0}r^{-β}μ(B(x,r))<\infty$$ provided that $1<p<q<\infty$ satisfies $n-\alpha p<\beta \le n,\alpha =\frac{n}{\lambda }-\frac{\beta }{{\lambda }_{\ast }}\text{ and }\frac{{\lambda }_{\ast }}{q}\le \frac{\lambda }{p}$. Our result provide a new class of functions spaces which is larger than previous ones, since we have strict continuous inclusions ${\dot{B}}_{p,\infty }^s\hookrightarrow L^{\lambda ,\infty }\hookrightarrow {\mathcal{M}}_p^{\lambda }\hookrightarrow {\mathcal{M}}_{p,\infty }^{\lambda }$ as $1<p<\lambda <\infty$ and $s\in \mathbb{R}$ satisfies $\frac{1}{p}-\frac{s}{n}=\frac{1}{\lambda }$. If $d\mu$ is concentrated on $\partial {\mathbb{R}}_{+}^n$, as a byproduct we get Sobolev-Morrey trace inequality on half-spaces ${\mathbb{R}}_{+}^n$ which recovers the well-known Sobolev-trace inequality in $L^p({\mathbb{R}}_{+}^n)$. Also, by a suitable analysis on non-doubling Caderón-Zygmund decomposition we show that \begin{equation} \Vert M_αf\Vert_{\mathcal{M}_{p, \ell}^λ(dμ)}\,\sim\, \Vert I_αf\Vert_{\mathcal{M}_{p, \ell}^λ(dμ)}\nonumber \end{equation} provided that $\mu (B_r(x))\sim r^{\beta }$ on support $\text{spt}(\mu )$ and $n-\alpha <\beta \le n$ with $0<\alpha <n$. This result extends the previous ones.

16 pages. This is the final version, incorporating referee report