Adams' trace principle on Morrey-Lorentz spaces over $β$-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 $I_α$ 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 $dμ$ supported in $Ω\subset \mathbb{R}^n$ 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 $n-αp<β\leq n,\; α=\frac{n}λ-\fracβ{λ_\ast}\; \text{ and }\;\frac{λ_\ast}{q}\leq \fracλ{p}\nonumber\,$. 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^{λ, \infty}\hookrightarrow \mathcal{M}_{p}^λ\hookrightarrow\mathcal{M}_{p, \infty}^λ \nonumber $ as $1<p<λ<\infty$ and $s\in\mathbb{R}$ satisfies $\frac{1}{p}-\frac{s}{n}=\frac{1}λ$. If $dμ$ 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 $μ(B_r(x))\sim r^β$ on support $\text{spt}(μ)$ and $n-α<β\leq n$ with $0<α<n$. This result extends the previous ones.