Adams’ trace principle in Morrey–Lorentz spaces on β-Hausdorff dimensional surfaces

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