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

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_{\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 if the Radon measure $d\mu$ supported in $\mathrm{\Omega }\subset {\mathbb{R}}^n$ is controlled by $$\text{Arrowvert}\mu {\text{Arrowvert}}_{\beta }=\underset{x\in {\mathbb{R}}^n,r>0}{\sup }{r}^{-\beta }\mu (B(x,r))<\mathrm{\infty }$$ provided that $1<p<q<\mathrm{\infty }$ satisfies $n-\alpha p<\beta\leq n,; \alpha=\frac{n}{\lambda}-\frac{\beta}{\lambda\ast}; \text{ and };\frac{\lambda_\ast}{q}\leq \frac{\lambda}{p}\nonumber,$. 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} \nonumber $as$1<p<\lambda<\infty$and$s\in$\mathbb{R}$$satisfies$\frac{1}{p}-\frac{s}{n}=\frac{1}{\lambda}$\mathrm{.}If$d\mu$isconcentratedon$\partial$\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 Cader'on-Zygmund decomposition we show that \begin{equation} \Vert M_{\alpha}f\Vert_{$\mathcal{M}${p, \ell}^{\lambda}(d\mu)},\sim, \Vert I{\alpha}f\Vert_{$\mathcal{M}$_{p, \ell}^{\lambda}(d\mu)}\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.