Adams’ trace principle in Morrey–Lorentz spaces on 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 $I_{\alpha}$ is continuous   $\Vert I_{\alpha }f{\Vert }_{{\mathcal{M}}_{q,\mathrm{\infty }}^{{\lambda }_{\ast }}(d\mu )}\lesssim \Vert \mu {\Vert }_{\beta }^{1/q}\Vert f{\Vert }_{{\mathcal{M}}_{p,\mathrm{\infty }}^{\lambda }(d\nu )}$   if and only if the Radon measure $d\mu$ supported in $\mathrm{\Omega }\subset {\mathbf{R}}^n$ is controlled by

$\Vert \mu {\Vert }_{\beta }=\underset{x\in {\mathbf{R}}^n,r>0}{sup}{r}^{-\beta }\mu (B(x,r))<\mathrm{\infty }$
provided that $1<p<q<\mathrm{\infty }$ satisfies $n-\alpha p<\beta \le n$, $\alpha =\frac{n}{\lambda }-\frac{\beta }{{\lambda }_{\ast }}$ 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 ${\dot{B}}_{p,\mathrm{\infty }}^s\hookrightarrow L^{\lambda ,\mathrm{\infty }}\hookrightarrow {\mathcal{M}}_p^{\lambda }\hookrightarrow {\mathcal{M}}_{p,\mathrm{\infty }}^{\lambda }$ as $1<p<\lambda <\mathrm{\infty }$ and $s\in \mathbf{R}$ satisfies $\frac{1}{p}-\frac{s}{n}=\frac{1}{\lambda }$. If $d\mu$ is concentrated on $\partial {\mathbf{R}}_{+}^n$, as a byproduct we get Sobolev-Morrey trace inequality on half-spaces ${\mathbf{R}}_{+}^n$ which recovers the well-known Sobolev-trace inequality in $L^p({\mathbf{R}}_{+}^n)$. Also, by a suitable analysis on non-doubling Calderón-Zygmund decomposition we show that   $\Vert M_{\alpha }f{\Vert }_{{\mathcal{M}}_{p,\ell }^{\lambda }(d\mu )}\sim \Vert I_{\alpha }f{\Vert }_{{\mathcal{M}}_{p,\ell }^{\lambda }(d\mu )}$
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.