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

Marcelo F. de Almeida¹ ; Lidiane S. M. Lima²

¹ Universidade Federal de Sergipe, Departamento de Matemática
² Universidade Federal de Goiás, IME - Departamento de Matemática

Annales Fennici Mathematici, Tome 46 (2021) no. 2, pp. 1161-1177.

Résumé

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   $\Vert I_{\alpha}f\Vert_{\mathcal{M}_{q, \infty}^{\lambda_{\ast}}(d\mu)}\lesssim \Arrowvert\mu\Arrowvert_{\beta}^{{1}/{q}}\,\Vert f\Vert_{\mathcal{M}_{p, \infty}^{\lambda}(d\nu)}$   if and only if the Radon measure $d\mu$ supported in $\Omega\subset \mathbf{R}^n$ is controlled by $\Vert\mu\Vert_{\beta}=\sup_{x\in\mathbf{R}^n,\,r>0}r^{-\beta}\mu(B(x,r))<\infty$ provided that $1<p<q<\infty$ satisfies $n-\alpha p<\beta\leq n$, $\alpha=\frac{n}{\lambda}-\frac{\beta}{\lambda_\ast}$ and $\frac{\lambda_\ast}{q}\leq \frac{\lambda}{p}$. 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 $s\in\mathbf{R}$ 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 $n-\alpha <\beta\leq n$ with $0<\alpha<n$. This result extends the previous ones.  

Détail

Keywords: Riesz potential, trace inequality, Morrey-Lorentz spaces, non-doubling measure