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

Abstrakti

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.