Carleson measure problems for parabolic Bergman spaces and homogeneous Sobolev spaces

TL;DR

This paper characterizes measures for parabolic Bergman and Sobolev spaces that ensure boundedness of solutions and their traces, establishing inequalities, decay properties, and capacity relations.

Contribution

It provides new characterizations of Carleson measures for parabolic Bergman and homogeneous Sobolev spaces, including trace and capacitary inequalities.

Findings

Characterization of measures for boundedness of solutions in parabolic spaces

Establishment of decay and iso-capacitary inequalities

Development of trace inequalities for solutions

Abstract

Let $b_{\alpha}^{p}(\mathbb{R}^{1+n}_{+})$ be the space of solutions to the parabolic equation $\partial_{t}u+(-\triangle)^{\alpha}u=0$ $(\alpha\in(0, 1])$ having finite $L^{p}(\mathbb{R}^{1+n}_{+})$ norm. We characterize nonnegative Radon measures $\mu$ on $\mathbb{R}^{1+n}_{+}$ having the property $\|u\|_{L^{q}(\mathbb{R}^{1+n}_{+},\mu)}\lesssim \|u\|_{\dot{W}^{1,p}(\mathbb{R}^{1+n}_{+})},$ $1\leq p\leq q<\infty,$ whenever $u(t,x)\in b_{\alpha}^{p}(\mathbb{R}^{1+n}_{+})\cap \dot{W}^{1.p}(\mathbb{R}^{1+n}_{+}).$ Meanwhile, denoting by $v(t,x)$ the solution of the above equation with Cauchy data $v_{0}(x),$ we characterize nonnegative Radon measures $\mu$ on $\mathbb{R}_{+}^{1+n}$ satisfying $\|v(t^{2\alpha},x)\|_{L^{q}(\mathbb{R}_{+}^{1+n}, \mu)}\lesssim\|v_{0}\|_{\dot{W}^{\beta,p}(\mathbb{R}^{n})},$ $\beta\in (0,n),$ $p\in [1, n/\beta],$ $q\in(0, \infty).$ Moreover, we obtain the decay of $v(t,x),$ an iso$-$capacitary inequality and a trace inequality.