Carleson measure problems for parabolic Bergman spaces and homogeneous Sobolev spaces

Abstract

Let $b_{\alpha }^p({\mathbb{R}}_{+}^{1+n})$ be the space of solutions to the parabolic equation ${\partial }_{t}u+(-△{)}^{\alpha }u=0$ $(\alpha \in (0,1])$ having finite $L^p({\mathbb{R}}_{+}^{1+n})$ norm. We characterize nonnegative Radon measures $\mu$ on ${\mathbb{R}}_{+}^{1+n}$ having the property $\Vert u{\Vert }_{L^q({\mathbb{R}}_{+}^{1+n},\mu )}\lesssim \Vert u{\Vert }_{{\dot{W}}^{1,p}({\mathbb{R}}_{+}^{1+n})},$ $1\le p\le q<\infty ,$ 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 $\Vert v(t^{2\alpha },x){\Vert }_{L^q({\mathbb{R}}_{+}^{1+n},\mu )}\lesssim \Vert v_0{\Vert }_{{\dot{W}}^{\beta ,p}({\mathbb{R}}^n)},$ $\beta \in (0,n),$ $p\in [1,n/\beta ],$ $q\in (0,\infty ).$ Moreover, we obtain the decay of $v(t,x),$ an iso$-$capacitary inequality and a trace inequality.

25 pages