Carleson measure problems for parabolic Bergman spaces and homogeneous Sobolev spaces

Details Carleson measure problems for parabolic Bergman spaces and homogeneous Sobolev spaces Carleson measure problems for parabolic Bergman spaces and homogeneous Sobolev spaces

2009-04-21

arxiv.org/abs/0904.3287v1

Mathematics

Analysis of PDEs

25 pages

Scientific paper

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.

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