Carleson measure problems for parabolic Bergman spaces and homogeneous Sobolev spaces

Let $b_{\alpha }^p({\mathbb{R}}_{+}^{1+n})$ be the space of solutions to the parabolic equation ${\mathrm{\partial }}_{t}u+(-\mathrm{△}{)}^{\alpha }u=0$ $(\alpha \in (0,1])$ 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}$^{1+n}{+})},$ $1\le p\le q<\mathrm{\infty },$ whenever $u(t,x)\in b_{\alpha}^{p}($\mathbb{R}$^{1+n}{+})\cap \dot{W}^{1.p}($\mathbb{R}$^{1+n}{+}).$Meanwhile,denotingby$v(t,x)$ 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}$^{n})},$ $\beta \in (0,n),$ $p\in [1,n\mathrm{/}\beta ],$ $q\in (0,\mathrm{\infty })\mathrm{.}$ Moreover, we obtain the decay of $v(t,x),$ an iso$-$capacitary inequality and a trace inequality.