Restriction of heat equation with Newton–Sobolev data on metric measure space

Abstract

On a complete doubling metric measure space $$({\mathcal {X}},d,\mu )$$ supporting the weak Poincare inequality, by establishing some capacitary strong-type inequalities for the Hardy–Littlewood maximal operator, we characterize such a measure $$\nu $$ on $${\mathcal {X}}\times {\mathbb {R}}_+$$ that $$f\mapsto \int _{{\mathcal {X}}} p_{t^2}(\cdot ,y)f(y)\,d\mu (y)$$ is bounded from Newton–Sobolev space $$N^{1,p}({\mathcal {X}})$$ under $$p\in [1,\infty )$$ into the Lebesgue space $$L^q({\mathcal {X}}\times {\mathbb {R}}_+,\nu )$$ with $$q\in {\mathbb {R}}_+$$ , where the kernel $$p_t$$ satisfies certain two-sided estimate. This offers a priori estimate for the solution to the heat equation with a Newton–Sobolev data on the given metric measure space $${\mathcal {X}}$$ . Via taking $$t\rightarrow 0$$ , a characterization of $$\nu $$ on $${\mathcal {X}}$$ ensuring the continuity of $$N^{1,p}({\mathcal {X}})\subset L^q({\mathcal {X}},\nu )$$ is also obtained.