On a complete doubling metric measure space $(X,d,/mu)$ supporting the weak Poincaré inequality, by  establishing some capacitary strong-type inequalities for the Hardy-Littlewood maximal operator, we characterize the measure $/nu$ on the space $X/times(0,/infty)$ so that the mapping $f/mapsto /int_{X} p_{t}(/cdot,y)f(y) d/mu(y)$, is bounded from the Newton-Sobolev space $N^{1,p}(X)$ with $p/in [1,/infty)$ into the Lebesgue space $L^q(X/times(0,/infty),/nu)$ with $q/in(0,/infty)$, where the kernels $p_t$  are some generalized heat kernels. This result generalizes the Carleson embeddings obtained in [J. Differential Equations 224 (2006), 277-295], and also provides a priori estimate of the solution of heat equations with Newton-Sobolev data on many metric measure spaces $X$ such as complete Riemannian manifolds and fractals.

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