Let \((\mathbb {M},d,\mu )\) be a locally compact metric measure space, where \(\mu \) is Ahlfors \(d_{H}\) -regular, i.e., there exist \(c_{1},c_{2},d_{H}\in (0,\infty )\) such that \(c_{1}r^{d_{H}}\le \mu (B(x,r))\le c_{2}r^{d_{H}}\) for any \(r\in [0,\infty ).\) Assume that \(K^{L}_{t}(\cdot ,\cdot )\) is a heat kernel on \(\mathbb {M}\) satisfying sub-Gaussian upper estimates and L is the generator of the semigroup associated with \(K^{L}_{t}(\cdot ,\cdot )\) . In this paper, we study the regularities of the kernel of the generalized Poisson operator \(P^{L}_{t,\sigma }\) , \(\sigma \in (0,1)\) , corresponding to the extension problem: By the \(L^{p}\) -capacities related with \(\{P^{L}_{t,\sigma }\}_{t>0}\) , we characterize the Carleson type embedding \(P^{L}_{t,\sigma }:\ L^{p}(\mathbb {M})\rightarrow L^{q}(\mathbb {M}_{+},v)\) . As an application, under the assumption that the heat kernel satisfies the weak Bakey–Émery non-negative curvature condition, we estimate the Hausdorff dimension of the blow-up set of the Eq. (1).
