Extension Problems Related to Fractional Operators on Metric Measure Spaces

Let {P _t ^α }_t>0 be the fractional semigroup equipped with the integral kernel {P _t ^α }_t>0 on a complete doubling metric measure space ( \(\mathbb{M}\) $$\mathbb{M}$$ , d, μ) supporting the weak Poincaré inequality. Our aim is to characterize such a measure ν on \(\mathbb{M}\times(0,\infty)\) $$\mathbb{M}\times (0,\mathrm{\infty })$$ that \(f\mapsto\int_{\mathbb{M}}p_{t^{\alpha d_{w}}}^{\alpha}(\cdot,y)f(y)d\mu(y)\) $$f\mapsto {\int }_{\mathbb{M}}{p}_{t^{\alpha d_w}}^{\alpha }(\cdot ,y)f(y)d\mu (y)$$ is bounded from the Newton–Sobolev spaces and the Lebesgue space into the Lebesgue space \(L^{q}(\mathbb{M}\times(0,\infty),\nu)\) $$L^q(\mathbb{M}\times (0,\mathrm{\infty }),\nu )$$ respectively, where the kernel P _t ^α satisfies certain two-sided estimate obtained from the assumption on the heat kernel p_t. Preliminary results including estimates, involving the variational p-capacity and the non-tangential maximal function are provided. For the extension of Lebesgue spaces, a new L^p-capacity associated to the semigroup {P _t ^α }_t>0 is introduced. Then some basic properties of the L^p-capacity, including its dual form, the L^p-capacity of fractional parabolic balls, and capacitary strong type inequalities, are established. Meanwhile, we also obtain the space-time estimate for the semigroup {P _t ^α }_t>0.