Extension Problems Related to Fractional Operators on Metric Measure Spaces

Abstract

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 ($\mathbb{M}$, d, μ) supporting the weak Poincaré inequality. Our aim is to characterize such a measure ν on $\mathbb{M}\times(0,\infty)$ that $f\mapsto\int_{\mathbb{M}}p_{t^{\alpha d_{w}}}^{\alpha}(\cdot,y)f(y)d\mu(y)$ is bounded from the Newton–Sobolev spaces and the Lebesgue space into the Lebesgue space $L^{q}(\mathbb{M}\times(0,\infty),\nu)$ 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.

Keywords

Sub-Gaussian estimates / capacities / metric measure spaces / extension / fractional Laplacian

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