Application of Capacities to Space-Time Fractional Dissipative Equations II: Carleson Measure Characterization for Lq(ℝ+n+1,μ) L^q (\mathbb{R}_ + ^{n + 1} ,\mu ) −Extension

Abstract

Abstract This paper provides the Carleson characterization of the extension of fractional Sobolev spaces and Lebesgue spaces to Lq(ℝ+n+1,μ) L^q (\mathbb{R}_ + ^{n + 1} ,\mu ) via space-time fractional equations. For the extension of fractional Sobolev spaces, preliminary results including estimates, involving the fractional capacity, measures, the non-tangential maximal function, and an estimate of the Riesz integral of the space-time fractional heat kernel, are provided. For the extension of Lebesgue spaces, a new Lp–capacity associated to the spatial-time fractional equations is introduced. Then, some basic properties of the Lp–capacity, including its dual form, the Lp–capacity of fractional parabolic balls, strong and weak type inequalities, are established.