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This paper provides the Carleson characterization of the extension of fractional Sobolev spaces and Lebesgue spaces toL   q   (    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  L p  –capacity associated to the spatial-time fractional equations is introduced. Then, some basic properties of the  L p  –capacity, including its dual form, the  L p  –capacity of fractional parabolic balls, strong and weak type inequalities, are established.

advances in nonlinear analysis

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