Abstract
Abstract This paper provides the Carleson characterization of the extension of fractional Sobolev spaces and Lebesgue spaces to L 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 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.
Highlights
In this paper, we study the Carleson characterization of Lq(R+n+, μ)-extensions of Sobolev spaces and Lebesgue spaces via the following space-time fractional equation:∂βt u(x, t) = −ν(−∆)α/ u(x, t), (x, t) ∈ n+ R+; u(x, ) = φ(x), x n R (1.1)with β ∈ (, ) and α >
This paper provides the Carleson characterization of the extension of fractional Sobolev spaces and Lebesgue spaces to Lq(R+n+, μ) via space-time fractional equations
For the extension of Lebesgue spaces, a new Lp−capacity associated to the spatial-time fractional equations is introduced
Summary
We study the Carleson characterization of Lq(R+n+ , μ)-extensions of Sobolev spaces and Lebesgue spaces via the following space-time fractional equation:. The symbol ∂βt denotes the Caputo fractional derivative de ned as t. Carleson [5] to characterize the interpolating sequences in the algebra H∞ of bounded analytic functions in the open unit disc and to give a solution to the corona problem. In the viewpoint of geometry, a Carleson measure on a domain Ω can be seen as a measure that does not vanish at the boundary ∂Ω when compared to the surface measure on the boundary of Ω.
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