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 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.

中文翻译：

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容量在时空分数耗散方程 II 中的应用：Lq(ℝ+n+1,μ) 的 Carleson 测量表征 - 扩展

本文提供了分数 Sobolev 空间和 Lebesgue 空间通过空间扩展到 Lq(ℝ+n+1,μ) L^q (\mathbb{R}_ + ^{n + 1} ,\mu ) 的 Carleson 表征-时间分数方程。对于分数Sobolev空间的扩展，提供了初步结果，包括估计，涉及分数容量、度量、非切线极大函数和时空分数热核的Riesz积分的估计。对于 Lebesgue 空间的扩展，引入了与时空分数方程相关的新 L p -容量。然后，建立了 L p 容量的一些基本性质，包括它的对偶形式、分数抛物线球的 L p 容量、强型和弱型不等式。