Weak trace measures on Hardy-Sobolev spaces

In this paper, we obtain a characterization of the weak trace measures for the Hardy–Sobolev spaces $H^p_s$, that is, the positive Borel measures on $S^n$ such that\[\underset{\lambda \gt 0}{sup} \, \lambda^{p}\mu ( \lbrace \zeta \in \mathbf{S}^{n}; \mathcal{M}_\mathsf{rad} [ f ] ( \zeta ) \gt \lambda \rbrace ) \leq C \lVert f \rVert ^p_{H^p_s},\]when $1 \lt p \lt + \infty$. Also, some partial results on weak $q$-trace measures for the non-diagonal case are obtained.

Keywords

Hardy-Sobolev spaces, Carleson measures

2010 Mathematics Subject Classification

32A35, 32A40, 46E35

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Published 3 December 2013