Fractional integrals on weighted 𝐻^{𝑝} spaces

Gatto, Angel E.; Gutiérrez, Cristian E.; Wheeden, Richard L.

doi:10.1090/S0002-9947-1985-0784004-1

Fractional integrals on weighted $H^ p$ spaces
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by Angel E. Gatto, Cristian E. Gutiérrez and Richard L. Wheeden PDF

Trans. Amer. Math. Soc. 289 (1985), 575-589 Request permission

Abstract:

We characterize the pairs of doubling weights $(u,v)$ on ${R^n}$ such that \[ \parallel {I_\alpha }f{\parallel _{H_u^q}} \leq c\parallel f{\parallel _{H_v^p}}, \quad 0 < p < q < \infty \] , where ${I_\alpha },\alpha > 0$, is the fractional integral operator. We also consider the behavior of an associated maximal function. Applications of the results to Sobolev inequalities in weighted ${L^p}$ spaces are given.

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Weighted Hardy spaces

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