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

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

DOI: https://doi.org/10.1090/S0002-9947-1985-0784004-1

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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.

References

Ernst Adams, On weighted norm inequalities for the Riesz transforms of functions with vanishing moments, Studia Math. 78 (1984), no. 2, 107–153. MR 766709, DOI 10.4064/sm-78-2-107-153
Ernst Adams, On the identification of weighted Hardy spaces, Indiana Univ. Math. J. 32 (1983), no. 4, 477–489. MR 703279, DOI 10.1512/iumj.1983.32.32034
L. Caffarelli, R. Kohn, and L. Nirenberg, First order interpolation inequalities with weights, Compositio Math. 53 (1984), no. 3, 259–275. MR 768824
A.-P. Calderón and A. Zygmund, Local properties of solutions of elliptic partial differential equations, Studia Math. 20 (1961), 171–225. MR 136849, DOI 10.4064/sm-20-2-181-225
C. Fefferman and E. M. Stein, Some maximal inequalities, Amer. J. Math. 93 (1971), 107–115. MR 284802, DOI 10.2307/2373450
I. M. Gel′fand and G. E. Shilov, Generalized functions. Vol. 2. Spaces of fundamental and generalized functions, Academic Press, New York-London, 1968. Translated from the Russian by Morris D. Friedman, Amiel Feinstein and Christian P. Peltzer. MR 0230128
Lars Inge Hedberg, On certain convolution inequalities, Proc. Amer. Math. Soc. 36 (1972), 505–510. MR 312232, DOI 10.1090/S0002-9939-1972-0312232-4

Weighted Hardy spaces

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Jan-Olov Strömberg and Richard L. Wheeden, Fractional integrals on weighted $H^p$ and $L^p$ spaces, Trans. Amer. Math. Soc. 287 (1985), no. 1, 293–321. MR 766221, DOI 10.1090/S0002-9947-1985-0766221-X
G. V. Welland, Weighted norm inequalities for fractional integrals, Proc. Amer. Math. Soc. 51 (1975), 143–148. MR 369641, DOI 10.1090/S0002-9939-1975-0369641-X