Boundedness of the fractional integral on weighted Lebesgue and Lipschitz spaces

Harboure, Eleonor; Salinas, Oscar; Viviani, Beatriz

doi:10.1090/S0002-9947-97-01644-9

Boundedness of the fractional integral on
weighted Lebesgue and Lipschitz spaces

Authors: Eleonor Harboure, Oscar Salinas and Beatriz Viviani
Journal: Trans. Amer. Math. Soc. 349 (1997), 235-255
MSC (1991): Primary 42B25
DOI: https://doi.org/10.1090/S0002-9947-97-01644-9
MathSciNet review: 1357395
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Abstract: Necessary and sufficient conditions are given for the fractional integral operator $I_\alpha$ to be bounded from weighted strong and weak $% L^p $ spaces within the range $p\geq n/\alpha$ into suitable weighted $BMO$ and Lipschitz spaces. We also characterize the weights for which $% I_\alpha$ can be extended to a bounded operator from weighted $BMO$ into a weighted Lipschitz space of order $\alpha$ . Finally, under an additional assumption on the weight, we obtain necessary and sufficient conditions for the boundedness of $I_\alpha$ between weighted Lipschitz spaces.

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Additional Information

Eleonor Harboure
Affiliation: Programa Especial de Matemática Aplicada and Facultad de Ingenier ía Qu í mica, Universidad Nacional del Litoral, Güemes 3450, 3000 Santa Fe, Rep. Argentina

Oscar Salinas
Affiliation: Programa Especial de Matemática Aplicada and Facultad de Ingenier ía Qu í mica, Universidad Nacional del Litoral, Güemes 3450, 3000 Santa Fe, Rep. Argentina

Beatriz Viviani
Affiliation: Programa Especial de Matemática Aplicada and Facultad de Ingenier ía Qu í mica, Universidad Nacional del Litoral, Güemes 3450, 3000 Santa Fe, Rep. Argentina

DOI: https://doi.org/10.1090/S0002-9947-97-01644-9
Keywords: Fractional integral, weighted Lebesgue and Lipschitz spaces, weighted BMO
Received by editor(s): June 26, 1995
Additional Notes: The authors were supported by the Consejo Nacional de Investigaciones Cient íficas y Técnicas de la República Argentina and by the Universidad Nacional del Litoral, CAI+D Program.
Article copyright: © Copyright 1997 American Mathematical Society