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
HTML articles powered by AMS MathViewer

by Eleonor Harboure, Oscar Salinas and Beatriz Viviani PDF

Trans. Amer. Math. Soc. 349 (1997), 235-255 Request permission

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.

References

Morton E. Harris, Finite groups with Sylow $2$-subgroups of type $PS\textrm {p}(6, q), q$ odd, Comm. Algebra 2 (1974), 181–232. MR 409647, DOI 10.1080/00927877408822010
F. W. Gehring, The $L^{p}$-integrability of the partial derivatives of a quasiconformal mapping, Acta Math. 130 (1973), 265–277. MR 402038, DOI 10.1007/BF02392268
José García-Cuerva, Weighted $H^{p}$ spaces, Dissertationes Math. (Rozprawy Mat.) 162 (1979), 63. MR 549091
A. Eduardo Gatto and Stephen Vági, Fractional integrals on spaces of homogeneous type, Analysis and partial differential equations, Lecture Notes in Pure and Appl. Math., vol. 122, Dekker, New York, 1990, pp. 171–216. MR 1044788
Richard Hunt, Benjamin Muckenhoupt, and Richard Wheeden, Weighted norm inequalities for the conjugate function and Hilbert transform, Trans. Amer. Math. Soc. 176 (1973), 227–251. MR 312139, DOI 10.1090/S0002-9947-1973-0312139-8
Eleonor Harboure, Oscar Salinas, and Beatriz Viviani, Boundedness of the fractional integral in Orlicz and $\phi$-bounded mean oscillation spaces, Proceedings of the Second “Dr. António A. R. Monteiro” Congress on Mathematics (Spanish) (Bahía Blanca, 1993) Univ. Nac. del Sur, Bahía Blanca, 1993, pp. 41–50 (Spanish). MR 1253076
Benjamin Muckenhoupt, Weighted norm inequalities for classical operators, Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978) Proc. Sympos. Pure Math., XXXV, Part, Amer. Math. Soc., Providence, R.I., 1979, pp. 69–83. MR 545240
Benjamin Muckenhoupt and Richard Wheeden, Weighted norm inequalities for fractional integrals, Trans. Amer. Math. Soc. 192 (1974), 261–274. MR 340523, DOI 10.1090/S0002-9947-1974-0340523-6
Benjamin Muckenhoupt and Richard L. Wheeden, Weighted bounded mean oscillation and the Hilbert transform, Studia Math. 54 (1975/76), no. 3, 221–237. MR 399741, DOI 10.4064/sm-54-3-221-237
Jaak Peetre, On the theory of ${\cal L}_{p},\,_{\lambda }$ spaces, J. Functional Analysis 4 (1969), 71–87. MR 0241965, DOI 10.1016/0022-1236(69)90022-6
E. M. Stein and A. Zygmund, Boundedness of translation invariant operators on Hölder spaces and $L^{p}$-spaces, Ann. of Math. (2) 85 (1967), 337–349. MR 215121, DOI 10.2307/1970445
A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776