Weighted norm inequalities for fractional integrals
Authors:
Benjamin Muckenhoupt and Richard Wheeden
Journal:
Trans. Amer. Math. Soc. 192 (1974), 261-274
MSC:
Primary 26A33
DOI:
https://doi.org/10.1090/S0002-9947-1974-0340523-6
MathSciNet review:
0340523
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: The principal problem considered is the determination of all nonnegative functions, , such that
where the functions are defined on
, C is a constant independent of f and
. The main result is that
is such a function if and only if
![$\displaystyle {\left( {\frac{1}{{\vert Q\vert}}\int_Q {{{[V(x)]}^q}dx} } \right... ...{\frac{1}{{\vert Q\vert}}\int_Q {{{[V(x)]}^{ - p'}}dx} } \right)^{1/p'}} \leq K$](images/img9.gif)




- [1] R. R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241–250. MR 358205, https://doi.org/10.4064/sm-51-3-241-250
- [2] Miguel de Guzmán, A covering lemma with applications to differentiability of measures and singular integral operators, Studia Math. 34 (1970), 299–317. (errata insert). MR 264022, https://doi.org/10.4064/sm-34-3-299-317
- [3] C. Fefferman and E. M. Stein, 𝐻^{𝑝} spaces of several variables, Acta Math. 129 (1972), no. 3-4, 137–193. MR 447953, https://doi.org/10.1007/BF02392215
- [4] Richard A. Hunt, On 𝐿(𝑝,𝑞) spaces, Enseign. Math. (2) 12 (1966), 249–276. MR 223874
- [5] 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, https://doi.org/10.1090/S0002-9947-1973-0312139-8
- [6] G. H. Hardy and J. E. Littlewood, Some properties of fractional integrals. I, Math. Z. 27 (1928), no. 1, 565–606. MR 1544927, https://doi.org/10.1007/BF01171116
- [7] Benjamin Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207–226. MR 293384, https://doi.org/10.1090/S0002-9947-1972-0293384-6
- [8] S. Sobolev, On a theorem in functional analysis, Mat. Sb. 4 (46) (1938), 471-497; English transl., Amer. Math. Soc. Transl. (2) 34 (1963), 39-68.
- [9] Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
- [10] E. M. Stein and Guido Weiss, Fractional integrals on 𝑛-dimensional Euclidean space, J. Math. Mech. 7 (1958), 503–514. MR 0098285, https://doi.org/10.1512/iumj.1958.7.57030
- [11] 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, https://doi.org/10.2307/1970445
- [12] T. Walsh, On weighted norm inequalities for fractional and singular integrals, Canadian J. Math. 23 (1971), 907–928. MR 287386, https://doi.org/10.4153/CJM-1971-100-1
- [13] A. Zygmund, Trigonometric series: Vols. I, II, Second edition, reprinted with corrections and some additions, Cambridge University Press, London-New York, 1968. MR 0236587
Retrieve articles in Transactions of the American Mathematical Society with MSC: 26A33
Retrieve articles in all journals with MSC: 26A33
Additional Information
DOI:
https://doi.org/10.1090/S0002-9947-1974-0340523-6
Article copyright:
© Copyright 1974
American Mathematical Society