## Weighted norm inequalities for fractional integrals

HTML articles powered by AMS MathViewer

- by Benjamin Muckenhoupt and Richard Wheeden PDF
- Trans. Amer. Math. Soc.
**192**(1974), 261-274 Request permission

## Abstract:

The principal problem considered is the determination of all nonnegative functions, $V(x)$, such that $\left \|{T_\gamma }f(x)V(x)\right \|_q \leq C\left \|f(x)V(x)\right \|_p$ where the functions are defined on ${R^n},0 < \gamma < n,1 < p < n/\gamma ,1/q = 1/p - \gamma /n$,*C*is a constant independent of

*f*and ${T_\gamma }f(x) = \smallint f(x - y)|y{|^{\gamma - n}}dy$. The main result is that $V(x)$ is such a function if and only if \[ {\left ( {\frac {1}{{|Q|}}\int _Q {{{[V(x)]}^q}dx} } \right )^{1/q}}{\left ( {\frac {1}{{|Q|}}\int _Q {{{[V(x)]}^{ - p’}}dx} } \right )^{1/p’}} \leq K\] where

*Q*is any

*n*dimensional cube, $|Q|$ denotes the measure of

*Q*, $p’ = p/(p - 1)$ and

*K*is a constant independent of

*Q*. Substitute results for the cases $p = 1$ and $q = \infty$ and a weighted version of the Sobolev imbedding theorem are also proved.

## References

- R. R. Coifman and C. Fefferman,
*Weighted norm inequalities for maximal functions and singular integrals*, Studia Math.**51**(1974), 241–250. MR**358205**, DOI 10.4064/sm-51-3-241-250 - 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**, DOI 10.4064/sm-34-3-299-317 - C. Fefferman and E. M. Stein,
*$H^{p}$ spaces of several variables*, Acta Math.**129**(1972), no. 3-4, 137–193. MR**447953**, DOI 10.1007/BF02392215 - Richard A. Hunt,
*On $L(p,\,q)$ spaces*, Enseign. Math. (2)**12**(1966), 249–276. MR**223874** - 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 - G. H. Hardy and J. E. Littlewood,
*Some properties of fractional integrals. I*, Math. Z.**27**(1928), no. 1, 565–606. MR**1544927**, DOI 10.1007/BF01171116 - Benjamin Muckenhoupt,
*Weighted norm inequalities for the Hardy maximal function*, Trans. Amer. Math. Soc.**165**(1972), 207–226. MR**293384**, DOI 10.1090/S0002-9947-1972-0293384-6
S. Sobolev, - Elias M. Stein,
*Singular integrals and differentiability properties of functions*, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR**0290095** - E. M. Stein and Guido Weiss,
*Fractional integrals on $n$-dimensional Euclidean space*, J. Math. Mech.**7**(1958), 503–514. MR**0098285**, DOI 10.1512/iumj.1958.7.57030 - 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 - T. Walsh,
*On weighted norm inequalities for fractional and singular integrals*, Canadian J. Math.**23**(1971), 907–928. MR**287386**, DOI 10.4153/CJM-1971-100-1 - A. Zygmund,
*Trigonometric series: Vols. I, II*, Cambridge University Press, London-New York, 1968. Second edition, reprinted with corrections and some additions. MR**0236587**

*On a theorem in functional analysis*, Mat. Sb.

**4**

**(46)**(1938), 471-497; English transl., Amer. Math. Soc. Transl. (2)

**34**(1963), 39-68.

## Additional Information

- © Copyright 1974 American Mathematical Society
- 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