Weighted Norm Inequalities for Fractional Integrals

Abstract

ABSTRACT. The principal problem considered is the determination of all nonnegative functions, V(x), such that ||7;/(jc)K(a:)||, < C||/(x)K(x)||, where the functions are defined on R", 0 < y < n, 1 < p < n/y, \/q = \/p -y/n, C is a constant independent of / and Tyf(x) = ff(x -yïiW'dy. The main result is that V(x) is such a function if and only if (Híírwp*r(aiirwr*rsjr where Q is any n dimensional cube, |ß| 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 9=00 and a weighted version of the Sobolev imbedding theorem are also proved.1. Introduction. The first norm inequality for fractional integrals was the one proved by Hardy and Littlewood in [6] for the one dimensional case with V(x) = 1; they also proved a result for V(x) = \x\". The result in n dimensions with V(x) = 1 was obtained by Sobolev in [8] and with V(x) = |x|a by Stein and G. Weiss in [10]. T. Walsh in [12] obtained a result for other weight functions and with a more general operator but did not characterize all such V 's.A slightly stronger result is obtained here than stated in the abstract. It is shown that

Heat Kernel estimates for some elliptic operators with unbounded diffusion coefficients

“…For reader's convenience we provide a direct proof following the methods of [1]). We point out that such type of inequalities follow also by more general results about Sobolev inequalities with respect to weights satisfying Muckenhoupt-type conditions, sse [11], [12].…”

A sharp subelliptic Sobolev embedding theorem with weights

“…We list some results that share a similar theme as ours: Capogna-Danielli-Garofalo [2], Cohn-Lu-Wang [5], Franchi-Gallot-Wheeden [7], Franchi-Lu-Wheeden [8][9][10], Franchi-Pérez-Wheeden [11], Jerison [15], Lu [17,18], 20], Muckenhoupt-Wheeden [21], Pérez-Wheeden [24,25] and Sawyer-Wheeden [28]. We list some results that share a similar theme as ours: Capogna-Danielli-Garofalo [2], Cohn-Lu-Wang [5], Franchi-Gallot-Wheeden [7], Franchi-Lu-Wheeden [8][9][10], Franchi-Pérez-Wheeden [11], Jerison [15], Lu [17,18], 20], Muckenhoupt-Wheeden [21], Pérez-Wheeden [24,25] and Sawyer-Wheeden [28].…”

New Proofs of Two-Weight Norm Inequalities for the Maximal Operator

“…The first arose from the fact that in light of both Theorem 1.1 and the A p -type conditions for the one-weight inequality for the maximal operator (see the results of Muckenhoupt and Wheeden in [11] and [12]), a natural problem was to find conditions for (1.2) to hold which do not involve the operator. We consider two.…”