F. J. Martín-Reyes

Weighted inequalities for one-sided maximal functions

Let M+ be the maximal operator defined by g MB f(x) = sup ( f(t)lg(t) dt) (f+g(t) dt) where g is a positive locally integrable function on R. We characterize the pairs of nonnegative functions (u, v) for which MB+ applies LP(v) in LP(u) or in weakLp(u) . Our results generalize Sawyers (case g = 1 ) but our proofs are different and we do not use Hardys inequalities, which makes the proofs of the inequalities self-contained.

On weighted inequalities for singular integrals

In this note we consider singular integrals associated to CalderonZygmund kernels. We prove that if the kernel is supported in (0, oo) then the one-sided Ap condition, A-, is a sufficient condition for the singular integral to be bounded in LP(w), 1 < p < oo, or from Ll(wdx) into weak-Ll(wdx) if p = 1. This one-sided Ap condition becomes also necessary when we require the uniform boundedness of the singular integrals associated to the dilations of a kernel which is not identically zero in (0, oo). The two-sided version of this result is also obtained: Muckenhoupts Ap condition is necessary for the uniform boundedness of the singular integrals associated to the dilations of a general Calder6n-Zygmund kernel which is not the function zero either in (-oo,cO) or in (0, oo). INTRODUCTION It is a classical result in the theory of weighted inequalities the fact that the Ap condition of B. Muckenhoupt on a weight w is equivalent to the LP(wdx) boundedness of the Hilbert transform. This result was proved in 1973 by Hunt, Muckenhoupt and Wheeden [HMW]. In 1974 Coifman and Fefferman [CF] gave a different proof which relies on a good-A inequality, producing an integral estimate of the singular integral in terms of the Hardy-Littlewood maximal operator. Since 1986 the work by E. Sawyer [S], Andersen and Sawyer [AS], Martin Reyes, Ortega Salvador and de la Torre [MOT], [MT] has shown that many positive operators of real analysis have one-sided versions for which the classes of weights are larger than Muckenhoupts ones. Our purpose here is to study the corresponding problems for singular integrals. The situation for one-sided singular integrals is different. The symmetry properties of the Hilbert kernel produce the necessary cancellation properties of a singular integral, so that, no one-sided truncation of l/x is expected to produce a one-sided singular integral. Nevertheless, as we show in Lemma (1.5), the class of general singular integral Calderon-Zygmund kernels supported on a half line is nontrivial. We ask for the more general class of weights w for which such singular integral operators are bounded in LP(wdx). It turns out (Theorem (2.1)) that the one-sided Ap condition is a sufficient condition which becomes also necessary when we require Received by the editors March 15, 1995 and, in revised form, January 30, 1996. 1991 Mathematics Subject Classification. Primary 42B25.

Two weight norm inequalities for fractional one-sided maximal operators

In this paper we introduce a new maximal function, the dyadic one-sided maximal function. We prove that this maximal function is equivalent to the one-sided maximal function studied by the authors and Ortega in Weighted inequalities for one-sided maximal functions (Trans. Amer. Math. Soc. 319 (1990)) and by Sawyer in Weighted inequalities for the one-sided HardyLittlewood maximal functions (Trans. Amer. Math. Soc. 297 (1986)), but our function, being dyadic, is much easier to deal with, and it allows us to study fractional maximal operators. In this way we obtain a geometric proof of the characterization of the good weights for fractional maximal operators, answering a question raised by Andersen and Sawyer in Weighted norm inequalities for the Riemann-Liouville and Weyl fractional integral operators (Trans. Amer. Math. Soc. 308 (1988)). Our methods, avoiding complex interpolation, give also the case of different weights for the fractional maximal operator, which was an open problem.

On weighted weak type inequalities for modified Hardy operators

We characterize the pairs of weights (w, v) for which the modified Hardy operator Tf(x) = g(x) ∫ x 0 f applies Lp(v) into weak-Lq(w) where g is a monotone function and 1 ≤ q < p <∞.

Boundedness of operators of Hardy type in Λ P,q spaces and weighted mixed inequalities for singular integral operators *

We consider certain n -dimensional operators of Hardy type and we study their boundedness in These spaces were introduced by M. J. Carro and J. Soria and include weighted L p, q spaces and classical Lorentz spaces. As an application, we obtain mixed weak-type inequalities for Calderon—Zygmund singular integrals, improving results due to K. Andersen and B. Muckenhoupt.

Weighted inequalities for the two-dimensional one-sided Hardy-Littlewood maximal function

In this work we characterize the pairs of weights (w, v) such that the one-sided Hardy-Littlewood maximal function in dimension two is of weak-type (p, p), 1 < p < oo, with respect to the pair (w, v). As an application of this result we obtain a generalization of the classic Dunford-Schwartz Ergodic Maximal Theorem for bi-parameter flows of null-preserving transformations.

Weights for One–Sided Operators

We present a survey about weights for one-sided operators, one of the areas in which Carlos Segovia made significant contributions. The classical Dunford–Schwartz ergodic theorem can be considered as the first result about weights for the one-sided Hardy–Littlewood maximal operator. From this starting point, we study weighted inequalities for one-sided operators: positive operators like the Hardy averaging operator, the one-sided Hardy–Littlewood maximal operator, singular approximations of the identity, one-sided singular integrals. We end with applications to ergodic theory and with some recent results in dimensions greater than 1.

Two weighted inequalities for convolution maximal operators

Let

Good and Bad Measures

\varphi\colon \mathbb{R} \to [0,\infty)

Singular integrals in the Cesàro sense

an integrable function such that