On weighted inequalities for singular integrals

Abstract

Abstract. In this note we consider singular integrals associated to Calderón-Zygmund kernels. We prove that if the kernel is supported in (0, ∞) then the one-sided Ap condition, A − p , is a sufficient condition for the singular integral to be bounded in L p (w), 1 < p < ∞, or from L 1 (wdx) into weak-L 1 (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, ∞). The two-sided version of this result is also obtained: Muckenhoupt's Ap condition is necessary for the uniform boundedness of the singular integrals associated to the dilations of a general Calderón-Zygmund kernel which is not the function zero either in (−∞, 0) or in (0, ∞).

Variation Inequalities for One-Sided Singular Integrals and Related Commutators

“…Recently, the variation inequalities and their weighed case for singular integrals and related operators have also been studied by many authors. The first work in this direction is due to Campbell et al [3] who proved that O(H) and V (H) with > 2 are of type (p, p) for 1 < p < ∞ and of weak type (1,1), where H = {H } >0 is the family of the truncated Hilbert transforms, i.e., H f (x) =´| x−y|> f (y)…”

“…The primary purpose of this paper is to study weighted boundedness of oscillation and variational operators for one-sided singular integrals and their commutators. We say a function K belongs to one-sided Calderón-Zygmund kernel OCZK(B 1…”

“…x log x χ (0,∞) ; see [1]. We mention here that the kernel of one-sided truncated Hilbert Transform, K 0 (x) = 1…”

“…The letter C will represent a positive constant that may vary at each occurrence but is independent of the essential variables. For a weight w, an interval I and a function f : R → R, we denote by w(I) =´I w(x)dx and f I = 1 |I|´I f (x)dx. We also use the convention ∑ i∈∅ a i = 0.…”

Two weight norm inequalities for communtators of one-sided singular integrals and the one-sided discrete square function

“…In [1], Aimar, Forzani and Martin-Reyes have studied these operators. They proved that the maximal operators which control them are the one-sided Hardy-Littlewood maximal operators M + and M~, and that the good weights for these operators are the one-sided weights introduced by Sawyer [12].…”