Weighted Norm Inequalities for the Hardy Maximal Function

Abstract

Abstract. The principal problem considered is the determination of all nonnegative functions, U(x), for which there is a constant, C, such that

Weak Type Estimates for Singular Integrals Related to a Dual Problem of Muckenhoupt–Wheeden

“…Note that by (2.4) along with (2.3), a weight (M w) 1−r(1− 1 2p ) belongs to A r with corresponding constants independent of w. Hence, by the Muckenhoupt theorem [12], S is bounded on L r M w for any r > 1. Therefore, by the Marcinkiewicz interpolation theorem [1, p. 225], S is bounded on L p ,1 M w .…”

“…Set A ∞ = ∪ p≥1 A p . We recall that Muckenhoupt's theorem [12] says that the maximal operator M is bounded on L p w , 1 < p < ∞, if and only if w ∈ A p . We mention several well-known facts about A p weights.…”

Some one-sided estimates for oscillatory singular integrals

“…The smallest constant C for which (1.1) is satisfied will be denoted by A + 1 (w) and [28]. This class consists of weight functions w for which…”

Weighted norm inequalities for general maximal operators

“…. Assuming (13) it is standard to see that w E AP L3, and hence o = Hence, by using the iteration technique of Rubio de Francia (cf. [7] p. 434, [9] p.392, or the previous work of Gagliardo in [6] ), there exists a nonnegative measurable function h E LP'(Rn) for which or, equivalently, Taking go = w -1/php'lp we obtain (14) .…”

“…In [13] Muckenhoupt proved the fundamental result characterizing all the weights for which the Hardy-Littlewood maximal operator is bounded ; the surprisingly simple necessary and sufficient condition is the so called AP-condition (see below). A different approach to this characterization was found by Jawerth (cf .…”