Weighted norm inequalities for averaging operators of monotone functions

Hardy's inequalities for monotone functions on partly ordered measure spaces

“…We will now prove that, as in the one-dimensional case (see [2] for the original result and [12] for a different proof, related to the one we will use),…”

Weighted norm inequalities for general operators on monotone functions

“…Using the notation of [7], we have This follows from Theorem 2.2, Remark after Theorem 5.4 with ß = 1 and methods developed in [6]. (5) We now compute some special sharp constants.…”

“…It is thus of interest to characterize the weights w : E+ -> R+ for which (1.2) IIVllp.«^C||/||pflB, as this gives extensions of the classical norm inequalities. This is the reason why the study of (1.2) has recently attracted a great deal of attention [3,4,[6][7][8][9], beginning with [1] Ariño and Muckenhoupt for the averaging operator Af(x) = j¿ fo f to the more general version of [3] for operators of the type S<j>f(x) = /0 <p(t)f(tx)dt. All of these operators are special cases of (1.1).…”

“…All of these operators are special cases of (1.1). In this paper we use extensions and refinements of the method introduced in [6] for Af to characterize those w : E+ -* K+ for which (1.2) holds for monotone functions. This will be done §2- §6.…”

Weighted variable Lpintegral inequalities for the maximal operator on non-increasing functions

“…Hf (x) = 1 A different proof of (1)⇔(2) was given by me in [7] where it is also apparent that in the implication (2)⇒(1) the constant c * can be taken to be (c + 1) p . For (1)⇒(2) one uses the test function f = χ [0,r] and (2) follows with c = c * .…”