Norm inequalities for the minimal and maximal operator, and differentiation of the integral

Abstract

We study the weighted norm inequalities for the minimal operator, a new operator analogous to the Hardy-Littlewood maximal operator which arose in the study of reverse Hölder inequalities. We characterize the classes of weights which govern the strong and weak-type norm inequalities for the minimal operator in the two weight case, and show that these classes are the same. We also show that a generalization of the minimal operator can be used to obtain information about the differentiability of the integral in cases when the associated maximal operator is large, and we give a new condition for this maximal operator to be weak (1, 1).

The minimal operator and the geometric maximal operator in Rn

“…Theorem 1.1 follows from Hölder's inequality and the one-weight norm inequalities for the maximal operator. Two-weight norm inequalities are significantly more difficult: working on the real line, Cruz-Uribe, Neugebauer and Olesen [8] were able to prove the following result. Remarks.…”

“…As was noted in [8], unlike the maximal operator, the minimal operator and the centered minimal operator are not equivalent. A simple example on the real line is given by e In [8] it was conjectured that a "centered W p " condition is necessary and sufficient for the weak-type inequality for m c .…”

“…As was noted in [8], unlike the maximal operator, the minimal operator and the centered minimal operator are not equivalent. A simple example on the real line is given by e In [8] it was conjectured that a "centered W p " condition is necessary and sufficient for the weak-type inequality for m c . Here we adapt the proof of Theorem 1.3 to m c and show that this condition (and the corresponding Sawyer-type condition) are necessary and sufficient for both the strong and weak-type norm inequalities.…”

“…The proof of Theorem 1.3 below or the proof of Theorem 1.2 in [8] are readily adapted to the dyadic minimal operator. The results in [6] also carry over to the dyadic geometric maximal operator with little change.…”

“…The proof of Theorem 1.2 in [8] depends heavily on the special covering lemmas for the real line, and does not extend in a satisfactory manner to higher dimensions. In particular, to prove the weak-type norm inequality in R n we had to assume that u was a doubling weight, but to prove the strong-type norm inequality we had to assume (among other things) that σ = v 1/(p+1) was a doubling weight.…”