Endpoint estimates and weighted norm inequalities for commutators of fractional integrals

Abstract

We prove that the commutator [b, Iα], b ∈ BMO, Iα the fractional integral operator, satisfies the sharp, modular weak-type inequality where B(t) = t log(e + t) and Ψ(t) = [t log(e + t α/n )] n/(n−α). These commutators were first considered by Chanillo, and our result complements his. The heart of our proof consists of the pointwise inequality,where M # is the sharp maximal operator, and M α,B is a generalization of the fractional maximal operator in the scale of Orlicz spaces. Using this inequality we also prove one-weight inequalities for the commutator; to do so we prove one and two-weight norm inequalities for M α,B which are of interest in their own right.

Generalized maximal functions and related operators on weighted Musielak-Orlicz spaces

“…By applying Lemma 3.4, we obtain In [10], the authors proved that, for any a > 1, the function A(y) = a y − 1 y χ (0,1] (y) + log a χ {y=0} (y)…”

“…The boundedness of many operators in harmonic analysis that appear in connection with the study of regularity properties of the solutions of partial differential equations were widely considered in the variable context by different authors, see for instance [9], [11], [14], [15], [16], [30], [31], [32], [38], [39] and [40] for the HardyLittlewood maximal function M, [5], [21], [22] and [28] for the fractional maximal function M α , [18] and [33] for Calderón-Zygmund operators and their commutators, and [1], [10], [24] and [28] for potential type operators (see [13] for other classical operators).…”

“…Following [10], and from the fact that p(·) satisfies the log-Hölder condition (1.3) and p + < ∞ we have that p(·) 2 and then, R(·) are also log-Hölder continuous functions. We now define r(x) = min(p(x), R(x)) which also satisfies the log-Hölder condition (1.3).…”

“…Fix 0 < ǫ < 1. We define r(·) and q(·) as in Theorem 1.5 of [10]. For the sake of completeness we include the definition and properties.…”

Weighted endpoint estimates for commutators of fractional integrals

“…Here we sketch the main steps, and we refer the reader to [3] for details. On the left-hand side of (1.1) (or (1.2)) we replace |[b, …”