The Effect of the Smoothness of Fractional Type Operators Over Their Commutators with Lipschitz Symbols on Weighted Spaces

Abstract

We prove boundedness results for integral operators of fractional type and their higher order commutators between weighted spaces, including L p -L q , L p -BM O and L p -Lipschitz estimates. The kernels of such operators satisfy certain size condition and a Lipschitz type regularity, and the symbol of the commutator belongs to a Lipschitz class. We also deal with commutators of fractional type operators with less regular kernels satisfying a Hörmander's type inequality. As far as we know, these last results are new even in the unweighted case. Moreover, we give a characterization result involving symbols of the commutators and continuity results for extreme values of p.

On two weighted problems for commutators of classical operators with optimal behaviour on the parameters involved and extrapolation results

“…As a consequence of Theorem 2.6 and Lemma 2.2 we have the following result, which was previously proved in [19].…”

“…In this paper we consider the optimal parameters that appear in connection with twoweighted problems of the type described above, but the range of p and β goes over the region where the classes of weights involved are satisfied by pairs (v, w) of non-trivial weights. The Lipschitz spaces are the same as in [19] and [34], but we can also prove the results for BMO and Morrey spaces, extending some previous theorems proved in [42] for the fractional integral operator. Concretely, we prove two-weighted norm estimates for singular integral and fractional type operators and their higher order commutators between weighted L p and certain generalized spaces related to a parameter β.…”

“…In this section we give some applications of Theorem 2.8. We first state some previous results proved in [34] and [19] in order to extrapolate them, by means of Theorem 2.8, to the variable context. We shall suppose that m ∈ N 0 = N ∪ {0}, and use the convention that β/0 = ∞ if β > 0.…”

“…The smoothness condition associated to these operators is defined as follows. Definition 6.2 ( [19]) Let m ∈ N 0 , 0 ≤ δ < min{1, n/m} and let Φ be a Young function. We say that K…”

“…More specifically, in [19] the authors obtained one-weight estimates for integral operators of fractional type and their higher order commutators between certain spaces. The results include weighted L p -Lipschitz(β) estimates, where β ≥ 0 depends on some parameter related with the symbol of the commutator.…”