Boundedness of Operators on Certain Weighted Morrey Spaces Beyond the Muckenhoupt Range
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Boundedness of Operators on Certain Weighted Morrey Spaces Beyond the Muckenhoupt Range Javier Duoandikoetxea1
· Marcel Rosenthal2
Received: 20 March 2018 / Accepted: 9 October 2019 / © Springer Nature B.V. 2019
Abstract We prove that for operators satistying weighted inequalities with Ap weights the boundedness on a certain class of Morrey spaces holds with weights of the form |x|α w(x) for w ∈ Ap . In the case of power weights the shift with respect to the range of Muckenhoupt weights was observed by N. Samko for the Hilbert transform, by H. Tanaka for the HardyLittlewood maximal operator, and by S. Nakamura and Y. Sawano for Calder´on-Zygmund operators and others. We extend the class of weights and establish the results in a very general setting, with applications to many operators. For weak type Morrey spaces, we obtain new estimates even for the Hardy-Littlewood maximal operator. Moreover, we prove the necessity of certain Aq condition. Keywords Morrey spaces · Muckenhoupt weights · Hardy-Littlewood maximal operator · Calder´on-Zygmund operators Mathematics Subject Classification (2010) 42B35 · 42B25 · 46E30 · 42B20
1 Introduction For 1 ≤ p < ∞ and 0 ≤ λ < n, let the Morrey space Lp,λ (w) be the collection of all measurable functions f such that f Lp,λ (w) :=
sup x∈Rn ,r>0
1 rλ
1/p
|f |p w
< ∞.
B(x,r)
The first author is supported by the grants MTM2014-53850-P of the Ministerio de Econom´ıa y Competitividad (Spain) and grant IT-641-13 of the Basque Gouvernment. Javier Duoandikoetxea
[email protected] 1
Departamento de Matem´aticas, Universidad del Pa´ıs Vasco/Euskal Herriko Unibertsitatea UPV/EHU, Apdo. 644, 48080 Bilbao, Spain
2
Stuttgart, Germany
(1.1)
J. Duoandikoetxea, M. Rosenthal
We also consider the weak Morrey space W Lp,λ (w), for which p t w({y ∈ B(x, r) : |f (y)| > t}) 1/p f W Lp,λ (w) := sup < ∞. rλ x∈Rn ,r>0,t>0 (Here and in what follows w(A) stands for the integral of w over A.) Clearly, Lp,λ (w) ⊂ W Lp,λ (w). N. Samko proved in [7] that the Hilbert transform is a bounded operator on Lp,λ (|x|α ) for 0 < λ < 1 and λ − 1 < α < λ + p − 1. This range of values of α shows a shift with respect to the corresponding range in the Ap class, which is −1 < α < p − 1. In [10], H. Tanaka explored the boundedness on Lp,λ (w) of the Hardy-Littlewood maximal operator and was able to describe necessary conditions and sufficient conditions, but not a characterization. Nevertheless, for power weights w(x) = |x|α he obtained the sharp range λ − n ≤ α < λ + n(p − 1), which in the one-dimensional case coincides with the range obtained by Samko for the Hilbert transform except at the endpoint α = λ − n. Later on, S. Nakamura and Y. Sawano in [6] studied the boundedness of the Riesz transforms and other singular integrals and obtained similar shifted ranges for the case of Lp,λ (|x|α ) (with open left endpoint). In [3] the authors of this paper proved a general result involving Muckenhoupt weights, under the assumptions of the extrapolation theorem for Ap weights. When partic
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