Weighted Fractional Leibniz-Type Rules for Bilinear Multiplier Operators

Weighted Fractional Leibniz-Type Rules for Bilinear Multiplier Operators Potential Anal https://doi.org/10.1007/s11118-018-9703-9 Weighted Fractional Leibniz-Type Rules for Bilinear Multiplier Operators 1 1 Joshua Brummer · Virginia Naibo Received: 8 June 2017 / Accepted: 26 April 2018 © Springer Science+Business Media B.V., part of Springer Nature 2018 Abstract We prove weighted fractional Leibniz-type rules for Coifman–Meyer and bipa- rameter Coifman–Meyer multiplier operators. Mapping properties of such operators in the scale of weighted Sobolev spaces then follow. Our results constitute natural extensions of the estimates corresponding to a multiplier identically equal to one and, even in this situation, they lead to new weighted inequalities. Keywords Kato–Ponce inequalities · Fractional Leibniz rules · Weights · Coifman–Meyer multipliers · Biparameter Coifman–Meyer multipliers Mathematics Subject Classification (2010) Primary: 42B25, 42B15 · Secondary: 42B20, 46E35 1 Introduction and Main Results Let s> 0 and consider D , the homogeneous differentiation operator of order s defined by s n  n D (f )(ξ ) =|ξ | f(ξ), ξ ∈ R ,f ∈ S (R ). 1 1 1 1 1 1 For 1 <p ,p ,q ,q ≤∞, <r ≤∞ such that = + = + , 1 2 1 2 2 r p q p q 1 1 2 2 s> max{0,n( −1)} http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Potential Analysis Springer Journals

Weighted Fractional Leibniz-Type Rules for Bilinear Multiplier Operators

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Publisher
Springer Journals
Copyright
Copyright © 2018 by Springer Science+Business Media B.V., part of Springer Nature
Subject
Mathematics; Potential Theory; Probability Theory and Stochastic Processes; Geometry; Functional Analysis
ISSN
0926-2601
eISSN
1572-929X
D.O.I.
10.1007/s11118-018-9703-9
Publisher site
See Article on Publisher Site

Abstract

Potential Anal https://doi.org/10.1007/s11118-018-9703-9 Weighted Fractional Leibniz-Type Rules for Bilinear Multiplier Operators 1 1 Joshua Brummer · Virginia Naibo Received: 8 June 2017 / Accepted: 26 April 2018 © Springer Science+Business Media B.V., part of Springer Nature 2018 Abstract We prove weighted fractional Leibniz-type rules for Coifman–Meyer and bipa- rameter Coifman–Meyer multiplier operators. Mapping properties of such operators in the scale of weighted Sobolev spaces then follow. Our results constitute natural extensions of the estimates corresponding to a multiplier identically equal to one and, even in this situation, they lead to new weighted inequalities. Keywords Kato–Ponce inequalities · Fractional Leibniz rules · Weights · Coifman–Meyer multipliers · Biparameter Coifman–Meyer multipliers Mathematics Subject Classification (2010) Primary: 42B25, 42B15 · Secondary: 42B20, 46E35 1 Introduction and Main Results Let s> 0 and consider D , the homogeneous differentiation operator of order s defined by s n  n D (f )(ξ ) =|ξ | f(ξ), ξ ∈ R ,f ∈ S (R ). 1 1 1 1 1 1 For 1 <p ,p ,q ,q ≤∞, <r ≤∞ such that = + = + , 1 2 1 2 2 r p q p q 1 1 2 2 s> max{0,n( −1)}

Journal

Potential AnalysisSpringer Journals

Published: May 31, 2018

References

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