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Commutators of Bilinear Pseudo-differential Operators on Local Hardy Spaces with Variable Exponents

Commutators of Bilinear Pseudo-differential Operators on Local Hardy Spaces with Variable Exponents The aim of this paper is to establish the boundedness of the commutator [b , b , T ] 1 2 σ generated by the bilinear pseudo-differential operator T with smooth symbols and b , b ∈ BMO(R ) on product of local Hardy spaces with variable exponents. By 1 2 applying the refined atomic decomposition result, the authors prove that the bilinear p n pseudo-differential operator T is bounded from the Lebesgue space L (R ) into p (·) n p (·) n 1 2 h (R ) × h (R ). Moreover, the boundedness of the commutator [b , b , T ] 1 2 σ on product of local Hardy spaces with variable exponents is also obtained. Keywords Bilinear pseudo-differential operator · Commutator · BMO function · Local Hardy spaces with variable exponent Mathematics Subject Classification 42B20 · 42B35 · 47G30 1 Introduction Hörmander (1967) introduced the class of symbols S with δ, ρ ≥ 0 and m ∈ R, ρ,δ n n composed of a smooth function σ(x,ξ) defined on R × R , such that for all multi- indices α and β, α β m−ρ|α|+δ|α| |D D σ(x,ξ)|≤ C (1 +|ξ |) , (1.1) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the Brazilian Mathematical Society, New Series Springer Journals

Commutators of Bilinear Pseudo-differential Operators on Local Hardy Spaces with Variable Exponents

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References (37)

Publisher
Springer Journals
Copyright
Copyright © 2019 by Sociedade Brasileira de Matemática
Subject
Mathematics; Mathematics, general; Theoretical, Mathematical and Computational Physics
ISSN
1678-7544
eISSN
1678-7714
DOI
10.1007/s00574-019-00184-7
Publisher site
See Article on Publisher Site

Abstract

The aim of this paper is to establish the boundedness of the commutator [b , b , T ] 1 2 σ generated by the bilinear pseudo-differential operator T with smooth symbols and b , b ∈ BMO(R ) on product of local Hardy spaces with variable exponents. By 1 2 applying the refined atomic decomposition result, the authors prove that the bilinear p n pseudo-differential operator T is bounded from the Lebesgue space L (R ) into p (·) n p (·) n 1 2 h (R ) × h (R ). Moreover, the boundedness of the commutator [b , b , T ] 1 2 σ on product of local Hardy spaces with variable exponents is also obtained. Keywords Bilinear pseudo-differential operator · Commutator · BMO function · Local Hardy spaces with variable exponent Mathematics Subject Classification 42B20 · 42B35 · 47G30 1 Introduction Hörmander (1967) introduced the class of symbols S with δ, ρ ≥ 0 and m ∈ R, ρ,δ n n composed of a smooth function σ(x,ξ) defined on R × R , such that for all multi- indices α and β, α β m−ρ|α|+δ|α| |D D σ(x,ξ)|≤ C (1 +|ξ |) , (1.1)

Journal

Bulletin of the Brazilian Mathematical Society, New SeriesSpringer Journals

Published: Nov 18, 2019

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