Double weighted commutators theorem for pseudo-differential operators with smooth symbols

Yu-long Deng; Zhi-tian Chen; Shun-chao Long

Czechoslovak Mathematical Journal (2021)

  • Issue: 1, page 173-190
  • ISSN: 0011-4642

Abstract

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Let - ( n + 1 ) < m ≤ - ( n + 1 ) ( 1 - ρ ) and let T a ∈ ℒ ρ , δ m be pseudo-differential operators with symbols a ( x , ξ ) ∈ ℝ n × ℝ n , where 0 < ρ ≤ 1 , 0 ≤ δ < 1 and δ ≤ ρ . Let μ , λ be weights in Muckenhoupt classes A p , ν = ( μ λ - 1 ) 1 / p for some 1 < p < ∞ . We establish a two-weight inequality for commutators generated by pseudo-differential operators T a with weighted BMO functions b ∈ BMO ν , namely, the commutator [ b , T a ] is bounded from L p ( μ ) into L p ( λ ) . Furthermore, the range of m can be extended to the whole m ≤ - ( n + 1 ) ( 1 - ρ ) .

How to cite

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Deng, Yu-long, Chen, Zhi-tian, and Long, Shun-chao. "Double weighted commutators theorem for pseudo-differential operators with smooth symbols." Czechoslovak Mathematical Journal (2021): 173-190. <http://eudml.org/doc/297164>.

@article{Deng2021,
abstract = {Let $-(n+1)<m\le -(n+1)(1-\rho )$ and let $T_\{a\}\in \mathcal \{L\}^\{m\}_\{\rho ,\delta \}$ be pseudo-differential operators with symbols $a(x,\xi )\in \mathbb \{R\}^n\times \mathbb \{R\}^n$, where $0<\rho \le 1$, $0\le \delta <1$ and $\delta \le \rho $. Let $\mu $, $\lambda $ be weights in Muckenhoupt classes $A_\{p\}$, $\nu =(\mu \lambda ^\{-1\})^\{1/p\}$ for some $1<p<\infty $. We establish a two-weight inequality for commutators generated by pseudo-differential operators $T_\{a\}$ with weighted BMO functions $b\in \{\rm BMO\}_\{\nu \}$, namely, the commutator $[b,T_\{a\}]$ is bounded from $L^\{p\}(\mu )$ into $L^\{p\}(\lambda )$. Furthermore, the range of $m$ can be extended to the whole $m\le -(n+1)(1-\rho )$.},
author = {Deng, Yu-long, Chen, Zhi-tian, Long, Shun-chao},
journal = {Czechoslovak Mathematical Journal},
keywords = {pseudo-differential operator; reverse Hölder inequality; $A_p$ weight; commutator},
language = {eng},
number = {1},
pages = {173-190},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Double weighted commutators theorem for pseudo-differential operators with smooth symbols},
url = {http://eudml.org/doc/297164},
year = {2021},
}

TY - JOUR
AU - Deng, Yu-long
AU - Chen, Zhi-tian
AU - Long, Shun-chao
TI - Double weighted commutators theorem for pseudo-differential operators with smooth symbols
JO - Czechoslovak Mathematical Journal
PY - 2021
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
IS - 1
SP - 173
EP - 190
AB - Let $-(n+1)<m\le -(n+1)(1-\rho )$ and let $T_{a}\in \mathcal {L}^{m}_{\rho ,\delta }$ be pseudo-differential operators with symbols $a(x,\xi )\in \mathbb {R}^n\times \mathbb {R}^n$, where $0<\rho \le 1$, $0\le \delta <1$ and $\delta \le \rho $. Let $\mu $, $\lambda $ be weights in Muckenhoupt classes $A_{p}$, $\nu =(\mu \lambda ^{-1})^{1/p}$ for some $1<p<\infty $. We establish a two-weight inequality for commutators generated by pseudo-differential operators $T_{a}$ with weighted BMO functions $b\in {\rm BMO}_{\nu }$, namely, the commutator $[b,T_{a}]$ is bounded from $L^{p}(\mu )$ into $L^{p}(\lambda )$. Furthermore, the range of $m$ can be extended to the whole $m\le -(n+1)(1-\rho )$.
LA - eng
KW - pseudo-differential operator; reverse Hölder inequality; $A_p$ weight; commutator
UR - http://eudml.org/doc/297164
ER -

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