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CHARACTERIZATIONS OF BMO AND LIPSCHITZ SPACES IN TERMS OF $A_{P,Q}$ WEIGHTS AND THEIR APPLICATIONS

Part of: Harmonic analysis in several variables Special classes of linear operators

Published online by Cambridge University Press:  30 January 2019

DINGHUAI WANG ,
JIANG ZHOU  and
ZHIDONG TENG
DINGHUAI WANG*
Affiliation:
School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China email [email protected]
JIANG ZHOU
Affiliation:
College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China email [email protected]
ZHIDONG TENG
Affiliation:
College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China email [email protected]
*
✉[email protected]
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Abstract

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Let $0<\unicode[STIX]{x1D6FC}<n,1\leq p<q<\infty$ with $1/p-1/q=\unicode[STIX]{x1D6FC}/n$, $\unicode[STIX]{x1D714}\in A_{p,q}$, $\unicode[STIX]{x1D708}\in A_{\infty }$ and let $f$ be a locally integrable function. In this paper, it is proved that $f$ is in bounded mean oscillation $\mathit{BMO}$ space if and only if

$$\begin{eqnarray}\sup _{B}\frac{|B|^{\unicode[STIX]{x1D6FC}/n}}{\unicode[STIX]{x1D714}^{p}(B)^{1/p}}\bigg(\int _{B}|f(x)-f_{\unicode[STIX]{x1D708},B}|^{q}\unicode[STIX]{x1D714}(x)^{q}\,dx\bigg)^{1/q}<\infty ,\end{eqnarray}$$
where $\unicode[STIX]{x1D714}^{p}(B)=\int _{B}\unicode[STIX]{x1D714}(x)^{p}\,dx$ and $f_{\unicode[STIX]{x1D708},B}=(1/\unicode[STIX]{x1D708}(B))\int _{B}f(y)\unicode[STIX]{x1D708}(y)\,dy$. We also show that $f$ belongs to Lipschitz space $Lip_{\unicode[STIX]{x1D6FC}}$ if and only if
$$\begin{eqnarray}\sup _{B}\frac{1}{\unicode[STIX]{x1D714}^{p}(B)^{1/p}}\bigg(\int _{B}|f(x)-f_{\unicode[STIX]{x1D708},B}|^{q}\unicode[STIX]{x1D714}(x)^{q}\,dx\bigg)^{1/q}<\infty .\end{eqnarray}$$
 As applications, we characterize these spaces by the boundedness of commutators of some operators on weighted Lebesgue spaces.

Keywords

Ap, q weightsBMO spacecharacterizecommutatorsLipschitz spaces

MSC classification

Primary: 47B47: Commutators, derivations, elementary operators, etc.
Secondary: 42B35: Function spaces arising in harmonic analysis
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 107 , Issue 3 , December 2019 , pp. 381 - 391
Copyright
© 2019 Australian Mathematical Publishing Association Inc. 

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