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4 results

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

  • Part of
  • DINGHUAI WANG, JIANG ZHOU, ZHIDONG TENG
  • Journal:
    Journal of the Australian Mathematical Society / Volume 107 / Issue 3 / December 2019
    Published online by Cambridge University Press:
    30 January 2019, pp. 381-391
    Print publication:
    December 2019

  • 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.

  • AN INTEGRAL REPRESENTATION FOR BESOV AND LIPSCHITZ SPACES

  • Part of
  • KEHE ZHU
  • Journal:
    Journal of the Australian Mathematical Society / Volume 98 / Issue 1 / February 2015
    Published online by Cambridge University Press:
    17 October 2014, pp. 129-144
    Print publication:
    February 2015
    • Article
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  • It is well known that functions in the analytic Besov space $B_{1}$ on the unit disk $\mathbb{D}$ admit an integral representation

    $$\begin{eqnarray}f(z)=\int _{\mathbb{D}}\frac{z-w}{1-z\overline{w}}\,d{\it\mu}(w),\end{eqnarray}$$
    where ${\it\mu}$ is a complex Borel measure with $|{\it\mu}|(\mathbb{D})<\infty$. We generalize this result to all Besov spaces $B_{p}$ with $0<p\leq 1$ and all Lipschitz spaces ${\rm\Lambda}_{t}$ with $t>1$. We also obtain a version for Bergman and Fock spaces.

  • On non-isotropic homogeneous Lipschitz spaces

  • Part of
  • Stefano Meda
  • Journal:
    Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics / Volume 47 / Issue 2 / October 1989
    Published online by Cambridge University Press:
    09 April 2009, pp. 240-255
    Print publication:
    October 1989
    • Article
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  • We prove that in a non-isotropic Euclidean space, homogeneous Lipschitz spaces of distributions, defined in terms of (generalized) Weierstrass integrals, can be characterized by means of higher order difference operators.