Another characterizations of Muckenhoupt $A_p$ class

Abstract

This manuscript addresses Muckenhoupt $A_p$ weight theory in connection to Morrey and BMO spaces. It is proved that $\omega$ belongs to Muckenhoupt $A_p$ class, if and only if Hardy-Littlewood maximal function $M$ is bounded from weighted Lebesgue spaces $L^p(\omega )$ to weighted Morrey spaces $M_q^p(\omega )$ for $1<q<p<\infty$. As a corollary, if $M$ is (weak) bounded on $M_q^p(\omega )$, then $\omega \in A_p$. The $A_p$ condition also characterizes the boundedness of the Riesz transform $R_j$ and convolution operators $T_{\epsilon }$ on weighted Morrey spaces. Finally, we show that $\omega \in A_p$ if and only if $\omega \in {\mathrm{BMO}}^{p^{\prime }}(\omega )$ for $1\le p<\infty$ and $1/p+1/p^{\prime }=1$.

19 pages

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• Characterizations of weighted BMO space and its application