Another characterizations of Muckenhoupt $A_{p}$ class

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 $M^{p}_{q}(\omega)$ for $1<q< p<\infty$. As a corollary, if $M$ is (weak) bounded on $M^{p}_{q}(\omega)$, 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 $\omega\in \mathrm{BMO}^{p'}(\omega)$ for $1\leq p< \infty$ and $1/p+1/p'=1$.