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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. 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