Another characterizations of Muckenhoupt Ap class

This manuscript addresses Muckenhoupt Ap weight theory in connection to Morrey and BMO spaces. It is proved that ω belongs to Muckenhoupt Ap class, if and only if Hardy-Littlewood maximal function M is bounded from weighted Lebesgue spaces Lp(ω) to weighted Morrey spaces Mqp(ω) for 1 < q < p < ∞. As a corollary, if M is (weak) bounded on Mqp(ω), then ω ∈ Ap. The Ap condition also characterizes the boundedness of the Riesz transform Rj and convolution operators Tɛ on weighted Morrey spaces. Finally, we show that ω ∈ Ap if and only if ω∈BMOp′(ω) for 1 ≤ p < ∞ and 1/p+1/p′=1.