Abstract:

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

Key words: characterization, Hardy-Littlewood maximal function, Muckenhoupt A_p class, weighted Morrey spaces, weighted BMO space