One-sided Cp estimates via M# function.

Abstract

We recall that w ∈ C+p if there exist " > 0 and C > 0 such that for any a < b < c with c − b < b − a and any measurable set E ⊂ (a, b), the following holds. This condition was introduced by Riveros and de la Torre as a one-sided counterpart of the Cp condition studied first by Muckenhoupt and Sawyer. In this paper we show that given 1<p<q<∞ if w∈C+q then ∥M+f∥Lp(w)≲∥M♯,+f∥Lp(w) and conversely if such an inequality holds, then w∈C+p.