On the Coifman type inequality for the oscillation of the one-sided averages

M. LORENTE, M.S. RIVEROS, A. DE LA TORRE

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

Año: 2007 vol. 336 p. 577 - 592

0022-247X

In this paper we study the Coifman type estimate for an oscillationoperator related to  the one-sided discrete square function, $S^+$.We prove that for any $A^+_infty$ weight $w,$ the $L^p(w)$-norm ofthis operator, and therefore the $L^p(w)$-norm of $S^+$, isdominated by a constant times the $L^p(w)$-norm of the one-sided Hardy-Littlewood maximal function iterated two times.For the  $k$-th commutator with a $BMO$ function  we show that$k+2$ iterates of the one-sided Hardy-Littlewood maximal functionare sufficient.