Discrete iterated Hardy-type inequalities with three weights

Abstract

Discrete, continuous Hardy-typ e inequalities are of great imp ortance and have numerous
applications in harmonic analysis, in the theory of integral, differential and difference op erators,
in the theory of emb eddings of function spaces and in other branches of mathematics. In recent
years, weighted estimates for multidimensional Hardy-typ e op erators have b een intensively studied,
which have an imp ortant application in the study of b oundedness prop erties of op erators from a
Leb esgue weighted space to a lo cal Morrey-typ e space, as well as in the study of bilinear op erators
in Leb esgue weighted spaces. The discrete case of Hardy typ e inequalities with three weights is an
op en problem. An inequality involving an iteration of the discrete Hardy op erator is traditionally
considered difficult to estimate b ecause it contains three indep endent weight sequences and three
parameters at their different ratios. In this pap er we prove some new discrete iterated Hardy-typ e
inequality involving three weights for the case 0 < p ≤ min { q , θ }.