Weighted norm inequalities for a family of one-sided minimal operators (Q1358717)

Given \(\mu>0\) and a real-valued, non-negative function \(f\) on \(\mathbb{R}\), we define the one-sided \(\mu\)-minimal function of \(f\), \(m^+_\mu f\), by \[ m^+_\mu f(x)= \inf {1 \over |J|} \int_Jdy, \] where the infimum is taken over all intervals \(J\) that lie to the right of \(x\) with the property that \(0\leq \text{dist} (x,J)< \mu|J |\). In this paper we consider the weighted norm inequalities that \(m^+_\mu\) satisfies. We are motivated by our earlier work with the one-sided minimal operator [Stud. Math. 116, No. 3, 255-270 (1995; Zbl 0851.42017)] and a one-sided maximal operator introduced by \textit{A. Martín}-\textit{Reyes} and \textit{F. J. de la Torre} [Proc. Am. Math. Soc. 117, No. 2, 483-489 (1993; Zbl 0769.42010)]. Our basic result determines the pairs of weights \((u,v)\) which govern the strong and weak-type norm inequalities for \(m^+_\mu\) and shows that these are the same. As an application we study the pointwise convergence of convolution operators with ``bad'' kernels.