Boundedness of some integral operators. (English) Zbl 0798.42010

Carro, María J.; Soria, Javier

Can. J. Math. 45, No. 6, 1155-1166 (1993).

The authors consider the integral operator given by \(Tf(x) = \int^ \infty_ 0 k(x,t) f(t)dt\), where \(k:{\mathcal M} \times \mathbb{R}^ + \to \mathbb{R}^ +\) and \(({\mathcal M},\mu)\) is some measure space. They determine the mapping properties of \(T\) mapping functions in \(L^ p_{dec}\), \(0<p \leq 1\), into functions in \(L^ q_ w\), \(q \geq p\); here \(L^ p_{dec}\) is the class of nonincreasing functions in \(L^ p\), and \(L^ q(w)\) is a weighted \(L^ q\) space. They also prove related weak-type inequalities and apply their results to generalized Hardy operators.

Reviewer: A.Seeger (Madison)

MSC:

42B25

Maximal functions, Littlewood-Paley theory

Keywords:

integral operator; weak-type inequalities; generalized Hardy operators