On a class of operators. (English) Zbl 0732.47033

Braverman, M. Sh.

J. Lond. Math. Soc., II. Ser. 47, No. 1, 119-128 (1993).

Let \(\phi\) be a non-negative function defined on (0,1) which satisfies \(\phi\) (xy)\(\leq B\phi (x)\phi (y)\) for some constant B. Put \[ (T_{\phi}f)(x)=\int^{1}_{0}f(xy)\phi (y)dy. \] The conditions are obtained under which \(T_{\phi}\) is bounded in various function spaces. In particular, the weights are characterized for which \(T_{\phi}\) is bounded in classical Lorentz space \(\Lambda_ p(v)\). For \(\phi \equiv 1\) this provides an alternate proof of a recent result of Arinio and Muckenhoupt.

Reviewer: M.Sh.Braverman

MSC:

47B38	Linear operators on function spaces (general)
46E30	Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

Keywords:

inequality; weights; Lorentz space