## On a class of operators.(English)Zbl 0732.47033

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
Full Text: