## Hokkaido Mathematical Journal

### Lorentz spaces as $L_1$-modules and multipliers

#### Abstract

Let $w$ be a weight function on a locally compact group $G$, so that the weighted $L_{1}$-space $L_{1}(w)$ forms a Banach algebra under convolution. Suppose that $G$ acts on a locally compact space $\Omega$, and that $B$ is a Banach space consisting of Radon measures on $\Omega$ which is also a left Banach $L_{1}(w)$-module. Under certain conditions on $B$, we shall characterize those bounded linear operators $T:L_{1}(w)\arrow B$ which satisfy $T(f*g)=f*T(g)$. We shall also show that there are numerous examples of Lorentz spaces which form left Banach $L_{1}(w)$-modules with respect to appropriate weight functions.

#### Article information

Source
Hokkaido Math. J., Volume 23, Number 1 (1994), 55-92.

Dates
First available in Project Euclid: 10 October 2013

https://projecteuclid.org/euclid.hokmj/1381412486

Digital Object Identifier
doi:10.14492/hokmj/1381412486

Mathematical Reviews number (MathSciNet)
MR1263824

Zentralblatt MATH identifier
0799.46031

#### Citation

SAEKI, Sadahiro; THOME, Edward L. Lorentz spaces as $L_1$-modules and multipliers. Hokkaido Math. J. 23 (1994), no. 1, 55--92. doi:10.14492/hokmj/1381412486. https://projecteuclid.org/euclid.hokmj/1381412486