We study convolution operators bounded on the non-normable Lorentz spaces L-1,L-q of the real line and the torus. Here 0 < q < 1. On the real line, such an operator is given by convolution with a discrete measure, but on the torus a convolutor can also be an integrable function. We then give some necessary and some sufficient conditions for a measure or a function to be a convolutor on L-1,L-q. In particular, when the positions of the atoms of a discrete measure are linearly independent over the rationals, we give a necessary and sufficient condition. This condition is, however, only sufficient in the general case
Colzani, L., Sjogren, P. (1999). Translation-invariant operators on Lorentz spaces $L(1,q)$ with $0<q><1$</q>. STUDIA MATHEMATICA, 132(2), 101-124.
Translation-invariant operators on Lorentz spaces $L(1,q)$ with $0<1$
Colzani, L;
1999
Abstract
We study convolution operators bounded on the non-normable Lorentz spaces L-1,L-q of the real line and the torus. Here 0 < q < 1. On the real line, such an operator is given by convolution with a discrete measure, but on the torus a convolutor can also be an integrable function. We then give some necessary and some sufficient conditions for a measure or a function to be a convolutor on L-1,L-q. In particular, when the positions of the atoms of a discrete measure are linearly independent over the rationals, we give a necessary and sufficient condition. This condition is, however, only sufficient in the general case
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