Translation-invariant operators on Lorentz spaces $L(1,q)$ with $0 < q < 1$

Volume 132 / 1999

Leonardo Colzani, Peter Sjögren Studia Mathematica 132 (1999), 101-124 DOI: 10.4064/sm-132-2-101-124


We study convolution operators bounded on the non-normable Lorentz spaces $L^{1,q}$ 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 $L^{1,q}$. 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.


  • Leonardo Colzani, Peter Sjögren

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