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

Studia Mathematica (1999)

• Volume: 132, Issue: 2, page 101-124
• ISSN: 0039-3223

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Abstract

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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.

How to cite

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Colzani, Leonardo, and Sjögren, Peter. "Translation-invariant operators on Lorentz spaces L(1,q) with 0 < q < 1." Studia Mathematica 132.2 (1999): 101-124. <http://eudml.org/doc/216589>.

@article{Colzani1999,
abstract = {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.},
author = {Colzani, Leonardo, Sjögren, Peter},
journal = {Studia Mathematica},
keywords = {Lorentz space; convolution operator; real line; torus},
language = {eng},
number = {2},
pages = {101-124},
title = {Translation-invariant operators on Lorentz spaces L(1,q) with 0 < q < 1},
url = {http://eudml.org/doc/216589},
volume = {132},
year = {1999},
}

TY - JOUR
AU - Colzani, Leonardo
AU - Sjögren, Peter
TI - Translation-invariant operators on Lorentz spaces L(1,q) with 0 < q < 1
JO - Studia Mathematica
PY - 1999
VL - 132
IS - 2
SP - 101
EP - 124
AB - 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.
LA - eng
KW - Lorentz space; convolution operator; real line; torus
UR - http://eudml.org/doc/216589
ER -

References

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1. [CO] L. Colzani, Translation invariant operators on Lorentz spaces, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 14 (1987), 257-276. Zbl0655.47025
2. [HU] R. Hunt, On L(p,q) spaces, Enseign. Math. 12 (1966), 249-287.
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5. [SH] A. M. Shteĭnberg, Translation-invariant operators in Lorentz spaces, Funktsional. Anal. i Prilozhen. 20 (1986), no. 2, 92-93 (in Russian); English transl.: Functional Anal. Appl. 20 (1986), 166-168.
6. [SJ-1] P. Sjögren, Translation-invariant operators on weak ${L}^{1}$, J. Funct. Anal. 89 (1990), 410-427.
7. [SJ-2] P. Sjögren, Convolutors on Lorentz spaces L(1,q) with 1 < q < ∞, Proc. London Math. Soc. 64 (1992), 397-417.
8. [SGW] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, Princeton, 1971.
9. [SNW] E. M. Stein and N. J. Weiss, On the convergence of Poisson integrals, Trans. Amer. Math. Soc. 140 (1969), 35-54. Zbl0182.10801
10. [TU] P. Turpin, Convexités dans les espaces vectoriels topologiques généraux, Dissertationes Math. 131 (1976).

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