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

Leonardo Colzani; Peter Sjögren

Studia Mathematica (1999)

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

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topColzani, 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

top- [CO] L. Colzani, Translation invariant operators on Lorentz spaces, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 14 (1987), 257-276. Zbl0655.47025
- [HU] R. Hunt, On L(p,q) spaces, Enseign. Math. 12 (1966), 249-287.
- [KA] Y. Katznelson, An Introduction to Harmonic Analysis, Wiley, New York, 1968. Zbl0169.17902
- [DLE] K. de Leeuw, On ${L}_{p}$ multipliers, Ann. of Math. 81 (1965), 364-379.
- [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.
- [SJ-1] P. Sjögren, Translation-invariant operators on weak ${L}^{1}$, J. Funct. Anal. 89 (1990), 410-427.
- [SJ-2] P. Sjögren, Convolutors on Lorentz spaces L(1,q) with 1 < q < ∞, Proc. London Math. Soc. 64 (1992), 397-417.
- [SGW] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, Princeton, 1971.
- [SNW] E. M. Stein and N. J. Weiss, On the convergence of Poisson integrals, Trans. Amer. Math. Soc. 140 (1969), 35-54. Zbl0182.10801
- [TU] P. Turpin, Convexités dans les espaces vectoriels topologiques généraux, Dissertationes Math. 131 (1976).

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