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

• Autorzy: Leonardo Colzani , Peter Sjögren
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Lorentz space; convolution operator

Kategorie tematyczne

• 42A45: Multipliers
• 46E30: Spaces of measurable functions ( L p -spaces, Orlicz spaces, K\"othe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
• 46A16: Not locally convex spaces (metrizable topological linear spaces, locally bounded spaces, quasi-Banach spaces, etc.)

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1999
• Tom: 132
• Numer: 2
• Strony: 101-124

Daty

wydano

1999

poprawiono

1995-06-05

poprawiono

1998-06-22

Twórcy

autor

Leonardo Colzani

• Dipartimento di Matematica, Università degli Studi di Milano, via C. Saldini 50, 20133 Milano, Italy

autor

Peter Sjögren

• Department of Mathematics, Chalmers University of Technology and Göteborg University, S-412 96 Göteborg, Sweden

Bibliografia

• [CO] L. Colzani, Translation invariant operators on Lorentz spaces, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 14 (1987), 257-276.
• [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.
• [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.
• [TU] P. Turpin, Convexités dans les espaces vectoriels topologiques généraux, Dissertationes Math. 131 (1976).