Multivariate rearrangements and Banach function spaces with mixed norms
HTML articles powered by AMS MathViewer
- by A. P. Blozinski PDF
- Trans. Amer. Math. Soc. 263 (1981), 149-167 Request permission
Abstract:
Multivariate nonincreasing rearrangement and averaging functions are defined for functions defined over product spaces. An investigation is made of Banach function spaces with mixed norms and using multivariate rearrangements. Particular emphasis is given to the $L(P,Q;\ast )$ spaces. These are Banach function spaces which are in terms of mixed norms, multivariate rearrangements and the Lorentz $L(p,g)$ spaces. Embedding theorems are given for the various function spaces. Several well-known theorems are extended to the $L(P,Q;\ast )$ spaces. Principal among these are the Strong Type (Riesz-Thorin) Interpolation Theorem and the Convolution (Youngโs inequality) Theorem.References
- A. Benedek and R. Panzone, The space $L^{p}$, with mixed norm, Duke Math. J. 28 (1961), 301โ324. MR 126155
- Colin Bennett and Karl Rudnick, On Lorentz-Zygmund spaces, Dissertationes Math. (Rozprawy Mat.) 175 (1980), 67. MR 576995
- A. P. Blozinski, On a convolution theorem for $L(p,q)$ spaces, Trans. Amer. Math. Soc. 164 (1972), 255โ265. MR 415293, DOI 10.1090/S0002-9947-1972-0415293-1
- A. P. Blozinski, Averaging operators and Lipschitz spaces, Indiana Univ. Math. J. 26 (1977), no.ย 5, 939โ950. MR 445285, DOI 10.1512/iumj.1977.26.26076
- Paul L. Butzer and Hubert Berens, Semi-groups of operators and approximation, Die Grundlehren der mathematischen Wissenschaften, Band 145, Springer-Verlag New York, Inc., New York, 1967. MR 0230022
- Michael Cwikel, On $(L^{po}(A_{o}),\,\ L^{p_{1}}(A_{1}))_{\theta },\,_{q}$, Proc. Amer. Math. Soc. 44 (1974), 286โ292. MR 358326, DOI 10.1090/S0002-9939-1974-0358326-0 C. Ballester de Pereyra, Sobre la continuidad debil y magra en ${L^p}({L^q})y$ y su aplicacion a operadores potencials, Tese de Doutorado, Universidad de Buenos Aires, 1963.
- Dicesar Lass Fernandez, Lorentz spaces, with mixed norms, J. Functional Analysis 25 (1977), no.ย 2, 128โ146. MR 0482127, DOI 10.1016/0022-1236(77)90037-4
- Paul R. Halmos, Measure Theory, D. Van Nostrand Co., Inc., New York, N. Y., 1950. MR 0033869 G. H. Hardy, J. E. Littlewood and G. Pรณlya, Inequalities, University Press, Cambridge, 1967.
- Richard A. Hunt, On $L(p,\,q)$ spaces, Enseign. Math. (2) 12 (1966), 249โ276. MR 223874 W. A. J. Luxemburg, Banach functions spaces, Thesis (Defft), Assen, 1955. โ, Rearrangement invariant Banach function spaces, Queenโs Papers in Pure and Applied Math. 10 (1967), 83-144. M. Milman, Embeddings of $L(p,q)$ spaces and Orlicz spaces with mixed norms, Notas de Math., No. 13, Univ. de Los Andes, 1977.
- Mario Milman, Some new function spaces and their tensor products, Notas de Matemรกtica [Mathematical Notes], vol. 20, Universidad de Los Andes, Mรฉrida, 1978. MR 695496 R. OโNeil, Convolution operators and $L(p,q)$ spaces, Duke Math. J. 30 (1963), 129-142.
- Richard OโNeil, Integral transforms and tensor products on Orlicz spaces and $L(p,\,q)$ spaces, J. Analyse Math. 21 (1968), 1โ276. MR 626853, DOI 10.1007/bf02787670 R. S. Schatten, A theory of Cross spaces, Ann. Math. Studies, no. 26, Princeton, N. J., 1950.
- Robert Sharpley, Spaces $\Lambda _{\alpha }(X)$ and interpolation, J. Functional Analysis 11 (1972), 479โ513. MR 0344872, DOI 10.1016/0022-1236(72)90068-7
- Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
- Alberto Torchinsky, Interpolation of operations and Orlicz classes, Studia Math. 59 (1976/77), no.ย 2, 177โ207. MR 438105, DOI 10.4064/sm-59-2-177-207
Additional Information
- © Copyright 1981 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 263 (1981), 149-167
- MSC: Primary 46E30; Secondary 46M35
- DOI: https://doi.org/10.1090/S0002-9947-1981-0590417-X
- MathSciNet review: 590417