Interpolation of operators on scales of generalized Lorentz-Zygmund spaces. (English) Zbl 0865.46016

Summary: We define generalized Lorentz-Zygmund spaces and obtain interpolation theorems for quasilinear operators on such spaces, using weighted Hardy inequalities. In the limiting cases of interpolation, we discover a certain scaling property of these spaces and use it to obtain fine interpolation theorems in which the source is a sum of spaces and the target is an intersection of spaces. This yields a considerable improvement of the known results which we demonstrate with examples. We prove sharpness of the interpolation theorems by showing that the constraints on parameters are necessary for the interpolation theorems.


46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46B70 Interpolation between normed linear spaces
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
47B38 Linear operators on function spaces (general)
47G10 Integral operators
26D10 Inequalities involving derivatives and differential and integral operators
Full Text: DOI


[1] and : On Lorentz- Zygmund Spaces, Dissertationes Math. 175 (1980), 1–72
[2] and : Interpolation of Operators, Pure and Appl Math Vol. 129, Academic Press, New York, 1988
[3] Edmunds, Indiana Univ. Math. J. 44 pp 19– (1995)
[4] Edmunds, Studia Math. 115 pp 151– (1995)
[5] , and : Some Remarks on Sobolev Imbeddings in Borderline Cases, Universita degli Studi di Napoli ”Federico II” Preprint, No. 25 (1993), pp. 7
[6] García Cuerva, North-Holland Math. Studies 116 (1985)
[7] Hunt, Bull. Amer. Math. Soc. 70 pp 803– (1964)
[8] Krejn, Transl. Math. Monogr. 54 (1982)
[9] and : Convex Functions and Orlicz Spaces, No-ordhoff. Groningen, 1961
[10] : Banach Function Spaces, Thesis, Delft, 1955
[11] O’Neil, Duke Math. J. 30 pp 129– (1963)
[12] and : Hardy-Type Inequalities, Pitman Research Notes in Math., Series 219, Longman Sci & Tech. Harlow, 1990
[13] Sawyer, Studia Math. 96 pp 145– (1990)
[14] Stein, Studia Math. 31 pp 305– (1969)
[15] : Real - Variable Methods in Harmonic Analysis, Pure and Appl. Math., Vol. 123, Academic Press New York, 1986
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.