Interpolation of operators when the extreme spaces are $L^{∞}$

• Autorzy: Jesús Bastero , Francisco J. Ruiz
• Języki publikacji: EN

Abstrakty

EN

Under some assumptions on the pair $(A_0,B_0)$, we study equivalence between interpolation properties of linear operators and monotonicity conditions for a pair (Y,Z) of rearrangement invariant quasi-Banach spaces when the extreme spaces of the interpolation are $L^∞$. Weak and restricted weak intermediate spaces fall within our context. Applications to classical Lorentz and Lorentz-Orlicz spaces are given.

Kategorie tematyczne

• 46E30: Spaces of measurable functions ( L p -spaces, Orlicz spaces, K\"othe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1993
• Tom: 104
• Numer: 2
• Strony: 133-150

Daty

wydano

1993

otrzymano

1992-01-24

Twórcy

autor

Jesús Bastero

• Departamento de Matemáticas, Facultad de Ciencias, Universidad de Zaragoza, 50009 Zaragoza, Spain.

autor

Francisco J. Ruiz

• Departamento de Matemáticas, Facultad de Ciencias, Universidad de Zaragoza, 50009 Zaragoza, Spain.

Bibliografia

• [1] M. A. Ariño and B. Muckenhoupt, Maximal functions on classical Lorentz spaces and Hardy's inequality with weights for nonincreasing functions, Trans. Amer. Math. Soc. 320 (2) (1990), 727-735.
• [2] C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, 1988.
• [3] J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Springer, 1976.
• [4] D. W. Boyd, Indices of function spaces and their relationship to interpolation, Canad. J. Math. 21 (1969), 1245-1254.
• [5] A. P. Calderón, Spaces between $L^1$ and $L^∞$ and the theorem of Marcinkiewicz, Studia Math. 26 (1966), 273-299.
• [6] M. Cwikel, K-divisibility of the K-functional and Calderón couples, Ark. Mat. 22 (1) (1984), 39-62.
• [7] J. García-Cuerva and J. L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North-Holland Math. Stud. 116, North-Holland, Amsterdam 1985.
• [8] N. J. Kalton, Endomorphisms of symmetric function spaces, Indiana Univ. Math. J. 34 (2) (1985), 225-247.
• [9] A. Kamińska, Some remarks on Orlicz-Lorentz spaces, Math. Nachr. 147 (1990), 29-38.
• [10] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces II, Springer, 1979.
• [11] G. Lorentz and T. Shimogaki, Interpolation theorems for the pairs of spaces $(L^p, L^∞)$ and $(L^1, L^q)$, Trans. Amer. Math. Soc. 139 (1971), 207-221.
• [12] L. Maligranda, A generalization of the Shimogaki theorem, Studia Math. 71 (1981), 69-83.
• [13] L. Maligranda, Indices and interpolation, Dissertationes Math. 234 (1985).
• [14] M. Mastyło, Interpolation of linear operators in Calderón-Lozanovskii spaces, Comment. Math. 26 (2) (1986), 247-256.
• [15] S. J. Montgomery-Smith, Comparison of Orlicz-Lorentz spaces, Studia Math. 103 (1992), 161-189.
• [16] B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207-226.
• [17] Y. Raynaud, On Lorentz-Sharpley spaces, in: Proc. Workshop "Interpolation Spaces and Related Topics", Haifa, June 1990, IMCP, Vol. 5 (1992), 207-228.
• [18] E. Sawyer, Boundedness of classical operators on classical Lorentz spaces, Studia Math. 96 (1990), 145-158.
• [19] R. Sharpley, Spaces $Λ_α (X)$ and interpolation, J. Funct. Anal. 11 (1972), 479-513.
• [20] T. Shimogaki, An interpolation theorem on Banach function spaces, Studia Math. 31 (1968), 233-240.
• [21] A. Torchinsky, Interpolation of operators and Orlicz classes, ibid. 59 (1976), 177-207.
• [22] M. Zippin, Interpolation of operators of weak type between rearrangement invariant function spaces, J. Funct. Anal. 7 (1971), 267-284.