Convolutions, tensor products and multipliers of the Orlicz-Lorentz spaces

Abstract

In this paper, the authors first give the properties of the convolutions of Orlicz-Lorentz spaces Λ_{φ1, w} and Λ_{φ2, w} on the locally compact abelian group. Secondly, the authors obtain the concrete representation as function spaces for the tensor products of Orlicz-Lorentz spaces Λ_{φ1, w} and Λ_{φ2, w} and get the space of multipliers from the space Λ_{φ1, w} to the space M _{φ2*, w} Finally, the authors discuss the homogeneous properties for the Orlicz-Lorentz space Λ_{φ, w} ^{p, q}.

Keywords

Orlicz-Lorentz spaces / Convolution / Tensor products / Multipliers / Hardy operator

References

[1] O’Neil R. Convolution operators and L(p, q) spaces. Duke Math. J., 1963, 30: 129-142

[2] Yap L Y H. Some remarks on convolution operators and L(p, q) spaces. Duke Math. J., 1969, 36: 647-658

[3] Kamińska A, Musielak J. On convolution operator in Orlicz spaces. Congress on Functional Analysis, Madrid, 1988. Rev. Mat. Univ. Complut. Madrid, 1989, 2: 157-178

[4] Hewitt E, Ross K A. Abstract Harmonic Analysis, Vol. I, 1963, Berlin: Springer-Verlag

[5] Hudzik H, Kamińska A, Musielak J. On Some Banach Algebras Given by a Modular, A. Haar memorial conference, Vol. I, II, Budapest, 1985, 1987, Amsterdam: North-Holland 445-463

[6] O’Neil R. Fractional integration in Orlicz spaces, I.. Trans. Amer. Math. Soc., 1965, 115: 300-328

[7] Zelazko W. On the algebras L _p of locally compact groups. Colloq. Math., 1961, 8: 115-120

[8] Rieffel M A. Induced Banach representations of Banach algebras and locally compact groups. J. Functional Analysis, 1967, 1: 443-491

[9] Rieffel M A. Multipliers and tensor products of L ^p-spaces of locally compact groups. Studia Math., 1969, 33: 71-82

[10] Mem. Amer. Math. Soc., 1955, 16 16

[11] Bonsall F F, Duncan J. Complete Normed Algebras, 1973, New York-Heidelberg: Springer-Verlag

[12] Schatten R. A Theory of Cross-Spaces, 1950, Princeton, N. J.: Princeton University Press

[13] Avci H, Gürkanli A T. Multipliers and tensor products of L (p, q) Lorentz spaces. Acta Math. Sci. Ser. B Engl. Ed., 2007, 27: 107-116

[14] Kerlin J E. Representations of generalized multipliers of L ^p-spaces of locally compact groups. Bull. Amer. Math. Soc., 1973, 79: 1223-1228

[15] Candeal Haro J C, Lai H C. Multipliers in vector-valued function spaces under convolution. Acta Math. Hungar., 1995, 67(3): 175-192

[16] Öztop S. Multipliers of Banach valued weighted function spaces. Int. J. Math. Math. Sci., 2000, 24(8): 511-517

[17] Özto S, Gürkanli A T. Multipliers and tensor products of weighted L ^p-spaces. Acta Math. Sci. Ser. B Engl. Ed., 2001, 21(1): 41-49

[18] Saǧir B. Multipliers and tensor products of vector valued L ^p(G,A) spaces. Taiwanese J. Math., 2003, 7(3): 493-501

[19] Bennett C, Sharpley R. Interpolation of operators, 1988, Boston, MA: Academic Press, Inc.

[20] Mastyło M. Interpolation of linear operators in Calderón-Lozanovskiĭ spaces. Comment. Math. Prace Mat., 1986, 26(2): 247-256

[21] Maligranda L. Indices and interpolation. Dissert. Math., 1984, 234: 1-49

[22] Orlicz W. Über eine gewisse Klasse von Räumen vom Typus B, Bull. Intern. Acad. Pol., 1932, 8: 207-220

[23] Luxemburg W A J. Banach function spaces, 1955, Delft: Delft Technical Univ.

[24] Lorentz G G. Some new functional spaces. Ann. of Math., 1950, 51: 37-55

[25] Lorentz G G. On the theory of spaces Λ. Pacific J. Math., 1951, 1: 411-429

[26] Kamińska A, Mastyło M. Abstract duality Sawyer formula and its applications. Monatsh. Math., 2007, 151(3): 223-245

[27] Li H L. Hardy-type inequalities on strong and weak Orlicz-Lorentz spaces. Sci. China Math., 2012, 55(12): 2493-2505

[28] Ariño M A, Muckenhoupt B. Maximal functions on classical Lorentz spaces and Hardy’s inequality with weights for nonincreasing functions. Trans. Amer. Math. Soc., 1990, 320: 727-735

[29] Sawyer E. Boundedness of classical operators on classical Lorentz spaces. Studia Math., 1990, 96: 145-158

[30] Carro M J, Raposo J A, Soria J. Recent developements in the theory of Lorentz spaces and weighted inequalities, 2007, Providence, RI: Amer. Math. Soc.

[31] Andô T. On products of Orlicz spaces. Math. Ann., 1960, 140: 174-186

[32] Conway J B. A Course in Functional Analysis, 1985, New York: Springer-Verlag

[33] Bloom S, Kerman R. Weighted L _φ integral inequalities for operators of Hardy type. Studia Math., 1994, 110(1): 35-52