A GENERALIZATION OF BOYD'S INTERPOLATION THEOREM

doi:10.1007/s10473-021-0414-8

摘要/Abstract

摘要： Boyd's interpolation theorem for quasilinear operators is generalized in this paper, which gives a generalization for both the Marcinkiewicz interpolation theorem and Boyd's interpolation theorem. By using this new interpolation theorem, we study the spherical fractional maximal functions and the fractional maximal commutators on rearrangementinvariant quasi-Banach function spaces. In particular, we obtain the mapping properties of the spherical fractional maximal functions and the fractional maximal commutators on generalized Lorentz spaces.

Abstract: Boyd's interpolation theorem for quasilinear operators is generalized in this paper, which gives a generalization for both the Marcinkiewicz interpolation theorem and Boyd's interpolation theorem. By using this new interpolation theorem, we study the spherical fractional maximal functions and the fractional maximal commutators on rearrangementinvariant quasi-Banach function spaces. In particular, we obtain the mapping properties of the spherical fractional maximal functions and the fractional maximal commutators on generalized Lorentz spaces.

中图分类号:

47B38

Kwok-Pun HO. A GENERALIZATION OF BOYD'S INTERPOLATION THEOREM[J]. 数学物理学报(英文版), 2021, 41(4): 1263-1274.

Kwok-Pun HO. A GENERALIZATION OF BOYD'S INTERPOLATION THEOREM[J]. Acta mathematica scientia,Series B, 2021, 41(4): 1263-1274.

参考文献

[1] Agora E, Antezanna J, Carro M, Soria J. Lorentz-Shimogaki and Boyd theorems for weighted Lorentz spaces. J London Math Soc, 2014, 89:321-336
[2] Ariño M, 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
[3] Bennett C, Rudnick K. On Lorentz-Zygmund spaces. Dissert Math, 1980, 175:1-72
[4] Bennett C, Sharpley R. Interpolations of Operators. New York:Academic Press, 1988
[5] Bergh J, Löfström J. Interpolation Spaces. Springer-Verlag, 1976
[6] Boyd D. A class of operators on Lorentz spaces M(φ). Canad J Math, 1967, 19:839-841
[7] Boyd D. Indices of function spaces and relationship to interpolation. Canad J Math, 1969, 21:1245-1254
[8] Calderón A. Spaces between L¹ and L^∞ and the theorem of Marcinkiewicz. Studia Math, 1964, 26:113-190
[9] Carro M, García del Amo A, Soria J. Weak-type weights and normable Lorentz spaces. Proc Amer Math Soc, 1996, 124849-857
[10] Carro M, Soria J. Weighted Lorentz space and the Hardy operator. J Funct Anal, 1993, 112:480-494
[11] Cwikel M. A counter example in nonlinear interpolation. Proc Amer Math Soc, 1977, 62:62-66
[12] Cwikel M, Kamińska A, Maligranda L, Pick L. Are generalized Lorentz "space" really spaces? Proc Amer Math Soc, 2004, 132:3615-3625
[13] Edmunds D E, Evans W D. Hardy Operators, Function Spaces and Embeddings. Springer, 2004
[14] Heikkinen T, Kinnunen J, Korvenpãã J, Tuominen H. Regularity of the local fractional maximal function. Ark Mat, 2015, 53:127-154
[15] Guliyev V, Deringoz F, Hasanov S. Fractional maximal function and its commutators on Orlicz spaces. Anal Math Phys, 2019, 9:165-179
[16] Ho K -P. Fourier integrals and Sobolev embedding on rearrangement invariant quasi-Banach function spaces. Ann Acad Sci Fenn Math, 2016, 41:897-922
[17] Ho K -P. Hardy's inequalities on Hardy spaces. Proc Japan Acad Ser A Math Sci, 2016, 92:125-130
[18] Ho K -P. Fourier type transforms on rearrangement-invariant quasi-Banach function spaces. Glasgow Math J, 2019, 61:231-248
[19] Ho K -P. Interpolation of sublinear operators which map into Riesz spaces and applications. Proc Amer Math Soc, 2019, 147:3479-3492
[20] Ho K -P. Sublinear operators on radial rearrangement-invariant quasi-Banach function spaces. Acta Math Hungar, 2020, 160:88-100
[21] Ho K -P. Martingale transforms and fractional integrals on rearrangement-invariant martingale Hardy spaces. Period Math Hung, 2020, 81:159-173. https://doi.org/10.1007/s10998-020-00318-1
[22] Ho K -P. Linear operators, Fourier integral operators and k-plane transforms on rearrangement-invariant quasi-Banach function spaces. Positivity, 2021, 25:73-96. https://doi.org/10.1007/s11117-020-00750-0
[23] Kalton N, Peck N, Roberts J. An F-space sampler, London Mathematical Society//Lecture Note Series, 89. Cambridge University Press, 1984
[24] Lorentz G G. On the theory of spaces Γ. Pacific J Math, 1951, 1:411-429
[25] Montgomery-Smith S. The Hardy operator and Boyd indices, Interaction between Functional analysis, Harmonic analysis, and Probability. Marcel Dekker, Inc, 1996
[26] Sawyer E. Boundedness of classical operators on classical Lorentz spaces. Studia Math, 1990, 96:145-158
[27] Schlag W, Sogge C. Local smoothing estimates related to the circular maximal theorem. Math Res Lett, 1997, 4:1-15
[28] Stein E. Maximal function. I. Spherical means. Proc Nat Acad Sci USA, 1976, 73:2174-2175
[29] Triebel H. Interpolation Theory, Function Spaces, Differential Operators. North-Holland, 1978
[30] Wang G, Yang J, Chen W. The endpoint estimate for fourier integral operators. Acta Mathematica Scientia, 2021, 41B(2):426-436