Publication: Interpolation of compact bilinear operators among quasi-Banach spaces and applications
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2018-07
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WILEY-VCH Verlag
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Abstract
We study the interpolation properties of compact bilinear operators by the general real method among quasi- Banach couples. As an application we show that commutators of Calderón-Zygmund bilinear operators S : Lp × Lq -? Lr are compact provided that 1/2 < r < 1, 1 < p, q < 8 and 1/p + 1/q = 1/r.
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[1] C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, New York, 1988.
[2] A . Bényi and T. Oh, Smoothing of commutators for a Hörmander class of bilinear
pseudodifferential operators, J. Fourier Anal. Appl. 20 (2014) 282–300.
[3] A . Bényi and R.H. Torres, Compact bilinear operators and commutators, Proc. Amer. Math.
Soc. 141 (2013) 3609–3621.
[4] J. Bergh and J. Lófström, Interpolation Spaces. An introduction, Springer, Berlin, 1976.
[5] Y. Brudny?i and N. Krugljak, Interpolation Functors and Interpolation Spaces, Vol. 1,
North-Holland, Amsterdam, 1991. [6] P.L. Butzer and H. Berens, Semi-Groups of Operators and
Approximation, Springer, Berlin, 1967.
[7] A.P. Calderón, Intermediate spaces and interpolation, the complex method, Studia Math. 24
(1964) 113–190.
[8] A.P. Calderón and A. Zygmund, A note on the interpolation of linear operations, Studia Math.
12 (1951) 194–204.
[9] F. Cobos, Interpolation theory and compactness, in: Function Spaces, Inequalities and
Interpolation (Paseky 2009). Edited by J. Luke?s and L. Pick, Matfyzpress, Prague, 2009, pp. 31–75.
[10] F. Cobos, L.M. Fernández-Cabrera, A. Manzano and A. Martínez, Real interpolation and closed
operator ideals, J. Math. Pures Appl. 83 (2004) 417–432.
[11] F. Cobos, L.M. Ferna´ndez-Cabrera and A. Mart´inez, Compact operators between K- and J
-spaces, Studia Math. 166 (2005) 199–220.
[12] F. Cobos, L.M. Ferna´ndez-Cabrera and A. Mart´inez, Abstract K and J spaces and measure of
non-compactness, Math.Nachr. 280 (2007) 1698-1708.
[13] F. Cobos, T. Ku¨hn and T. Schonbek, One-sided compactness results for Aronszajn-Gagliardo
functors, J. Funct. Anal. 106 (1992) 274-313.
[14] F. Cobos and J. Peetre, Interpolation of compactness using Aronszajn-Gagliardo functors,
Israel J. Math. 68 (1989) 220-240.
[15] F. Cobos and L.-E. Persson, Real interpolation of compact operators between quasi-Banach
spaces, Math. Scand. 82 (1998) 138–160.
[16] F. Cobos and A. Segurado, Description of logarithmic interpolation spaces by means of the J
-functional and applica-
tions, J. Funct. Anal. 268 (2015) 2906–2945.
[17] M. Cwikel, Real and complex interpolation and extrapolation of compact operators, Duke Math.
J. 65 (1992) 333-343. [18] M. Cwikel and J. Peetre, Abstract K and J spaces, J. Math. Pures Appl.
60 (1981) 1-50.
[19] D. E. Edmunds and B. Opic, Limiting variants of Krasnosel’ski?i’s compact interpolation
theorem, J. Funct. Anal. 266
(2014) 3265–3285.
[20] W.D. Evans and B. Opic, Real interpolation with logarithmic functors and reiteration, Canad.
J. Math. 52 (2000) 920– 960.
[21] W.D. Evans, B. Opic and L. Pick, Real interpolation with logarithmic functors, J. Inequal.
Appl. 7 (2002) 187–269. [22] D.L. Fernandez and E.B. da Silva, Interpolation of bilinear operators
and compactness, Nonlinear Anal. 73 (2010)
526–537.
[23] L.M. Fernández-Cabrera and A. Martínez, On interpolation properties of compact bilinear
operators, Math. Nachr. 290 (2017) 1663–1677.
[24] L.M. Ferna´ndez-Cabrera and A. Mart´inez, Real interpolation of compact bilinear operators,
J. Fourier Anal. Appl. (2017). https://doi.org/10.1007/s00041-017-9561-7.
[25] L. Grafakos and R.H. Torres, Multilinear Calderón-Zygmund theory, Adv. Math. 165 (2002)
124–164.
[26] J. Gustavsson, A function parameter in connection with interpolation of Banach spaces, Math.
Scand. 42 (1978) 289– 305.
[27] G. Hu, Compactness of the commutator of bilinear Fourier multiplier operator, Taiwanese J.
Math. 18 (2014) 661–675. [28] S. Janson, Minimal and maximal methods of interpolation, J. Funct.
Anal. 44 (1981) 50–73.
[29] S. Janson, On interpolation of multi-linear operators, in Function Spaces and Applications.
Edited by M. Cwikel, J. Peetre, Y. Sagher and H. Wallin, Springer Lect. Notes Math. 1302, Berlin,
1988, pp.290–302.
[30] G.E. Karadzhov, The interpolation method of ”means” for quasinormed spaces, Doklady Acad. Nauk
SSSR 209 (1973) 33–36.
[31] H. König, On the tensor stability of s-number ideals, Math. Ann. 269 (1984) 77–93. [32] H.
Ko¨nig, Eigenvalue Distribution of Compact Operators, Birkha¨user, Basel, 1986. [33] G. Ko¨the,
Topological Vector Spaces I, Springer, Berlin, 1969.
[34] M.A. Krasnosel’ski?i, On a theorem of M. Riesz, Soviet Math. Dokl. 1 (1960) 229–231.
[35] M.A. Krasnosel’ski?i, P.P. Zabreiko, E.I. Pustylnik and P.E. Sbolevskii Integral operators in
spaces of summable func- tions, Noordhoff, Leyden, 1976.
[36] A.K. Lerner, S. Ombrosi, C. Pe´rez, R.H. Torres and R. Trujillo-González, New maximal
functions and multiple weights for the multilinear Caldero´n-Zygmund theory, Adv. Math. 220 (2009)
1222–1264.
[37] J.-L. Lions and J. Peetre, Sur une classe d´espaces d´interpolation, Inst. Hautes E´ tudes
Sci. Publ. Math. 19 (1964) 5–68.
[38] P. Nilsson, Reiteration theorems for real interpolation and approximation spaces, Ann. Mat.
Pura Appl. 132 (1982) 291–330.
[39] P. Nilsson, Interpolation of Caldero´n and Ov?cnnikov pairs, Ann. Mat. Pura Appl. 134 (1983)
201–232. [40] J. Peetre, A theory of interpolation of normed spaces, Lecture Notes, Brasilia, 1978.
[41] C. Pe´rez, G. Pradolini, R.H. Torres and R. Trujillo-Gonza´lez, End-point estimates for
iterated commutators of multilin- ear singular integrals, Bull. London Math. Soc. 46 (2014) 26–42.
[42] A. Persson, Compact linear mappings between interpolation spaces, Ark. Mat. 5 (1964) 215–219.
[43] L.-E. Persson, Interpolation with a parameter function, Math. Scand. 59 (1986) 199–222.
[44] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland,
Amsterdam, 1978. [45] H. Triebel, Theory of Function Spaces II, Birkha¨user, Basel, 1992.