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Double singular integrals: interpolation and correction. (English. Russian original) Zbl 1089.42006
St. Petersbg. Math. J. 16, No. 5, 747-772 (2005); translation from Algebra Anal. 16, No. 5, 1-33 (2004).
Let \((\Omega,\mu)\) be a measure space. Suppose that \(Q\) is an operator acting in all spaces \(L^p(\mu)\), \(1\leq p\leq \infty\), and is a projection in each of these spaces. Put \(X_p= \{ f\in L^p(\mu): Qf=f\}.\) A subcouple \((F_0, F_1)\) of an interpolation couple \((E_0, E_1)\) is said to be \(K\)-closed if for every \(f\in F_0+ F_1\) and every decomposition \(f=e_1+e_2\) of \(f\) with \(e_0\in E_0\), \(e_1\in E_1\), there exists another decomposition \(f=f_0+f_1\), where \(f_0\in F_0\), \(f_1\in F_1\) and \(\| f_i\| \leq c \| e_i\| \), \(i=0,1\). In the paper interpolation problems between \(X_p\) spaces and \(K\)-closedness of the couple \((X_{p_0}, X_{p_1})\) in \((L^{p_0}, L^{p_1})\) are studied for the following operators: Calderón-Zygmund operator acting between \(L^p\) spaces of functions with values in Banach spaces, double singular integrals, Hardy-Littlewood square functions, Littlewood-Paley function, the area integral etc. Correction theorems for these operators and \(K\)-closedness with weights are also discussed.

MSC:
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B25 Maximal functions, Littlewood-Paley theory
46B70 Interpolation between normed linear spaces
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[1] J. Bourgain, Some consequences of Pisier’s approach to interpolation, Israel J. Math. 77 (1992), no. 1-2, 165 – 185. · Zbl 0788.46070
[2] R. R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241 – 250. · Zbl 0291.44007
[3] Ronald R. Coifman and Guido Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), no. 4, 569 – 645. · Zbl 0358.30023
[4] E. M. Dyn\(^{\prime}\)kin, Methods of the theory of singular integrals. II. The Littlewood-Paley theory and its applications, Current problems in mathematics. Fundamental directions, Vol. 42 (Russian), Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1989, pp. 105 – 198, 233 (Russian). E. M. Dyn\(^{\prime}\)kin, Methods of the theory of singular integrals: Littlewood-Paley theory and its applications [ MR1027848 (91j:42015)], Commutative harmonic analysis, IV, Encyclopaedia Math. Sci., vol. 42, Springer, Berlin, 1992, pp. 97 – 194.
[5] José García-Cuerva and José L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland Mathematics Studies, vol. 116, North-Holland Publishing Co., Amsterdam, 1985. Notas de Matemática [Mathematical Notes], 104. · Zbl 0578.46046
[6] P. W. Jones and P. F. X. Müller, Conditioned Brownian motion and multipliers into \(SL^{\infty}\), Preprint, 2003. · Zbl 1059.43002
[7] S. V. Kisljakov, Quantitative aspect of correction theorems, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 92 (1979), 182 – 191, 322 (Russian, with English summary). Investigations on linear operators and the theory of functions, IX. · Zbl 0434.42017
[8] Serguei V. Kislyakov, Fourier coefficients of continuous functions and a class of multipliers, Ann. Inst. Fourier (Grenoble) 38 (1988), no. 2, 147 – 183 (English, with French summary). · Zbl 0607.42004
[9] S. V. Kisliakov, A sharp correction theorem, Studia Math. 113 (1995), no. 2, 177 – 196. · Zbl 0833.42009
[10] S. V. Kislyakov, Some more spaces for which an analogue of the Grothendieck theorem holds, Algebra i Analiz 7 (1995), no. 1, 62 – 91 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 7 (1996), no. 1, 53 – 76. · Zbl 0847.46008
[11] S. V. Kisliakov, Interpolation of \?^{\?}-spaces: some recent developments, Function spaces, interpolation spaces, and related topics (Haifa, 1995) Israel Math. Conf. Proc., vol. 13, Bar-Ilan Univ., Ramat Gan, 1999, pp. 102 – 140. · Zbl 0956.46018
[12] S. V. Kislyakov, On BMO-regular couples of lattices of measurable functions, Studia Math. 159 (2003), no. 2, 277 – 290. Dedicated to Professor Aleksander Pełczyński on the occasion of his 70th birthday (Polish). · Zbl 1063.46014
[13] Serguei V. Kisliakov and Quan Hua Xu, Interpolation of weighted and vector-valued Hardy spaces, Trans. Amer. Math. Soc. 343 (1994), no. 1, 1 – 34. · Zbl 0806.46026
[14] S. V. Kislyakov and Kuankhua Shu, Real interpolation and singular integrals, Algebra i Analiz 8 (1996), no. 4, 75 – 109 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 8 (1997), no. 4, 593 – 615. · Zbl 0908.42007
[15] Gilles Pisier, Interpolation between \?^{\?} spaces and noncommutative generalizations. I, Pacific J. Math. 155 (1992), no. 2, 341 – 368. · Zbl 0747.46050
[16] Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. · Zbl 0207.13501
[17] V. I. Vasyunin, The exact constant in the inverse Hölder inequality for Muckenhoupt weights, Algebra i Analiz 15 (2003), no. 1, 73 – 117 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 15 (2004), no. 1, 49 – 79.
[18] Thomas H. Wolff, A note on interpolation spaces, Harmonic analysis (Minneapolis, Minn., 1981) Lecture Notes in Math., vol. 908, Springer, Berlin-New York, 1982, pp. 199 – 204.
[19] Quan Hua Xu, Some properties of the quotient space \?\textonesuperior (\?^{\?})/\?\textonesuperior (\?^{\?}), Illinois J. Math. 37 (1993), no. 3, 437 – 454.
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