# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [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.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.