×

Fredholmness of singular integral operators with piecewise continuous coefficients on weighted Banach function spaces. (English) Zbl 1044.45004

Let \(L_w^{p(\cdot)}\) denote the weighted Lebesgue space with variable exponent \(p(\cdot)\) and weight \(w\) and let \(R_a=aP_++P_-\) be the singular integral operator where \(P\pm=\frac 12 (I+S)\) and \(S\) be the Cauchy singular operator: \(S\varphi=\frac{1}{\pi i} \int_\Gamma \frac{\varphi(\tau) d\tau}{\tau-t}, t\in\Gamma\). Recently, Fredholmness of the singular integral operators \(R_a\) whose coefficient \(a(t)\) is piece-wise continuous, was studied in the generalized Lebesgue spaces \(L^{p(\cdot)}(\Gamma)\) by V. Kokilashvili and S. Samko [Singular integrals in weighted Lebesgue spaces with variable exponent, Georgian Math. J. 10, No. 1, 145–156 (2003; Zbl 1046.42006); Singular integral equations in the Lebesgue spaces with variable exponent, Proc. Razmadze Math. Inst. 131, 61–78 (2003; Zbl 1159.45302)]. It was shown that the Fredholmness nature of the operator \(R_a\) in the spaces \(L^{p(\cdot)}(\Gamma)\) is related only to values of the exponent \(p(t)\) at the points of discontinuity of the coefficient \(a(t)\), not depending on values of \(p(t)\) at the points of continuity of \(a(t).\) The boundedness of the singular operator in weighted spaces \(L_w^{p(\cdot)}\) with power weight also was given.
In this paper the author bases on the previous results and considers the more general case of bad behaved coefficients and curves. The main result is the proof of the necessity of Fredholmness condition within the framework of general approach of Banach function spaces including the spaces \(L^{p(\cdot)}_w(\Gamma)\) with a general weight. In case of power weight and “nice” curves they are also shown to be sufficient. The case of weighted Banach space is considered by using the terms of indices of submultiplicative functions associated with local properties of the curve, of the weight, and of the space.

MSC:

45P05 Integral operators
47G10 Integral operators
45E05 Integral equations with kernels of Cauchy type
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
47A53 (Semi-) Fredholm operators; index theories
47A68 Factorization theory (including Wiener-Hopf and spectral factorizations) of linear operators
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] C. Bennett and R. Sharpley, Interpolation of operators , Pure Appl. Math., vol. 129, Academic Press, Boston, 1988. · Zbl 0647.46057
[2] E.I. Berezhnoi, Two-weighted estimations for the Hardy-Littlewood maximal function in ideal Banach spaces , Proc. Amer. Math. Soc. 127 (1999), 79-87. JSTOR: · Zbl 0918.42011 · doi:10.1090/S0002-9939-99-04998-9
[3] A. Böttcher and Yu.I. Karlovich, Carleson curves, Muckenhoupt weights, and Toeplitz operators , Progr. Math., vol. 154, Birkhäuser Verlag, Basel, 1997. · Zbl 0889.47001
[4] ——–, Cauchy’s singular integral operator and its beautiful spectrum , in Systems, approximation, singular integral operators, and related topics (Bordeaux, 2000), Oper. Theory Adv. Appl., vol. 129, Birkhäuser Verlag, Basel, 2001, pp. 109-142. · Zbl 1030.47031
[5] A. Böttcher and B. Silbermann, Analysis of Toeplitz operators , Springer-Verlag, Berlin, 1990. · Zbl 0732.47029
[6] K.F. Clancey and I. Gohberg, Factorization of matrix functions and singular integral operators , Oper. Theory Adv. Appl., vol. 3, Birkhäuser Verlag, Basel, 1981. · Zbl 0474.47023
[7] L.A. Coburn, Weyl’s theorem for nonnormal operators , Michigan Math. J. 13 (1966), 285-288. · Zbl 0173.42904 · doi:10.1307/mmj/1031732778
[8] I.I. Danilyuk, Nonregular boundary value problems in the plane , Nauka, Moscow, 1975 (in Russian). · Zbl 0302.45007
[9] I.I. Danilyuk and V.Yu. Shelepov, Boundedness in \(L_p\) of a singular operator with Cauchy kernel along a curve of bounded rotation , Dokl. Akad. Nauk SSSR 174 (1967), 514-517 (in Russian). English transl.: Soviet Math. Dokl. 8 (1967), 654-657. · Zbl 0155.17902
[10] L. Diening, Maximal functions on generalized Lebesgue spaces \(L^p(x)\) , Mathematische Fakultät, Albert-Ludvigs-Universität Freiburg (2002), Preprint Nr. 02/2002-16.01.2002. Math. Inequal. Appl.,
[11] E.M. Dynkin, Methods of the theory of singular integrals. ( Hilbert transform and Calderón-Zygmund theory ), Itogi nauki i tehniki VINITI, Ser. Sovrem. probl. mat., 15 (1987), 197-292 (in Russian). English transl.: Commutative harmonic analysis I. General survey. Classical aspects , Encyclopaedia Math. Sci. 15 (1991), 167-259.
[12] D.E. Edmunds, J. Lang and A. Nekvinda, On \(L^p(x)\) norms , Proc. Roy. Soc. London Ser. A 455 (1999), 219-225. JSTOR: · Zbl 0953.46018 · doi:10.1098/rspa.1999.0309
[13] R.J. Fleming, J.E. Jamison and A. Kamińska, Isometries of Musielak-Orlicz spaces , in Function spaces , Edwardsville, IL, 1990, pp. 139-154; Lecture Notes in Pure and Appl. Math., vol. 136, Dekker, New York, 1992. · Zbl 0769.46018
[14] D. Gaier, Lectures on complex approximation , Birkhäuser Boston, Inc., Boston, MA, 1987. · Zbl 0612.30003
[15] I. Genebashvili, A. Gogatishvili, V. Kokilashvili and M. Krbec, Weight theory for integral transforms on spaces of homogeneous type , Pitman Monographs Surveys Pure Appl. Math., vol. 92, Addison Wesley Longman, Harlow, 1998. · Zbl 0955.42001
[16] I. Gohberg and N. Krupnik, One-dimensional linear singular integral equations , Vols. 1, 2, Oper. Theory Adv. Appl., vols. 53, 54, Birkhäuser Verlag, Basel, 1992. Russian original: Shtiintsa, Kishinev, 1973. · Zbl 0781.47038
[17] S.M. Grudsky, Singular integral equations and the Riemann boundary value problem with an infinite index in the space \(L_p(\Gamma, \omega)\) , Izv. Akad. Nauk SSSR Ser. Mat. 49 (1985), 55-80 (in Russian). English transl.: Math. USSR-Izv. 26 (1986), 53-76.
[18] A. Kamińska, Indices, convexity and concavity in Musielak-Orlicz spaces , Funct. Approx. Comment. Math. 26 (1998), 67-84. · Zbl 0914.46024
[19] A. Kamińska and B. Turett, Type and cotype in Musielak-Orlicz spaces , in Geometry of Banach spaces , Strobl, 1989, pp. 165-180; London Math. Soc. Lecture Note Ser., vol. 158, Cambridge Univ. Press, Cambridge, 1990. · Zbl 0770.46009
[20] L.V. Kantorovich and G.P. Akilov, Functional analysis , Nauka, Moscow, 3rd ed., 1984 (in Russian). English transl.: Pergamon Press, 2nd ed., Oxford, 1982. · Zbl 0484.46003
[21] N. Karapetiants and S. Samko, Equations with involutive operators , Birkhäuser Boston, Inc., Boston, MA, 2001. · Zbl 0990.47011
[22] A.Yu. Karlovich, Algebras of singular integral operators with piecewise continuous coefficients on reflexive Orlicz spaces , Math. Nachr. 179 (1996), 187-222. · Zbl 0873.47020 · doi:10.1002/mana.19961790112
[23] ——–, Algebras of singular integral operators with piecewise continuous coefficients in rearrangement-invariant spaces with weight on Carleson curves , Ph.D. Thesis, Odessa, Ukraine, 1998 (in Russian). Available at http://www.math.ist.utl.pt/\(^\sim\)akarlov/theses.html.
[24] ——–, Singular integral operators with piecewise continuous coefficients in reflexive rearrangement-invariant spaces , Integral Equations Operator Theory 32 (1998), 436-481. · Zbl 0923.47016 · doi:10.1007/BF01194990
[25] ——–, On the essential norm of the Cauchy singular integral operator in weighted rearrangement-invariant spaces , Integral Equations Operator Theory 38 (2000), 28-50. · Zbl 0998.47031 · doi:10.1007/BF01192300
[26] ——–, Algebras of singular integral operators with PC coefficients in rearrangement-invariant spaces with Muckenhoupt weights , J. Operator Theory 47 (2002), 303-323. · Zbl 1019.47051
[27] B.V. Khvedelidze, Linear discontinuous boundary problems in the theory of functions, singular integral equations and some of their applications , Akad. Nauk Gruzin. SSR. Trudy Tbiliss. Mat. Inst. Razmadze 23 (1956), 3-158 (in Russian). · Zbl 0083.30002
[28] ——–, The method of the Cauchy type integrals for discontinuous boundary value problems of the theory of holomorphic functions of one complex variable , Itogi nauki i tehniki VINITI, Ser. Sovrem. probl. mat. 7 (1975), 5-162 (in Russian). English transl.: J. Soviet Math. 7 (1977), 309-414. · Zbl 0406.30034 · doi:10.1007/BF01091836
[29] V. Kokilashvili and M. Krbec, Weighted inequalities in Lorentz and Orlicz spaces , World Scientific, New Jersey, 1991. · Zbl 0751.46021
[30] V. Kokilashvili and S. Samko, Singular integrals in weighted Lebesgue spaces with variable exponent , Georgian Math. J. 10 (2003), 145-156. · Zbl 1046.42006
[31] ——–, Singular integral equations in the Lebesgue spaces with variable exponent , Proc. A. Razmadze Math. Inst. 131 (2003), 61-78. · Zbl 1159.45302
[32] O. Kováčik and J. Rákosník, On spaces \(L^p(x)\) and \(W^k,p(x)\) , Czechoslovak Math. J. 41 (1991), 592-618.
[33] M.A. Krasnoselskii and Ya.B. Rutickii, Convex functions and Orlicz spaces , Fizmatgiz, Moscow, 1958 (in Russian). English transl.: Noordhoff Ltd., Groningen, 1961.
[34] M. Krbec, B. Opic, L. Pick and J. Rákosník, Some recent results on Hardy type operators in weighted function spaces and related topics , in Function spaces, differential operators and nonlinear analysis , Friedrichroda, 1992, pp. 158-184; Teubner-Texte Math., vol. 133, Teubner, Stuttgart, 1993. · Zbl 0803.46036
[35] S.G. Krein, Ju.I. Petunin and E.M. Semenov, Interpolation of linear operators , Nauka, Moscow, 1978 (in Russian). English transl.: Amer. Math. Soc. Transl. Ser. 2, vol. 54, Providence, RI, 1982.
[36] N.Ya. Krupnik, Banach algebras with symbol and singular integral operators , Oper. Theory Adv. Appl., vol. 26, Birkhäuser Verlag, Basel, 1987. · Zbl 0641.47031
[37] J. Lang, A. Nekvinda and L. Pick, Boundedness and compactness of general kernel integral operators from a weighted Banach function space into \(L_\infty\) , Dept. of Math. Analysis (KMA), Faculty of Math. and Phys., Charles University, Praha, Preprint MATH-KMA -2003/94. Available at http://adela.karlin.mff.cuni.cz/\(^\sim\)rokyta/preprint/.
[38] G.S. Litvinchuk and I.M. Spitkovsky, Factorization of measurable matrix functions , Oper. Theory Adv. Appl., vol. 25, Birkhäuser Verlag, Basel, 1987.
[39] L. Maligranda, Indices and interpolation , Dissert. Math. (Rozprawy Mat.) 234 (1985), 1-49. · Zbl 0566.46038
[40] S.G. Mikhlin and S. Prössdorf, Singular integral operators , Springer-Verlag, Berlin, 1986.
[41] J. Musielak, Orlicz spaces and modular spaces , Lecture Notes in Math., vol. 1034-1983. · Zbl 0557.46020 · doi:10.1007/BFb0072210
[42] J. Musielak and W. Orlicz, On modular spaces , Studia Math. 18 (1959), 49-65. · Zbl 0086.08901
[43] H. Nakano, Modulared semi-ordered linear spaces , Maruzen Co., Ltd., Tokyo, 1950. · Zbl 0041.23401
[44] ——–, Topology of linear topological spaces , Maruzen Co., Ltd., Tokyo, 1951. · Zbl 0043.17801
[45] W. Orlicz, Über konjugierte Exponentenfolgen , Studia Math. 3 (1931), 200-211. Reprinted in Wladyslaw Orlicz, Collected Papers , PWN, Warsaw, 1988, pp. 200-213. · Zbl 0003.25203
[46] L. Pick and M. Ružička, An example of a space \(L ^p(x)\) on which the Hardy-Littlewood maximal operator is not bounded , Exposition. Math. 19 (2001), 369-371. · Zbl 1003.42013 · doi:10.1016/S0723-0869(01)80023-2
[47] I.I. Privalov, Boundary properties of analytic functions , Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow-Leningrad, 1950 (in Russian). · Zbl 0045.34703
[48] M. Ružička, Electrorheological fluids : modeling and mathematical theory , Lecture Notes in Math., vol. 1748, Springer-Verlag, Berlin, 2000. · Zbl 0962.76001
[49] I.B. Simonenko, The Riemann boundary value problem for \(n\) pairs functions with measurable coefficients and its application to the investigation of singular integral operators in the spaces \(L^p\) with weight , Izv. AN SSSR, Ser. Matem. 28 (1964), 277-306 (in Russian). · Zbl 0136.06901
[50] ——–, A new general method of investigating linear operator equations of singular integral equation type , I-II, Izv. Akad. Nauk SSSR Ser. Mat. 29 (1965), 567-586 (Part I), 757-782 (Part II) (in Russian).
[51] ——–, Some general questions in the theory of the Riemann boundary value problem , Izv. AN SSSR, Ser. Matem. 32 (1968), 1138-1146 (in Russian). English transl.: Math. USSR Izv. 2 (1968), 1091-1099. · Zbl 0186.13601 · doi:10.1070/IM1968v002n05ABEH000706
[52] I. Spitkovsky, Singular integral operators with \(PC\) symbols on the spaces with general weights , J. Funct. Anal. 105 (1992), 129-143. · Zbl 0761.45001 · doi:10.1016/0022-1236(92)90075-T
[53] M. Zippin, Interpolation of operators of weak type between rearrangement invariant spaces , J. Funct. Anal. 7 (1971), 267-284. · Zbl 0224.46038 · doi:10.1016/0022-1236(71)90035-8
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.