Abstract
Let V (ℝ) denote the Banach algebra of absolutely continuous functions of bounded total variation on ℝ. We study an algebra \( \mathfrak{B} \) of pseudodifferential operators of zero order with compound slowly oscillating V (ℝ)-valued symbols (x, y) ↦ a(x, y, ·) of limited smoothness with respect to x, y ∈ ℝ. Sufficient conditions for the boundedness and compactness of pseudodifferential operators with compound symbols on Lebesgue spaces L p(ℝ) are obtained. A symbol calculus for the algebra \( \mathfrak{B} \) is constructed on the basis of an appropriate approximation of symbols by infinitely differentiable ones and by use of the techniques of oscillatory integrals. A Fredholm criterion and an index formula for pseudodifferential operators A ∈ \( \mathfrak{B} \) are obtained. These results are carried over to Mellin pseudodifferential operators with compound slowly oscillating V (ℝ)-valued symbols. Finally, we construct a Fredholm theory of generalized singular integral operators on weighted Lebesgue spaces L p with slowly oscillating Muckenhoupt weights over slowly oscillating Carleson curves.
Partially supported by the CONACYT (México) Project No. 47220.
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References
A. Böttcher and Yu.I. Karlovich, Carleson Curves, Muckenhoupt Weights, and Toeplitz Operators. Progress in Mathematics 154, Birkhäuser, Verlag, Basel, Boston, Berlin, 1997.
A. Böttcher, Yu.I. Karlovich, and V.S. Rabinovich, Mellin pseudodifferential operators with slowly varying symbols and singular integral on Carleson curves with Muckenhoupt weights. Manuscripta Math. 95 (1998), 363–376.
A. Böttcher, Yu.I. Karlovich, and V.S. Rabinovich, The method of limit operators for one-dimensional singular integrals with slowly oscillating data. J. Operator Theory 43 (2000), 171–198.
A. Böttcher, Yu.I. Karlovich, and V.S. Rabinovich, Singular integral operators with complex conjugation from the viewpoint of pseudodifferential operators. Operator Theory: Advances and Applications 121 (2001), 36–59.
A. Böttcher and B. Silbermann, Analysis of Toeplitz Operators. Akademie-Verlag, Berlin, 1989 and Springer-Verlag, Berlin, Heidelberg, New York, 1990.
R.R. Coifman and Y. Meyer, Au delà des opérateurs pseudodifférentiels. Astérisque 57 (1978), 1–184.
H.O. Cordes, On compactness of commutators of multiplications and convolutions, and boundedness of pseudodifferential operators. J. Funct. Anal. 18 (1975), 115–131.
H.O. Cordes, Elliptic Pseudo-Differential Operators — An Abstract Theory. Lecture Notes in Math. 756, Springer, Berlin, 1979.
D. David, Opérateurs integraux singuliers sur certaines courbes du plan complexe. Ann. Sci. École Norm. Sup. 17 (1984), 157–189.
J. Duoandikoetxea, Fourier Analysis. American Mathematical Society, Providence, RI, 2000.
E.M. Dynkin and B.P. Osilenker, Weighted norm estimates for singular integrals and their applications. J. Soviet Math. 30 (1985), 2094–2154.
I. Gohberg and N. Krupnik, One-Dimensional Linear Singular Integral Equations. Vols. 1 and 2, Birkhäuser, Basel, 1992; Russian original, Shtiintsa, Kishinev, 1973.
L. Hörmander, The Analysis of Linear Partial Differential Operators. Vols. 1–4, Springer, Berlin, 1983–1985.
R. Hunt, B. Muckenhoupt, and R. Wheeden, Weighted norm inequalities for the conjugate function and Hilbert transform. Trans. Amer. Math. Soc. 176 (1973), 227–251.
A.Yu. Karlovich, Yu.I. Karlovich, and A.B. Lebre, Invertibility of functional operators with slowly oscillating non-Carleman shifts. Operator Theory: Advances and Applications 142 (2003), 147–174.
Yu.I. Karlovich, An algebra of pseudodifferential operators with slowly oscillating symbols. Proc. London Math. Soc., to appear.
Yu.I. Karlovich and A.B. Lebre, Algebra of singular integral operators with a Carleman backward slowly oscillating shift. Integral Equations and Operator Theory 41 (2001), 288–323.
Yu.I. Karlovich and E. Ramírez de Arellano, Singular integral operators with fixed singularities on weighted Lebesgue spaces. Integral Equations and Operator Theory 48 (2004), 331–363.
Yu.I. Karlovich and B. Silbermann, Fredholmness of singular integral operators with discrete subexponential groups of shifts on Lebesgue spaces. Math. Nachr. 272 (2004), 55–94.
C.E. Kenig and P.A. Tomas, Maximal operators defined by Fourier multipliers. Studia Math. 68 (1980), 79–83.
V.G. Kravchenko and G.S. Litvinchuk, Introduction to the Theory of Singular Integral Operators with Shift. Mathematics and its Applications. Kluwer Academic Publishers, v. 289, Dordrecht, Boston, London, 1994.
H. Kumano-go, Pseudodifferential Operators. MIT Press, Cambridge, MA, 1974.
J.E. Lewis and C. Parenti, Pseudodifferential operators of Mellin type. Comm. Part. Diff. Equ. 8 (1983), 477–544.
J. Marschall, Weighted L p-estimates for pseudo-differential operators with nonregular symbols. Z. Anal. Anwendungen. 10 (1991), 493–501.
R. Melrous, Transformation of boundary value problems. Singularities in Boundary Value Problems. Proc. NATO Adv. Stud. Inst., Dordrecht, 1981, 133–168.
R. Melrous and J. Sjöstrand, Singularities of boundary value problems, I. Comm. Pure Appl. Math. 31 (1978), 593–617.
I.P. Natanson, Theory of Functions of Real Variable. Vol. 1, Frederick Ungar Publ. Co., New York, 1964.
S.C. Power, Fredholm Toeplitz operators and slow oscillation. Canad. J. Math. XXXII (1980), 1058–1071.
V.S. Rabinovich, Algebras of singular integral operators on compound contours with nodes that are logarithmic whirl points. Russ. Acad. Sci. Izv.Math. 60 (1996), 1261–1292.
V.S. Rabinovich, An introductory course on pseudodifferential operators, Textos de Matemática-1, Centro de Matemática Aplicada, Instituto Superior Técnico, Lisboa, 1998.
V.S. Rabinovich, Pseudodifferential operators on R n with variable shifts. Zeitschrift für Analysis und ihre Adwendungen, Journal for Analysis and its Applications 22 (2003), No. 2, 315–338.
V. Rabinovich, S. Roch, and B. Silbermann, Limit Operators and Their Applications in Operator Theory. Birkhäuser, Basel, 2004.
M. Reed and B. Simon, Methods of Modern Mathematical Physics. 1. Functional Analysis. Academic Press, New York, 1972.
D. Sarason, Toeplitz operators with piecewise quasicontinuous symbols. Indiana Univ. Math. J. 26 (1977), 817–838.
B.W. Schulze, Pseudo-Differential Operators on Manifolds with Singularities. North-Holland, Amsterdam, 1991.
B.W. Schulze, Pseudo-Differential Boundary Value Problems, Conical Singularities and Asymptotics. Akademie-Verlag, Berlin, 1994.
L. Schwartz, Analyse Mathématique. Vol. 1, Hermann, 1967.
M.A. Shubin, Pseudodifferential Operators and Spectral Theory. Springer, Berlin, 1987; Russian original, Nauka, Moscow, 1978.
E.M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Univ. Press, Princeton, NJ, 1993.
M.E. Taylor, Pseudodifferential Operators. Princeton Univ. Press, Princeton, NJ, 1981.
M.E. Taylor, Tools for PDE. Pseudodifferential Operators, Paradifferential Operators, and Layer Potentials. American Mathematical Society, Providence, RI, 2000.
F. Treves, Introduction to Pseudodifferential and Fourier Integral Operators. Vols. 1 and 2, Plenum Press, New York, 1982.
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Karlovich, Y.I. (2006). Pseudodifferential Operators with Compound Slowly Oscillating Symbols. In: Dritschel, M.A. (eds) The Extended Field of Operator Theory. Operator Theory: Advances and Applications, vol 171. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-7980-3_10
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