## Communications in Mathematical Analysis

### Algebras of Pseudodifferential Operators with Discontinuous Oscillating Symbols.

#### Abstract

Non-closed algebras ${\mathfrak{A}}_{\delta,\gamma}$ of pseudodifferential operators with slowly oscillating Lipschitz symbols in $\Lambda^{SO}_{\delta,\gamma}({\mathbb R}\times{\mathbb R})$ with $\delta,\gamma\in(1/2,1]$ and the minimal $C^*$-algebra ${\mathfrak{A}}$ containing all ${\mathfrak{A}}_{\delta,\gamma}$ are studied on the Lebesgue space $L^2({\mathbb R})$. Applying results on the boundedness and compactness of pseudodifferential operators $A\in{\mathfrak{A}}$, a commutative algebra of their Fredholm symbols is described. A Fredholm criterion and an index formula for the operators $A\in{\mathfrak{A}}$ are obtained. Then we study the Fredholmness for the $C^*$-algebra ${\mathfrak{B}}$ generated by the operators $A\in{\mathfrak{A}}$ with multiplicatively slowly oscillating Lipschitz symbols and by the operators of multiplication $aI$ and convolution operators $W^0(b)$ with piecewise continuous functions $a,b:\overline{{\mathbb R}}\to{\mathbb C}$. The algebra of Fredholm symbols for the operators $A\in{\mathfrak{B}}$ is not commutative. A Fredholm criterion for the operators $A\in{\mathfrak{B}}$ is established.

#### Article information

Source
Commun. Math. Anal., Conference 3 (2011), 108-130.

Dates
First available in Project Euclid: 25 February 2011

https://projecteuclid.org/euclid.cma/1298670007

Mathematical Reviews number (MathSciNet)
MR2772056

Zentralblatt MATH identifier
1218.47135

#### Citation

Karlovich, Yu. I.; Rabinovich, V. S.; Vasilevski, N. L. Algebras of Pseudodifferential Operators with Discontinuous Oscillating Symbols. Commun. Math. Anal. (2011), no. 3, 108--130. https://projecteuclid.org/euclid.cma/1298670007

#### References

• M. A. Bastos, A. Bravo, and Yu. I. Karlovich, Convolution type operators with symbols generated by slowly oscillating and piecewise continuous matrix functions. Operator Theoretical Methods and Applications to Mathematical Physics. Oper. Theory: Adv. Appl. 147 (2004), pp 151-174.
• M. A. Bastos, A. Bravo, and Yu. I. Karlovich, Symbol calculus and Fredholmness for a Banach algebra of convolution type operators with slowly oscillating and piecewise continuous data. Math. Nachr. 269-270 (2004), pp 11-38.
• A. Böttcher and Yu. I. Karlovich, Carleson Curves, Muckenhoupt Weights, and Toeplitz Operators. Progr. Math. 154, Birkhäuser Verlag, Basel 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), pp 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), pp 171-198.
• A. Böttcher, Yu. I. Karlovich, and V. S. Rabinovich, Singular integral operators with complex conjugation from the viewpoint of pseudodifferential operators, Problems and methods in mathematical physics. Problems and Methods in Mathematical Physics. Oper. Theory: Adv. Appl. 121 (2001), pp 36-59.
• A. G. Childs, On the $L^2$-boundedness of pseudo-differential operators. Proc. Amer. Math. Soc. 61 (1976), pp 252-254 (1977).
• R. R. Coifman and Y. Meyer, Au delà des opérateurs pseudodifférentiels. Astérisque 57 (1978), pp 1-184.
• H. O. Cordes, On compactness of commutators of multiplications and convolutions, and boundedness of pseudodifferential operators. J. Funct. Anal. 18 (1975), pp 115-131.
• H. O. Cordes, Elliptic Pseudo-Differential Operators - An Abstract Theory. Lecture Notes in Math. 756, Springer, Berlin 1979.
• Yu. I. Karlovich, An algebra of pseudodifferential operators with slowly oscillating symbols. Proc. London Math. Soc. (3) 92 (2006), pp 713-761.
• Yu. I. Karlovich, Pseudodifferential operators with compound slowly oscillating symbols. New Aspects of Operator Theory and Its Applications. Oper. Theory: Adv. Appl. 171 Birkhäuser, Basel (2006), pp 189-224.
• Yu. I. Karlovich, Algebras of pseudodifferential operators with discontinuous symbols. Modern Trends in Pseudo-Differential Operators. Oper. Theory: Adv. Appl. 172 Birkhäuser, Basel (2007), pp 207-233.
• Yu. I. Karlovich, Pseudodifferential operators with compound non-regular symbols. Math. Nachr. 280 (2007), No. 9–10, pp 1128-1144.
• T. Kato, Boundedness of some pseudodifferential operators. Osaka J. Math. 13 (1976), pp 1-9.
• J. Marschall, Weighted $L^p$-estimates for pseudo-differential operators with nonregular symbols. Z. Anal. Anwendungen 10 (1991), pp 493-501.
• 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), pp 1261-1292.
• V. Rabinovich, S. Roch, and B. Silbermann, Limit Operators and Their Applications in Operator Theory. Birkhäuser, Basel 2004.
• V. Rabinovich and N. Vasilevski, Bergman-Toeplitz and pseudodifferential operators. Complex Analysis and Related Topics. Oper. Theory: Adv. Appl. 114 (2000), pp 207-234.
• 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 Math. Soc., Providence, RI 2000.
• F. Treves, Introduction to Pseudodifferential and Fourier Integral Operators. Vols. 1 and 2. Plenum Press, New York 1982.
• N. L. Vasilevski, Commutative Algebras of Toeplitz Operators on the Bergman Space. Birkhäuser, Basel 2008.