## Maximal properties of the normalized Cauchy transform

HTML articles powered by AMS MathViewer

- by Alexei Poltoratski;
- J. Amer. Math. Soc.
**16**(2003), 1-17 - DOI: https://doi.org/10.1090/S0894-0347-02-00403-4
- Published electronically: August 27, 2002
- PDF | Request permission

## Abstract:

We study the normalized Cauchy transform in the unit disk. Our goal is to find an analog of the classical theorem by M. Riesz for the case of arbitrary weights. Let $\mu$ be a positive finite measure on the unit circle of the complex plane and $f\in L^{1}(\mu )$. Denote by $K\mu$ and $Kf\mu$ the Cauchy integrals of the measures $\mu$ and $f\mu$, respectively. The normalized Cauchy transform is defined as $C_{\mu }: f\mapsto \frac {Kf\mu }{K\mu }$. We prove that $C_{\mu }$ is bounded as an operator in $L^{p}(\mu )$ for $1<p\leq 2$ but is unbounded (in general) for $p>2$. The associated maximal non-tangential operator is bounded for $1<p<2$ and has weak type $(2,2)$ but is unbounded for $p>2$.## References

- A. B. Aleksandrov,
*Multiplicity of boundary values of inner functions*, Izv. Akad. Nauk Armyan. SSR Ser. Mat.**22**(1987), no. 5, 490–503, 515 (Russian, with English and Armenian summaries). MR**931885** - A. B. Aleksandrov,
*Inner functions and related spaces of pseudocontinuable functions*, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI)**170**(1989), no. Issled. Lineĭn. Oper. Teorii Funktsiĭ. 17, 7–33, 321 (Russian, with English summary); English transl., J. Soviet Math.**63**(1993), no. 2, 115–129. MR**1039571**, DOI 10.1007/BF01099304 - A. B. Aleksandrov,
*On the existence of angular boundary values of pseudocontinuable functions*, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI)**222**(1995), no. Issled. po Lineĭn. Oper. i Teor. Funktsiĭ. 23, 5–17, 307 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (New York)**87**(1997), no. 5, 3781–3787. MR**1359992**, DOI 10.1007/BF02355824 - A. B. Aleksandrov,
*Isometric embeddings of co-invariant subspaces of the shift operator*, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI)**232**(1996), no. Issled. po Lineĭn. Oper. i Teor. Funktsiĭ. 24, 5–15, 213 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (New York)**92**(1998), no. 1, 3543–3549. MR**1464420**, DOI 10.1007/BF02440138 - A. B. Aleksandrov,
*On the maximum principle for pseudocontinuable functions*, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI)**217**(1994), no. Issled. po Lineĭn. Oper. i Teor. Funktsiĭ. 22, 16–25, 218 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (New York)**85**(1997), no. 2, 1767–1772. MR**1327511**, DOI 10.1007/BF02355285 - A. B. Aleksandrov,
*Invariant subspaces of the backward shift operator in the space $H^{p}$ $(p\in (0,\,1))$*, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI)**92**(1979), 7–29, 318 (Russian, with English summary). Investigations on linear operators and the theory of functions, IX. MR**566739** - Douglas N. Clark,
*One dimensional perturbations of restricted shifts*, J. Analyse Math.**25**(1972), 169–191. MR**301534**, DOI 10.1007/BF02790036 - Guy David,
*Analytic capacity, Cauchy kernel, Menger curvature, and rectifiability*, Harmonic analysis and partial differential equations (Chicago, IL, 1996) Chicago Lectures in Math., Univ. Chicago Press, Chicago, IL, 1999, pp. 183–197. MR**1743862** - Albert Eagle,
*Series for all the roots of the equation $(z-a)^m=k(z-b)^n$*, Amer. Math. Monthly**46**(1939), 425–428. MR**6**, DOI 10.2307/2303037 - John B. Garnett,
*Bounded analytic functions*, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR**628971**
[N]N N. K. Nikolski, - F. Nazarov, S. Treil, and A. Volberg,
*Weak type estimates and Cotlar inequalities for Calderón-Zygmund operators on nonhomogeneous spaces*, Internat. Math. Res. Notices**9**(1998), 463–487. MR**1626935**, DOI 10.1155/S1073792898000312
[P1]P1 A. Poltoratski, - Alexei G. Poltoratski,
*On the distributions of boundary values of Cauchy integrals*, Proc. Amer. Math. Soc.**124**(1996), no. 8, 2455–2463. MR**1327037**, DOI 10.1090/S0002-9939-96-03363-1
[P3]P3 A. Poltoratski, - Alexei G. Poltoratski,
*Finite rank perturbations of singular spectra*, Internat. Math. Res. Notices**9**(1997), 421–436. MR**1443321**, DOI 10.1155/S1073792897000299 - Alexei G. Poltoratski,
*Equivalence up to a rank one perturbation*, Pacific J. Math.**194**(2000), no. 1, 175–188. MR**1756633**, DOI 10.2140/pjm.2000.194.175 - Donald Sarason,
*Sub-Hardy Hilbert spaces in the unit disk*, University of Arkansas Lecture Notes in the Mathematical Sciences, vol. 10, John Wiley & Sons, Inc., New York, 1994. A Wiley-Interscience Publication. MR**1289670** - Elias M. Stein,
*Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals*, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR**1232192**
[T]T Tolsa X.,

*Treatise on the shift operator*, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 273, Springer-Verlag, Berlin-New York, 1986.

*On the boundary behavior of pseudocontinuable functions*, St. Petersburg Math. J.

**5**(1994), 389-406.

*Properties of Exposed Points in the Unit Ball of $H^{1}$*, Indiana Univ. Math. J.

**50**(2001), 1789–1806. [P4]P4 A. Poltoratski,

*The Krein spectral shift and rank one perturbations of spectra*, Algebra i Analiz

**10**(1998 No 5), 143-183 (Russian; English translation to appear in St. Petersburg Math. J.).

*Littlewood-Paley theory and the $T(1)$ theorem with non-doubling measures*, Adv. Math.

**164**(2001), 57–116.

## Bibliographic Information

**Alexei Poltoratski**- Affiliation: Department of Mathemathcs, Texas A&M University, College Station, Texas 77843
- MR Author ID: 292108
- Email: alexeip@math.tamu.edu
- Received by editor(s): June 12, 2000
- Published electronically: August 27, 2002
- Additional Notes: The author is supported in part by N.S.F. grant DMS 9970151
- © Copyright 2002 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**16**(2003), 1-17 - MSC (2000): Primary 30E20
- DOI: https://doi.org/10.1090/S0894-0347-02-00403-4
- MathSciNet review: 1937196