Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)

 
 

 

Maximal properties of the normalized Cauchy transform


Author: Alexei Poltoratski
Journal: J. Amer. Math. Soc. 16 (2003), 1-17
MSC (2000): Primary 30E20
DOI: https://doi.org/10.1090/S0894-0347-02-00403-4
Published electronically: August 27, 2002
MathSciNet review: 1937196
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [A1] A. B. Aleksandrov, Multiplicity of boundary values of inner functions, Izv. Acad. Nauk. Arm. SSR, Matematica 22 5 (1987), 490-503 (Russian). MR 89e:30058
  • [A2] A. B. Aleksandrov, Inner functions and related spaces of pseudocontinuable functions, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 170 (1989), 7-33 (Russian). MR 91c:30063
  • [A3] A. B. Aleksandrov, On the existence of angular boundary values for pseudocontinuable functions, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 222 (1995) (Russian). MR 97a:30046
  • [A4] A. B. Aleksandrov, Isometric embeddings of coinvariant subspaces of the shift operator, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 232 (1996) (Russian). MR 98k:30050
  • [A5] A. B. Aleksandrov, On the maximum principle for pseudocontinuable functions, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 217 (1994) (Russian). MR 96c:30033
  • [A6] A. B. Aleksandrov, Invariant subspaces of the backward shift operator in the space $H\sp{p}$ $(p\in (0,\,1))$, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 92 (1979) (Russian). MR 81h:46018
  • [C] D. Clark, One dimensional perturbations of restricted shifts, J. Anal. Math. 25 (1972), 169-91. MR 46:692
  • [D] G. David, Analytic capacity, Cauchy kernel, Menger curvature and rectifiability, Harmonic analysis and partial differential equations, Chicago Lectures in Math., Univ. of Chicago Press (1999). MR 2001a:30027
  • [F] O. Frostman, Sur les produits de Blaschke, Kungl. Fysiogr. Sällsk. i Lund Förh. 12, 15 (1942), 169-182. MR 6:262e
  • [G] J. B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, 96, vol. 273, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 83g:30037
  • [N] N. K. Nikolski, Treatise on the shift operator, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 273, Springer-Verlag, Berlin-New York, 1986.
  • [NTV] F. Nazarov , S. Treil and A. Volberg, Weak type estimates and Cotlar inequalities for Calderon-Zygmund operators on nonhomogeneous spaces, J. Amer. Math. Soc. 12 (1999), 909-929. MR 99f:42035
  • [P1] A. Poltoratski, On the boundary behavior of pseudocontinuable functions, St. Petersburg Math. J. 5 (1994), 389-406.
  • [P2] A. Poltoratski, On the distributions of boundary values of Cauchy integrals, Proc. Amer. Math. Soc. 124 No 8 (1996), 2455-2463. MR 96j:30057
  • [P3] A. Poltoratski, Properties of Exposed Points in the Unit Ball of $H^{1}$, Indiana Univ. Math. J. 50 (2001), 1789-1806.
  • [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.).
  • [P5] A. Poltoratski, Finite rank perturbations of singular spectra, Internat. Math. Res. Notices no. 9 (1997), 421-436. MR 98d:47035
  • [P6] A. Poltoratski, Equivalence up to a Rank One Perturbation, Pacific J. of Math. 194 (2000), 175-188. MR 2001j:47013
  • [S] D. Sarason, Sub-Hardy Hilbert Spaces in the unit disk, The University of Arkansas lecture notes in the mathematical sciences; v. 10, J. Wiley and Sons, New York, 1994. MR 96k:46039
  • [St] E. Stein, Harmonic Analysis, Princeton Univ. Press, 1993. MR 95c:42002
  • [T] Tolsa X., Littlewood-Paley theory and the $T(1)$ theorem with non-doubling measures, Adv. Math. 164 (2001), 57-116.

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 30E20

Retrieve articles in all journals with MSC (2000): 30E20


Additional Information

Alexei Poltoratski
Affiliation: Department of Mathemathcs, Texas A&M University, College Station, Texas 77843
Email: alexeip@math.tamu.edu

DOI: https://doi.org/10.1090/S0894-0347-02-00403-4
Keywords: Cauchy integrals, boundary convergence, non-tangential maximal function
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
Article copyright: © Copyright 2002 American Mathematical Society

American Mathematical Society