Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

 

 

A uniqueness theorem for Riesz potentials


Author: K. A. Izyurov
Translated by: the author
Original publication: Algebra i Analiz, tom 19 (2007), nomer 4.
Journal: St. Petersburg Math. J. 19 (2008), 577-595
MSC (2000): Primary 31A15, 31A20
Published electronically: May 9, 2008
MathSciNet review: 2381935
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The existence is proved of a nonzero Hölder function $ f:\mathbb{R}\rightarrow\mathbb{R}$ that vanishes together with its M. Riesz potential $ f\ast \frac{1}{\vert x\vert^{1-\alpha}}$ at all points of some set of positive length. This result improves that of D. Beliaev and V. Havin.


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

  • 1. A. B. Aleksandrov and P. P. Kargaev, Hardy classes of functions that are harmonic in a half-space, Algebra i Analiz 5 (1993), no. 2, 1–73 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 5 (1994), no. 2, 229–286. MR 1223170
  • 2. Dmitri B. Beliaev and Victor P. Havin, On the uncertainty principle for M. Riesz potentials, Ark. Mat. 39 (2001), no. 2, 223–243. MR 1861059, 10.1007/BF02384555
  • 3. I. A. Binder, A theorem on correction by gradients of harmonic functions, Algebra i Analiz 5 (1993), no. 2, 91–107 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 5 (1994), no. 2, 301–315. MR 1223172
  • 4. J. Bourgain and T. Wolff, A remark on gradients of harmonic functions in dimension ≥3, Colloq. Math. 60/61 (1990), no. 1, 253–260. MR 1096375
  • 5. V. P. Khavin, The uncertainty principle for one-dimensional Riesz potentials, Dokl. Akad. Nauk SSSR 264 (1982), no. 3, 559–563 (Russian). MR 659763
  • 6. Victor Havin and Burglind Jöricke, The uncertainty principle in harmonic analysis, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 28, Springer-Verlag, Berlin, 1994. MR 1303780
  • 7. Thomas H. Wolff, Counterexamples with harmonic gradients in 𝑅³, Essays on Fourier analysis in honor of Elias M. Stein (Princeton, NJ, 1991), Princeton Math. Ser., vol. 42, Princeton Univ. Press, Princeton, NJ, 1995, pp. 321–384. MR 1315554
  • 8. A. Zygmund, Trigonometric series. Vol. I, II, 3rd ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2002. With a foreword by Robert A. Fefferman. MR 1963498

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2000): 31A15, 31A20

Retrieve articles in all journals with MSC (2000): 31A15, 31A20


Additional Information

K. A. Izyurov
Affiliation: Mathematics and Mechanics Department, St. Petersburg State University, Universitetskii Prospekt 28, Staryi Peterhof, St. Petersburg 198504, Russia
Email: k.izyurov@gmail.com

DOI: http://dx.doi.org/10.1090/S1061-0022-08-01011-X
Keywords: Riesz potential, uncertainty principle, H\"older condition
Received by editor(s): February 8, 2007
Published electronically: May 9, 2008
Additional Notes: Partially supported by RFBR (grant no. 06-01-00313).
Article copyright: © Copyright 2008 American Mathematical Society