Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Hörmander's condition and a convolution operator generalizing Riesz potentials


Author: Jong-Guk Bak
Journal: Proc. Amer. Math. Soc. 120 (1994), 647-649
MSC: Primary 42B20; Secondary 47B35
DOI: https://doi.org/10.1090/S0002-9939-1994-1195475-4
MathSciNet review: 1195475
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Under certain hypotheses including a Hörmander-type condition on the convolution kernel $ K$ we show that $ K{\ast}f$ belongs to the space $ \operatorname{BMO} ({{\mathbf{R}}^n})$ whenever $ f$ belongs to the space $ {L^{p,\infty }}({{\mathbf{R}}^n})$ (weak $ {L^p}$) for certain $ p$.


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

  • [A] D. Adams, A note on Riesz potentials, Duke Math. J. 42 (1975), 765-778. MR 0458158 (56:16361)
  • [GR] J. Garcia-Cuerva and J. L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland Math. Stud., vol. 116, North-Holland, Amsterdam and New York, 1986.
  • [H] L. Hörmander, Estimates for translation invariant operators in $ {L^p}$ spaces, Acta Math. 104 (1960), 93-140. MR 0121655 (22:12389)
  • [O] R. O'Neil, Convolution operators and $ L(p,q)$ spaces, Duke Math. J. 30 (1963), 129-142. MR 0146673 (26:4193)
  • [S1] E. M. Stein, Singular integrals, harmonic functions, and differentiability properties of functions of several variables, Proc. Sympos. Pure Math., vol. 10, Amer. Math. Soc., Providence, RI, 1967, pp. 316-335. MR 0482394 (58:2467)
  • [S2] -, Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton, NJ, 1970. MR 0290095 (44:7280)
  • [SW] E. M. Stein and G. Weiss, An introduction to harmonic analysis on Euclidean spaces, Princeton Univ. Press, Princeton, NJ, 1971. MR 0304972 (46:4102)
  • [SZ] E. M. Stein and A. Zygmund, Boundedness of translation invariant operators on Hölder and $ {L^p}$ spaces, Ann. of Math. (2) 85 (1967), 337-349. MR 0215121 (35:5964)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 42B20, 47B35

Retrieve articles in all journals with MSC: 42B20, 47B35


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1994-1195475-4
Article copyright: © Copyright 1994 American Mathematical Society

American Mathematical Society