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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
MathSciNet review: 1195475
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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$.


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  • [A] David R. Adams, A note on Riesz potentials, Duke Math. J. 42 (1975), no. 4, 765–778. MR 0458158
  • [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] Lars Hörmander, Estimates for translation invariant operators in 𝐿^{𝑝} spaces, Acta Math. 104 (1960), 93–140. MR 0121655
  • [O] Richard O’Neil, Convolution operators and 𝐿(𝑝,𝑞) spaces, Duke Math. J. 30 (1963), 129–142. MR 0146673
  • [S1] E. M. Stein, Singular integrals, harmonic functions, and differentiability properties of functions of several variables, Singular integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966), Amer. Math. Soc., Providence, R.I., 1967, pp. 316–335. MR 0482394
  • [S2] Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
  • [SW] Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton University Press, Princeton, N.J., 1971. Princeton Mathematical Series, No. 32. MR 0304972
  • [SZ] E. M. Stein and A. Zygmund, Boundedness of translation invariant operators on Hölder spaces and 𝐿^{𝑝}-spaces, Ann. of Math. (2) 85 (1967), 337–349. MR 0215121

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