Hörmander’s condition and a convolution operator generalizing Riesz potentials

Bak, Jong-Guk

doi:10.1090/S0002-9939-1994-1195475-4

Hörmander’s condition and a convolution operator generalizing Riesz potentials
HTML articles powered by AMS MathViewer

by Jong-Guk Bak PDF

Proc. Amer. Math. Soc. 120 (1994), 647-649 Request permission

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

David R. Adams, A note on Riesz potentials, Duke Math. J. 42 (1975), no. 4, 765–778. MR 458158

Weighted norm inequalities and related topics

Lars Hörmander, Estimates for translation invariant operators in $L^{p}$ spaces, Acta Math. 104 (1960), 93–140. MR 121655, DOI 10.1007/BF02547187
Richard O’Neil, Convolution operators and $L(p,\,q)$ spaces, Duke Math. J. 30 (1963), 129–142. MR 146673
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
Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971. MR 0304972
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 215121, DOI 10.2307/1970445