Carleson conditions for asymptotic weights

Korey, Michael

doi:10.1090/S0002-9947-98-01931-X

Carleson conditions for asymptotic weights
HTML articles powered by AMS MathViewer

by Michael Brian Korey

Trans. Amer. Math. Soc. 350 (1998), 2049-2069

DOI: https://doi.org/10.1090/S0002-9947-98-01931-X

PDF | Request permission

Abstract:

The doubling and $A_\infty$ conditions are characterized in terms of convolution with rapidly decreasing kernels. The Carleson-measure criterion for $A_\infty$ of Fefferman, Kenig, and Pipher is extended to the case when all bounds become optimally small in the asymptotic limit.

References

Raoul Bott and Loring W. Tu, Differential forms in algebraic topology, Graduate Texts in Mathematics, vol. 82, Springer-Verlag, New York-Berlin, 1982. MR 658304
Stephen M. Buckley, Estimates for operator norms on weighted spaces and reverse Jensen inequalities, Trans. Amer. Math. Soc. 340 (1993), no. 1, 253–272. MR 1124164, DOI 10.1090/S0002-9947-1993-1124164-0
Michael Christ, A $T(b)$ theorem with remarks on analytic capacity and the Cauchy integral, Colloq. Math. 60/61 (1990), no. 2, 601–628. MR 1096400, DOI 10.4064/cm-60-61-2-601-628
R. R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241–250. MR 358205, DOI 10.4064/sm-51-3-241-250
C. Fefferman and E. M. Stein, $H^{p}$ spaces of several variables, Acta Math. 129 (1972), no. 3-4, 137–193. MR 447953, DOI 10.1007/BF02392215
R. A. Fefferman, C. E. Kenig, and J. Pipher, The theory of weights and the Dirichlet problem for elliptic equations, Ann. of Math. (2) 134 (1991), no. 1, 65–124. MR 1114608, DOI 10.2307/2944333
José García-Cuerva and José L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland Mathematics Studies, vol. 116, North-Holland Publishing Co., Amsterdam, 1985. Notas de Matemática [Mathematical Notes], 104. MR 807149
Sergei V. Hruščev, A description of weights satisfying the $A_{\infty }$ condition of Muckenhoupt, Proc. Amer. Math. Soc. 90 (1984), no. 2, 253–257. MR 727244, DOI 10.1090/S0002-9939-1984-0727244-4
David S. Jerison and Carlos E. Kenig, The logarithm of the Poisson kernel of a $C^{1}$ domain has vanishing mean oscillation, Trans. Amer. Math. Soc. 273 (1982), no. 2, 781–794. MR 667174, DOI 10.1090/S0002-9947-1982-0667174-2
F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), 415–426. MR 131498, DOI 10.1002/cpa.3160140317
M. B. Korey, Ideal weights: doubling and absolute continuity with asymptotically optimal bounds, Ph.D. Thesis, University of Chicago, 1995.
Benjamin Muckenhoupt, The equivalence of two conditions for weight functions, Studia Math. 49 (1973/74), 101–106. MR 350297, DOI 10.4064/sm-49-2-101-106
S. C. Power, Vanishing Carleson measures, Bull. London Math. Soc. 12 (1980), no. 3, 207–210. MR 572103, DOI 10.1112/blms/12.3.207
Hans Martin Reimann and Thomas Rychener, Funktionen beschränkter mittlerer Oszillation, Lecture Notes in Mathematics, Vol. 487, Springer-Verlag, Berlin-New York, 1975 (German). MR 0511997
Donald Sarason, Functions of vanishing mean oscillation, Trans. Amer. Math. Soc. 207 (1975), 391–405. MR 377518, DOI 10.1090/S0002-9947-1975-0377518-3
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, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192