Abstract
Sharp inequalities between weight bounds (from the doubling, Ap, and reverse Hölder conditions) and the BMO norm are obtained when the former are near their optimal values. In particular, the BMO norm of the logarithm of a weight is controlled by the square root of the logarithm of its A∞ bound. These estimates lead to a systematic development of asymptotically sharp higher integrability results for reverse Hölder weights and extend Coifman and Fefferman's formulation of the A∞ condition as an equivalence relation on doubling measures to the setting in which all bounds become optimal over small scales.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Beurling, A. and Ahlfors, L. (1956). The boundary correspondence under quasiconformal mappings,Acta Math.,96, 125–142.
Buckley, S.M. (1993). Estimates for operator norms on weighted spaces and reverse Jensen inequalities,Trans. Amer. Math. Soc.,340, 253–272.
Caffarelli, L., Fabes, E., Mortola, S., and Salsa, S. (1981). Boundary behavior of non-negative solutions of elliptic operators in divergence form,Indiana U. Math. J.,30, 621–640.
Carleson, L. (1967). On mappings, conformal at the boundary,J. Analyse Math.,19, 1–13.
Carleson, L. (1981). BMO—10 years' development,18th Scandanavian Congress of Mathematicians (Aarhus, 1980), Progress in Mathematics,11, Birkhäuser, 3–21.
Chandrasekharan, K. (1968)Introduction to Analytic Number Theory, Grund. der math. Wiss.,148, Springer-Verlag, New York, NY.
Coifman, R.R. and Fefferman, C. (1974). Weighted norm inequalities for maximal functions and singular integrals,Studia Math.,51, 241–250.
Dahlberg, B. (1977) On estimates for harmonic measure,Arch. Rat. Mech. Anal.,65, 272–288.
Fefferman, C. and Muckenhoupt, B. (1974) Two nonequivalent conditions for weight functions,Proc. Amer. Math. Soc.,45, 99–104.
Fefferman, R., Kenig, C.E., and Pipher, J. (1991). The theory of weights and the Dirichlet problem for elliptic equations,Ann. of Math.,134(2), 65–124.
García-Cuerva, J. and Rubio de Francia, J.L. (1985).Weighted Norm Inequalities and Related Topics, North-Holland, Amsterdam and New York, NY.
Gehring, F.W. (1973). TheL p-integrability of the partial derivatives of a quasiconformal mapping,Acta Math.,130, 265–277.
Hardy, G.H., Littlewood, J.E., and Pólya, G. (1934).Inequalities, Cambridge University Press, Cambridge.
Hruŝĉev, S.V. (1984). A description of weights satisfying theA ∞ condition of Muckenhoupt,Proc. Amer. Math. Soc.,90, 253–257.
Hunt, R., Muckenhoupt, B., and Wheeden, R. (1973). Weighted norm inequalities for the conjugate function and Hilbert transform,Trans. Amer. Math. Soc.,176, 227–251.
Jerison, D., and Kenig, C.E. (1982). The logarithm of the Poisson kernel for aC 1 domain has vanishing mean oscillation,Trans. Amer. Math. Soc.,176, 781–794.
John, F. (1965). Quasi-isometric mappings,Seminari 1962/63, Instituto Nazionali di Alta Matematica, 462–473.
John, F. and Nirenberg, L. (1961). On functions of bounded mean oscillation,Comm. Pure Appl. Math.,14, 415–426.
Kenig, C.E. (1994).Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems, CBMS Regional Conf. Ser. in Math.,83, American Math. Society, Providence, RI.
Kenig, C.E. and Toro, T. (1997). Harmonic measure on locally flat domains,Duke Math. J.,87, 509–551.
Kinnunen, J. (1994). Sharp results on reverse Hölder inequalities,Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes,95.
Korey, M.B. (1995). Ideal weights: doubling and absolute continuity with asymptotically optimal bounds, Ph.D. Thesis, University of Chicago, Illinois.
Korey, M.B. (1998). Carleson conditions for asymptotic weights,Trans. Amer. Math. Soc.,350, 2049–2069.
Moser, J. (1961). On Harnack's theorem for elliptic partial differential equations,Comm. Pure Appl. Math.,14, 577–591.
Muckenhoupt, B. (1972). Weighted norm inequalities for the Hardy maximal function,Trans. Amer. Math. Soc.,165, 207–226.
Muckenhoupt, B. (1974). The equivalence of two conditions for weight functions,Studia Math.,49, 101–106
Muckenhoupt, B. and Wheeden, R. (1976). Weighted bounded mean oscillation and the Hilbert transform,Studia Math.,54, 221–237.
Neri, U. (1977). Some properties of functions with bounded mean oscillation,Studia Math.,61, 63–75.
Politis, A. (1995). Sharp results on the relation between weight spaces and BMO, Ph.D. Thesis, University of Chicago, Illinois.
Reimann, H.M. and Rychener, T. (1975) Funktionen beschränkter mittlerer Oszillation.Lecture Notes in Math.,487, Springer-Verlag, Berlin and New York, NY.
Sarason, D. (1975). Functions of vanishing mean oscillation,Trans. Amer. Math. Soc.,207, 391–405.
Stein, E.M. (1993).Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, NJ.
Strömberg, J.-O. (1979). Bounded mean oscillation with Orlicz norms and duality of Hardy spaces,Indiana U. Math. J.,28, 511–544.
Strömberg, J.-O. and Torchinsky, A. (1980). Weights, sharp maximal functions and Hardy spaces,Bull. Amer. Math. Soc.,3, 1053–1056.
Strömberg, J.-O. and Torchinsky, A. (1989). Weighted Hardy Spaces,Lecture Notes in Math. 1381, Springer-Verlag, Berlin and New York, NY.
Wik, I. (1990). Reverse Hölder inequalities with constant close to 1,Ricerche Mat.,39, 151–157.
Author information
Authors and Affiliations
Additional information
Oh! the little more, and how much it is! And the little less, and what worlds away!
Acknowledgements and Notes. Supported by the Max-Planck-Gesellschaft. This work is a revised form of part of the author's dissertation, which was written under Professor Carlos E. Kenig at the University of Chicago.
Rights and permissions
About this article
Cite this article
Korey, M.B. Ideal weights: Asymptotically optimal versions of doubling, absolute continuity, and bounded mean oscillation. The Journal of Fourier Analysis and Applications 4, 491–519 (1998). https://doi.org/10.1007/BF02498222
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02498222