Ideal weights: Asymptotically optimal versions of doubling, absolute continuity, and bounded mean oscillation

  • Michael Brian Korey


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.

Math Subject Classifications

Primary 42B25, 26D15 secondary 26B35 

Keywords and Phrases

Doubling measure bounded mean oscillation A condition reverse Hölder inequality MuckenhouptAp condition arithmetic-geometric inequality 


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  1. [1]
    Beurling, A. and Ahlfors, L. (1956). The boundary correspondence under quasiconformal mappings,Acta Math.,96, 125–142.MATHMathSciNetCrossRefGoogle Scholar
  2. [2]
    Buckley, S.M. (1993). Estimates for operator norms on weighted spaces and reverse Jensen inequalities,Trans. Amer. Math. Soc.,340, 253–272.MATHMathSciNetCrossRefGoogle Scholar
  3. [3]
    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.MATHMathSciNetCrossRefGoogle Scholar
  4. [4]
    Carleson, L. (1967). On mappings, conformal at the boundary,J. Analyse Math.,19, 1–13.MATHMathSciNetCrossRefGoogle Scholar
  5. [5]
    Carleson, L. (1981). BMO—10 years' development,18th Scandanavian Congress of Mathematicians (Aarhus, 1980), Progress in Mathematics,11, Birkhäuser, 3–21.Google Scholar
  6. [6]
    Chandrasekharan, K. (1968)Introduction to Analytic Number Theory, Grund. der math. Wiss.,148, Springer-Verlag, New York, NY.MATHGoogle Scholar
  7. [7]
    Coifman, R.R. and Fefferman, C. (1974). Weighted norm inequalities for maximal functions and singular integrals,Studia Math.,51, 241–250.MATHMathSciNetGoogle Scholar
  8. [8]
    Dahlberg, B. (1977) On estimates for harmonic measure,Arch. Rat. Mech. Anal.,65, 272–288.MathSciNetCrossRefGoogle Scholar
  9. [9]
    Fefferman, C. and Muckenhoupt, B. (1974) Two nonequivalent conditions for weight functions,Proc. Amer. Math. Soc.,45, 99–104.MATHMathSciNetCrossRefGoogle Scholar
  10. [10]
    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.MATHMathSciNetCrossRefGoogle Scholar
  11. [11]
    García-Cuerva, J. and Rubio de Francia, J.L. (1985).Weighted Norm Inequalities and Related Topics, North-Holland, Amsterdam and New York, NY.MATHGoogle Scholar
  12. [12]
    Gehring, F.W. (1973). TheL p-integrability of the partial derivatives of a quasiconformal mapping,Acta Math.,130, 265–277.MATHMathSciNetCrossRefGoogle Scholar
  13. [13]
    Hardy, G.H., Littlewood, J.E., and Pólya, G. (1934).Inequalities, Cambridge University Press, Cambridge.Google Scholar
  14. [14]
    Hruŝĉev, S.V. (1984). A description of weights satisfying theA condition of Muckenhoupt,Proc. Amer. Math. Soc.,90, 253–257.MathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    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.MATHMathSciNetCrossRefGoogle Scholar
  16. [16]
    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.MathSciNetCrossRefGoogle Scholar
  17. [17]
    John, F. (1965). Quasi-isometric mappings,Seminari 1962/63, Instituto Nazionali di Alta Matematica, 462–473.Google Scholar
  18. [18]
    John, F. and Nirenberg, L. (1961). On functions of bounded mean oscillation,Comm. Pure Appl. Math.,14, 415–426.MATHMathSciNetGoogle Scholar
  19. [19]
    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.MATHGoogle Scholar
  20. [20]
    Kenig, C.E. and Toro, T. (1997). Harmonic measure on locally flat domains,Duke Math. J.,87, 509–551.MATHMathSciNetCrossRefGoogle Scholar
  21. [21]
    Kinnunen, J. (1994). Sharp results on reverse Hölder inequalities,Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes,95.Google Scholar
  22. [22]
    Korey, M.B. (1995). Ideal weights: doubling and absolute continuity with asymptotically optimal bounds, Ph.D. Thesis, University of Chicago, Illinois.Google Scholar
  23. [23]
    Korey, M.B. (1998). Carleson conditions for asymptotic weights,Trans. Amer. Math. Soc.,350, 2049–2069.MATHMathSciNetCrossRefGoogle Scholar
  24. [24]
    Moser, J. (1961). On Harnack's theorem for elliptic partial differential equations,Comm. Pure Appl. Math.,14, 577–591.MATHMathSciNetGoogle Scholar
  25. [25]
    Muckenhoupt, B. (1972). Weighted norm inequalities for the Hardy maximal function,Trans. Amer. Math. Soc.,165, 207–226.MATHMathSciNetCrossRefGoogle Scholar
  26. [26]
    Muckenhoupt, B. (1974). The equivalence of two conditions for weight functions,Studia Math.,49, 101–106MATHMathSciNetGoogle Scholar
  27. [27]
    Muckenhoupt, B. and Wheeden, R. (1976). Weighted bounded mean oscillation and the Hilbert transform,Studia Math.,54, 221–237.MATHMathSciNetGoogle Scholar
  28. [28]
    Neri, U. (1977). Some properties of functions with bounded mean oscillation,Studia Math.,61, 63–75.MATHMathSciNetGoogle Scholar
  29. [29]
    Politis, A. (1995). Sharp results on the relation between weight spaces and BMO, Ph.D. Thesis, University of Chicago, Illinois.Google Scholar
  30. [30]
    Reimann, H.M. and Rychener, T. (1975) Funktionen beschränkter mittlerer Oszillation.Lecture Notes in Math.,487, Springer-Verlag, Berlin and New York, NY.MATHGoogle Scholar
  31. [31]
    Sarason, D. (1975). Functions of vanishing mean oscillation,Trans. Amer. Math. Soc.,207, 391–405.MATHMathSciNetCrossRefGoogle Scholar
  32. [32]
    Stein, E.M. (1993).Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, NJ.MATHGoogle Scholar
  33. [33]
    Strömberg, J.-O. (1979). Bounded mean oscillation with Orlicz norms and duality of Hardy spaces,Indiana U. Math. J.,28, 511–544.CrossRefGoogle Scholar
  34. [34]
    Strömberg, J.-O. and Torchinsky, A. (1980). Weights, sharp maximal functions and Hardy spaces,Bull. Amer. Math. Soc.,3, 1053–1056.MATHMathSciNetCrossRefGoogle Scholar
  35. [35]
    Strömberg, J.-O. and Torchinsky, A. (1989). Weighted Hardy Spaces,Lecture Notes in Math. 1381, Springer-Verlag, Berlin and New York, NY.MATHGoogle Scholar
  36. [36]
    Wik, I. (1990). Reverse Hölder inequalities with constant close to 1,Ricerche Mat.,39, 151–157.MATHMathSciNetGoogle Scholar

Copyright information

© Birkhäuser Boston 1998

Authors and Affiliations

  • Michael Brian Korey
    • 1
  1. 1.Institut für MathematikUniversität PotsdamPotsdamGermany

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