On the radial and nontangential maximal functions for the disc

Wheeden, Richard L.

doi:10.1090/S0002-9939-1974-0333194-1

On the radial and nontangential maximal functions for the disc
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by Richard L. Wheeden

Proc. Amer. Math. Soc. 42 (1974), 418-422

DOI: https://doi.org/10.1090/S0002-9939-1974-0333194-1

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Abstract:

Positive powers of the radial and nontangential maximal functions of a function which is harmonic or analytic in the unit disc are shown to have equivalent integrals with respect to Borel measures satisfying the growth condition $\mu (2I) \leqq c\mu (I)$ for every interval I.

References

D. L. Burkholder and R. F. Gundy, Boundary behaviour of harmonic functions in a half-space and Brownian motion, Ann. Inst. Fourier (Grenoble) 23 (1973), no. 4, 195–212 (English, with French summary). MR 365691
D. L. Burkholder and R. F. Gundy, Distribution function inequalities for the area integral, Studia Math. 44 (1972), 527–544. MR 340557, DOI 10.4064/sm-44-6-527-544
Charles Fefferman and Benjamin Muckenhoupt, Two nonequivalent conditions for weight functions, Proc. Amer. Math. Soc. 45 (1974), 99–104. MR 360952, DOI 10.1090/S0002-9939-1974-0360952-X
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. F. Gundy and R. L. Wheeden, Weighted integral inequalities for the nontangential maximal function, Lusin area integral, and Walsh-Paley series, Studia Math. 49 (1973/74), 107–124. MR 352854, DOI 10.4064/sm-49-2-107-124
Miguel de Guzmán, A covering lemma with applications to differentiability of measures and singular integral operators, Studia Math. 34 (1970), 299–317. (errata insert). MR 264022, DOI 10.4064/sm-34-3-299-317
Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095