• DOI:10.2307/1995838
• Corpus ID: 53996980

A MAXIMAL FUNCTION CHARACTERIZATION OF THE CLASS H ' i 1 )

@inproceedings{Gundy2010AMF,
  title={A MAXIMAL FUNCTION CHARACTERIZATION OF THE CLASS H ' i 1 )},
  author={Richard F. Gundy and Martin L. Silverstein},
  year={2010}
}

• R. Gundy, M. Silverstein
• Published 2010
• Mathematics

Let u be harmonic in the upper half-plane and 0 < p < co. Then « = ReF for some analytic function F of the Hardy class Hp if and only if the nontangential maximal function of « is in L". A general integral inequality between the nontangential maximal function of u and that of its conjugate function is established. Hardy and Littlewood have shown [6] ([12,1, p. 278]) that if F(z) is an analytic function in the unit disc \z\ < 1, and 0,(0) is the Stolz domain given by the interior of the smallest convex set containing the disc \z\ < a and the point ew, then sup \ReF(z)\"d9 ^ Qp sup \F(reie)\p d9 Jo zeCla(B) ' 0<r<l Jo for all/? > 0,0 < a < 1. In this paper we prove the converse inequality which, together with the above theorem of Hardy and Littlewood, gives a maximal function characterization of the Hardy class Hp. We also prove the analogous inequality for the upper half-plane. Since the latter is the more difficult case, we treat it first. In order to state our principal result we require some definitions. Recall that an analytic function F(z) belongs to the class Hp, p>0, in the upper half-plane Im z>0 if it is defined there and /»oo sup \F(x + iy)\pdx !/>0 J -oo < oo. If f(z) is any function defined in the upper half-plane, its nontangential maximal function is given by Na(f)(x)= sup |/(z)| zera(x) where Ta(x) is the cone {z = s + iy : \s — x\ <ay}. The function Na(f) belongs to L", p>0, if /: \Na(f)\* dx < co. Received by the editors September 1, 1970. AMS 1969 subject classifications. Primary 3067, 3110, 6062.