Characterizations of spaces of holomorphic functions in the ball

Abstract

Let / be holomorphic in the unit ball of C n . Several equivalent criteria for / to belong to the Hardy space H? as well as the weighted Bergman space A\, 0<p<co, q>0, of the ball are established. In the one variable case, some of the above conditions reduce to those of Yamashita, characterizing Hardy spaces of the unit disk. In addition, various identities for the norm of /, in terms of a certain integrated counting function and certain Lusin characteristics, are obtained.

Boundary Continuity of Holomorphic Functions in the Ball

“…Following [2] we define for each q > 0 a measure dVq on B by dVqiz) = Gnj9(l -¡z^^^'-dViz), where dV is the usual Lebesgue measure on C™ and the constant CnA is chosen so that dVq is a probability measure on B. Although the definitions of Sobolev norms given above are standard, it will be convenient for our purposes to use equivalent norms which involve differentiation in only the radial direction.…”

“…Thus we may use this formula as the definition of Vsf for an arbitrary real number s. By analogy with the above definitions, we define The proof of the following result can be found in [2] License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use (1.1) THEOREM. For 0 < q, 0 < p < oo and any nonnegative integer s the norms || \\p,q,s and \\ \\\p,q,s are equivalent on 0(5).…”

“…The continuity portion of Theorem 1.4 is based on the following representation formula which is a special case of Theorem 1.9 and Corollary 2.4 of [2]. Note that the integral in (iii) is well defined since the function Dnf is in the Hardy class A¿, and hence it has a boundary value function in L1idB;a).…”

“…The result is a special case of Theorems 5.12 and 5.13 of [2], and the proof will not be included here.…”

“…It can be shown (see [2]) that for s > n/p, any holomorphic function / satisfying (3)PjS is in the Lipschitz class As_"/P, and that in the limiting case s = n/p, any function satisfying (3)PiS is in the class BMOA. Moreover, we will show (see Theorem 1.3 below) that for any p > 1 there are unbounded holomorphic functions on Bn satisfying (3)Pi"/p.…”