Hardy-Sobolev spaces and algebras of holomorphic functions on the unit ball in Cn

For 0<p <= infinity let H-p (B-n) denote the usual Hardy space of holomorphic functions on the unit ball B-n in C-n, n >= 2. If f is a holomorphic function on B-n, the radial derivative Rf is defined by Rf(z) = Sigma(n)(i=1)z(i)partial derivative f/partial derivative z(i), and for m = 2, 3, ... , the mth order radial derivative R-m is defined by R(m)f = R(R(m-1)f). The Hardy-Sobolev space H-m(p)(B-n) of order m is defined as the set of holomorphic functions f on B-n for which R(m)f is an element of H-p(B-n). The main result is as follows: Theorem: Let m is an element of {1, 2, 3, ... }. If (a) m >= n and n/m <= p <= infinity, or (b) 1 <= m < n (n >= 2) and n/m < p <= infinity, then H-m(p)(B-n) is an algebra. Furthermore, parallel to parallel to(p,m,lambda) defined for p >= 1 by parallel to f parallel to(p,m,lambda) = parallel to f parallel to(infinity) + Sigma(m)(k=1)lambda(k) parallel to R(k)f parallel to((m/k)p), f is an element of H-m(p)(B-n), where {lambda(k)}(k=1)(m) is a sequence of positive numbers satisfying lambda(k)((k)(j)) <= lambda j lambda(k-j,) j = 1, ... , k, is a Banach algebra norm on H-m(p)(B-n) whenever p, m with p >= 1 satisfy one of (a) or (b) above. In the article we also prove that if m > n and (n/m) <= p 1, then H-m(p)(B-n) is a p-Banach algebra under a suitable p-norm on H-m(p)(B-n). By examples it is shown that the results are best possible.