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Holomophic Sobolev spaces on the ball. (English) Zbl 0691.46024

The authors develop some aspects of the theory of the weighted Sobolev spaces of holomorphic functions on the unit ball B in \({\mathbb{C}}^ n\), one of the main results being comparison of various weighted Sobolev norms.
Let \(dv_ q(z)=(\Gamma (n+q)/\pi^ n\Gamma (q))(1-\| z\|^ 2)^{q-1}dv(z),\) \(q>0\), \(\| f\|_{p,q}=\{\int_{B}| f|^ pdv_ q\}^{1/p}\) and let \(W^ p_{q,s}\) be the completion of \(C^{\infty}(\bar B)\) with respect to \[ (1)\quad \| f\|_{p,q;s}=\{\int_{| \alpha | +| \beta | \leq s}\frac{| \alpha | !}{\alpha !}\| \partial^{\alpha}{\bar \partial}^{\beta}f\|^ p_{p,q}\}^{1/p}. \] The authors consider the “radial derivative operator” \(R=\sum^{n}_{j=1}z_ j\partial /\partial z_ j\) and the operator \(D=R+E\), E being the identity operator. The operator D allows an easy definition of the fractional power \(D^ sf=\sum (| \alpha | +1)^ sa_{\alpha}z^{\alpha}\) for holomorphic functions \(f(z)=\sum a_{\alpha}z^{\alpha}\). The authors define \(A^ p_{q,s}\) to be the space of holomorphic functions with \(\| f\|_{p,q;s}=\| D^ sf\|_{p,q}<\infty\) and show that \(A^ p_{q,s}=W^ p_{q,s}\), \(0<p\leq \infty\), \(0\leq q<\infty\) and the norms are equivalent (the case \(q=0\) denoting Hardy space, in fact) if s is an integer. This result was known earlier in the case \(q=1\) and \(p=2\) (H. Boas). It leads in particular to a Lipschitz estimate for holomorphic functions f with \(\| f\|_{p,q;s}<\infty\). BMO and Hardy-Littlewood-type estimates are also given, which extends previous results of I. Graham and S. G. Krantz. We also point out some estimates of \(| \partial^{\alpha}f(z)|\) via \(\| f\|_{p,q,s}\) in the case of non-integer s.
The proof of the above results is based on some assertions which are obtained by the authors for the fractional powers of the Bergman and the Poisson-Szegö kernels.
Reviewer: S.G.Samko

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
42B30 \(H^p\)-spaces
32A35 \(H^p\)-spaces, Nevanlinna spaces of functions in several complex variables
32A10 Holomorphic functions of several complex variables
32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)
30D55 \(H^p\)-classes (MSC2000)
26A16 Lipschitz (Hölder) classes