Beatrous, Frank; Burbea, Jacob Holomophic Sobolev spaces on the ball. (English) Zbl 0691.46024 Diss. Math. 276, 57 p. (1989). 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 Cited in 1 ReviewCited in 36 Documents 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 Keywords:weighted Sobolev spaces of holomorphic functions; comparison of various weighted Sobolev norms; radial derivative operator; fractional power; Lipschitz estimate; BMO and Hardy-Littlewood-type estimates; fractional powers of the Bergman and the Poisson-Szegö kernels PDFBibTeX XML