Let Ω be a domain of finite type in ℂ² and let f be a function holomorphic in Ω and belonging to ${}^{k,\alpha }\left(\Omega ̅\right)$. We prove the existence of boundary values for some suitable derivatives of f of order greater than k. The gain of derivatives holds in the complex-tangential direction and it is precisely related to the geometry of ∂Ω. Then we prove a property of non-isotropic Hölder regularity for these boundary values. This generalizes some results given by J. Bruna and J. M. Ortega for the unit ball.

Let $D$ be a domain in ${ℂ}^2$. For $w\in ℂ$, let $D_w=\{z\in ℂ\mid (z,w)\in D\}$. If $f$ is a holomorphic and square-integrable function in $D$, then the set $E(D,f)$ of all $w$ such that $f(.,w)$ is not square-integrable in $D_w$ is of measure zero. We call this set the exceptional set for $f$. In this note we prove that for every $0<r<1$,and every $G_{\delta }$-subset $E$ of the circle $C(0,r)=\{z\in ℂ\mid |z|=r\}$,there exists a holomorphic square-integrable function $f$ in the unit ball $B$ in ${ℂ}^2$ such that $E(B,f)=E.$

In this paper, we limit our analysis to the difference of the weighted composition operators acting from the Hardy space to weighted-type space in the unit ball of ${ℂ}^N$, and give some necessary and sufficient conditions for their boundedness or compactness. The results generalize the corresponding results on the single weighted composition operators and on the differences of composition operators, for example, M. Lindström and E. Wolf: Essential norm of the difference of weighted composition operators....

Let D be a bounded strictly pseudoconvex domain of Cn with C ∞ boundary and Y = {z; u1(z) = ... = ul(z) = 0} a holomorphic submanifold in the neighbourhood of D', of codimension l and transversal to the boundary of D.In this work we give a decomposition formula f = u1f1 + ... + ulfl for functions f of the Bergman-Sobolev space vanishing on M = Y ∩ D. Also we give necessary and sufficient conditions on a set of holomorphic functions {fα}|α|≤m on M, so that there exists a holomorphic function in the...