Pseudoconvex domains are exhausted in such a way that we keep a part of the boundary fixed in all the domains of the exhaustion. This is used to solve a problem concerning whether the generators for the ideal of either the holomorphic functions continuous up to the boundary or the bounded holomorphic functions, vanishing at a point in ${ℂ}^n$ where the fibre is nontrivial, has to exceed $n$. This is shown not to be the case.

On introduit une classe de domaines dans ${\mathbf{C}}_{\left(z\right)}^n={\mathbf{R}}_{\left(x\right)}^n\times {\mathbf{R}}_{\left(y\right)}^n$ appelés tuboïdes. Un tuboïde $D={\cup }_{x\in \Omega }(x,D_x)$ de profil $\Lambda ={\cup }_{x\in \Omega }(x,{\Lambda }_x)$ est un domaine de ${\mathbf{C}}_{\left(z\right)}^n$ dont chaque fibre $D_x$ (dans ${\mathbf{R}}_{\left(y\right)}^n)$ admet ${\Lambda }_x$ comme cône tangent à l’origine.On montre dans la première partie que l’enveloppe d’holomorphie d’un tuboïde $\widehat{D}$ de profil $\widehat{\Lambda }={\cup }_{x\in \Omega }(x,{\widehat{\Lambda }}_x)$ où ${\widehat{\Lambda }}_x$ est pour tout $x$ l’enveloppe convexe de ${\Lambda }_x$. dans la deuxième partie, l’on montre alors que tout tuboïde $D$ dont le profil $\Lambda$ a toutes ses fibres ${\Lambda }_x$ convexes contient un tuboïde $D^{\text{'}}$ de même profil qui est de plus un domaine d’holomorphie....

Let $Z\left(\mathrm{Ann}\right(r,R\left)\right)$ be the class of all continuous functions $f$ on the annulus $\mathrm{Ann}(r,R)$ in ${ℂ}^n$ with twisted spherical mean $f\times {\mu }_s\left(z\right)=0,$ whenever $z\in {ℂ}^n$ and $s\>0$ satisfy the condition that the sphere $S_s\left(z\right)\subseteq \mathrm{Ann}(r,R)$ and ball $B_r\left(0\right)\subseteq B_s\left(z\right).$ In this paper, we give a characterization for functions in $Z\left(\mathrm{Ann}\right(r,R\left)\right)$ in terms of their spherical harmonic coefficients. We also prove support theorems for the twisted spherical means in ${ℂ}^n$ which improve some of the earlier results.

It is shown that the weak multidimensional Suita conjecture fails for any bounded non-pseudoconvex domain with $C^{1+\epsilon }$-smooth boundary. On the other hand, it is proved that the weak converse to the Suita conjecture holds for any finitely connected planar domain.

We give some explicit values of the constants $C_1$ and $C_2$ in the inequality $C_1/sin\left(\frac{\pi }{p}\right)\le {\left|P\right|}_p\le C_2/sin\left(\frac{\pi }{p}\right)$ where ${\left|P\right|}_p$ denotes the norm of the Bergman projection on the $L^p$ space.