The questions considered in this paper arose from the study [KS] of I. Fredholm's (insufficient) proof that the gap series Σ0∞ an ζn2 (where 0 < |a| < 1) is nowhere continuable across {|ζ| = 1}. The interest of Fredholm's method ([F],[ML]) is not so much its efficacy in proving gap theorems (indeed, much more general results can be got by other means, cf. the Fabry gap theorem in [Di]) as in the connection it made between certain special gap series and partial differential equations...

We solve the following Dirichlet problem on the bounded balanced domain $\Omega$ with some additional properties: For $p>0$ and a positive lower semi-continuous function $u$ on $\partial \Omega$ with $u\left(z\right)=u\left(\lambda z\right)$ for $\left|\lambda \right|=1$, $z\in \partial \Omega$ we construct a holomorphic function $f\in 𝕆\left(\Omega \right)$ such that $u\left(z\right)={\int }_{𝔻z}{\left|f\right|}^{p}d{𝔏}_{𝔻z}^2$ for $z\in \partial \Omega$, where $𝔻=\{\lambda \in ℂ|\lambda |<1\}$.

Extending previous work by Meise and Vogt, we characterize those convolution operators, defined on the space ${}_{\left(\omega \right)}\left(ℝ\right)$ of (ω)-quasianalytic functions of Beurling type of one variable, which admit a continuous linear right inverse. Also, we characterize those (ω)-ultradifferential operators which admit a continuous linear right inverse on ${}_{\left(\omega \right)}[a,b]$ for each compact interval [a,b] and we show that this property is in fact weaker than the existence of a continuous linear right inverse on ${}_{\left(\omega \right)}\left(ℝ\right)$.

We characterize stability under composition of ultradifferentiable classes defined by weight sequences M, by weight functions ω, and, more generally, by weight matrices , and investigate continuity of composition (g,f) ↦ f ∘ g. In addition, we represent the Beurling space ${}^{\left(\omega \right)}$ and the Roumieu space ${}^{\omega }$ as intersection and union of spaces ${}^{\left(M\right)}$ and ${}^M$ for associated weight sequences, respectively.

We study a rigidity property, at the vertex of some plane sector, for Gevrey classes of holomorphic functions in the sector. For this purpose, we prove a linear continuous version of Borel-Ritt's theorem with Gevrey conditions

The problem of the existence of extension maps from 0 to ℝ in the setting of the classical ultradifferentiable function spaces has been solved by Petzsche [9] by proving a generalization of the Borel and Mityagin theorems for $C^{\infty }$-spaces. We get a Ritt type improvement, i.e. from 0 to sectors of the Riemann surface of the function log for spaces of ultraholomorphic functions, by first establishing a generalization to some nonclassical ultradifferentiable function spaces.

Motivated by some recent results by Li and Stević, in this paper we prove that a two-parameter family of Cesàro averaging operators ${}^{b,c}$ is bounded on the Dirichlet spaces ${}_{p,a}$. We also give a short and direct proof of boundedness of ${}^{b,c}$ on the Hardy space $H^p$ for 1 < p < ∞.

We generalize some criteria of boundedness of $𝐋$-index in joint variables for in a bidisc analytic functions. Our propositions give an estimate the maximum modulus on a skeleton in a bidisc and an estimate of $(p+1)$th partial derivative by lower order partial derivatives (analogue of Hayman’s theorem).

We give a characterization for two different concepts of quasi-analyticity in Carleman ultraholomorphic classes of functions of several variables in polysectors. Also, working with strongly regular sequences, we establish generalizations of Watson’s Lemma under an additional condition related to the growth index of the sequence.

We develop a theory of removable singularities for the weighted Bergman space ${𝒜}_{\mu }^p\left(\Omega \right)=\{f\text{analytic}\text{in}\Omega {\int }_{\Omega }{\left|f\right|}^p\mathrm{d}\mu <\infty \}$, where $\mu$ is a Radon measure on $ℂ$. The set $A$ is weakly removable for ${𝒜}_{\mu }^p(\Omega \setminus A)$ if ${𝒜}_{\mu }^p(\Omega \setminus A)\subset \text{Hol}\left(\Omega \right)$, and strongly removable for ${𝒜}_{\mu }^p(\Omega \setminus A)$ if ${𝒜}_{\mu }^p(\Omega \setminus A)={𝒜}_{\mu }^p\left(\Omega \right)$. The general theory developed is in many ways similar to the theory of removable singularities for Hardy $H^p$ spaces, $\mathrm{B}MO$ and locally Lipschitz spaces of analytic functions, including the existence of counterexamples to many plausible properties, e.g. the union of two compact removable singularities needs not be removable....