Related papers: Corona problem with data in ideal spaces of sequen…

We characterize the sequences of complex numbers $(z_{n})_{n \in \mathbb{N}}$ and the locally complete $(DF)$-spaces $E$ such that for each $(e_{n})_{n \in \mathbb{N}} \in E^\mathbb{N}$ there exists an $E$-valued function $\mathbf{f}$ on…

We extend representation formulas that generalize the similarity principle of solutions to the Vekua equation to certain classes of meta-analytic functions. Also, we solve a generalization of the higher-order Schwarz boundary value problem…

Let ${\mathbb{D}}=\{z\in \mathbb{C}:|z|<1\}$ and for an integer $d\geq 1$, let $S_d$ denote the symmetric group, consisting of of all permutations of the set $\{1,\cdots, d\}$. A function $f:{\mathbb{D}}^d\rightarrow \mathbb{C}$ is…

A classical problem posed in 1992 by Huijsmans and de Pagter asks whether, for every positive operator $T$ on a Banach lattice with spectrum $\sigma(T) = \{1\}$, the inequality $T \ge \operatorname{id}$ holds true. While the problem remains…

Given a probability measure space $(X,\Sigma,\mu)$, it is well known that the Riesz space $L^0(\mu)$ of equivalence classes of measurable functions $f: X \to \mathbf{R}$ is universally complete and the constant function $\mathbf{1}$ is a…

We study the ideal membership problem in $H^\infty$ on the unit disc. Thus, given functions $f,f_1,\ldots,f_n$ in $H^\infty$, we seek sufficient conditions on the size of $f$ in order for $f$ to belong to the ideal of $H^\infty$ generated…

The Cluster Value Theorem is known for being a weak version of the classical Corona Theorem. Given a Banach space $X$, we study the Cluster Value Problem for the ball algebra $A_u(B_X)$, the Banach algebra of all uniformly continuous…

For a non-empty locally compact Hausdorff space $X$ and a Dedekind complete normal vector lattice $E$, we show that the vector lattice of norm to order bounded operators from ${\text C}_{\text c}(X)$ or ${\text C}_0(X)$ into $E$ is…

Let $\Omega$ be a convex domain in the complex plane ${\mathbb C}$ with $\Omega \not= {\mathbb C}$, and $P$ be a conformal map of the unit disk ${\mathbb D}$ onto $\Omega$. Let ${\mathcal F}_\Omega$ be the class of analytic functions $g$ in…

We continue the work of \cite{TLNT}. Let $E$ be a non-Blaschke subset of the unit disc $\mathbb{D}$ of the complex plane $\mathbb{C}$. Fixed $1\leq p\leq \infty$, let $H^p(\mathbb{D})$ be the Hardy space of holomorphic functions in the disk…

Let $K$ be a square Cantor set, i.e. the Cartesian product $K=E\times E$ of two linear Cantor sets. Let $\delta_n$ denote the proportion of the intervals removed in the $n$th stage of the construction of $E$. It is shown that if…

In this paper, we study an extremal problem concerning best approximation in the Hardy space $H^1$ on the unit disk $\mathbb D$. Specifically, we consider weighted combinations of the Cauchy-Szeg\"o kernel and its derivative, parametrized…

It is well known that non-negative solutions to the Dirichlet problem $\Delta u =f$ in a bounded domain $\Omega$, where $f\in L^q(\Omega)$, $q>\frac{n}2$, satisfy $\|u\|_{L^\infty(\Omega)} \leq C\|f\|_{L^q(\Omega)}$. We generalize this…

The aim of this paper is investigating the existence of one or more critical points of a family of functionals which generalizes the model problem \[ \bar J(u)\ =\ \frac1p\ \int_\Omega \bar A(x,u)|\nabla u|^p dx - \int_\Omega G(x,u) dx \]…

Let $\mathbb D^n\subset\mathbb C^n$ be the open unit polydisk, $K\subset\mathbb D^n$ be an $n$-ary Cartesian product of planar sets, and $\hat U\subset \mathfrak M^n$ be an open neighbourhood of the closure $\bar K$ of $K$ in $\mathfrak…

In this paper, we consider the H\'enon problem in the setting of Orlicz-Sobolev spaces: \begin{equation*} \begin{cases} -\Delta_g u= |x|^\alpha h( u) \quad \text{in }B\\ u>0 \quad \text{in }B\\ u= 0 \quad \text{on }\partial B\\ \end{cases}…

The Cauchy problem for the higher order equations in the mKdV hierarchy is investigated with data in the spaces $\hat{H}^r_s(\R)$ defined by the norm $$\n{v_0}{\hat{H}^r_s(\R)} := \n{< \xi > ^s\hat{v_0}}{L^{r'}_{\xi}},\quad < \xi…

We investigate the problem of a characterization of extreme points of the unit ball of a Hardy-Lorentz space $H(\Lambda(\varphi))$, posed by Semenov in 1978. New necessary and sufficient conditions, under which a normalized function $f$ in…

Let $0<p<\infty$, $\Gamma$ be a Lipschitz curve on the complex plane~$\mathbb{C}$ and $\Omega_+$ is the domain above $\Gamma$, we define Hardy space $H^p(\Omega_+)$ as the set of analytic functions $F$ satisfying…

We develop arguments on convexity and minimization of energy functionals on Orlicz-Sobolev spaces to investigate existence of solution to the equation $\displaystyle -\mbox{div} (\phi(|\nabla u|) \nabla u) = f(x,u) + h \mbox{in} \Omega$…