arXiv subjects
Publications and source records attributed to Evgueni Doubtsov.
For arbitrary radial weights $w$ and $u$, we study the integration operator between the growth spaces $H_w^\infty$ and $H_u^\infty$ on the complex plane. Also, we investigate the differentiation operator on the Hardy growth spaces $H_w^p$, $0<p<\infty$, defined on the unit disk or on the complex plane. As in the case $p=\infty$, the log-convex weights $w$ play a special role in the problems under consideration.
Let $\mathbb{D}$ denote the unit disk of $\mathbb{C}$ and let $Λ^α(\mathbb{D})$ denote the scale of holomorphic Lipschitz spaces extended to all $α\in\mathbb{R}$. For arbitrary $α, β\in\mathbb{R}$, we characterize the bounded weighted composition operators from $Λ^β(\mathbb{D})$ into $Λ^α(\mathbb{D})$ and estimate their essential norms.
Let $\mathcal{H}ol(B_d)$ denote the space of holomorphic functions on the unit ball $B_d$ of $\mathbb{C}^d$, $d\ge 1$. Given a log-convex strictly positive weight $w(r)$ on $[0,1)$, we construct a function $f\in\mathcal{H}ol(B_d)$ such that the standard integral means $M_p(f, r)$ and $w(r)$ are equivalent for any $0<p\le\infty$. Also, we obtain similar results related to volume integral means.
Let $w$ be an unbounded radial weight on the complex plane. We study the following approximation problem: find a proper holomorphic map $f: \mathbb{C}\to\mathbb{C}^n$ such that $|f|$ is equivalent to $w$. We give several characterizations of those $w$ for which the problem is solvable. In particular, a constructive characterization is given in terms of tropical power series. Moreover, the following natural objects and properties are involved: essential weights on the complex plane, approximation by power series with positive coefficients, approximation by the maximum of a holomorphic function modulus. Extensions to several complex variables and approximation by harmonic maps are also considered.
Let $ω$ be an unbounded radial weight on $\mathbb{C}^d$, $d\ge 1$. Using results related to approximation of $ω$ by entire maps, we investigate Volterra type and weighted composition operators defined on the growth space $\mathcal{A}^ω(\mathbb{C}^d)$. Special attention is given to the operators defined on the growth Fock spaces.
Let $Ω\subset \mathbb{C}$ be a bounded domain with $\mathcal{C}^{1,ω}$-smooth boundary, where $ω$ is a Dini-smooth modulus of continuity. We prove that the restricted Beurling transform is bounded on the Campanato space $\mathrm{BMO}_ω(Ω)$.
Let $H(D)$ denote the space of holomorphic functions on the unit disk $D$. We characterize those radial weights $w$ on $D$, for which there exist functions $f, g \in H(D)$ such that the sum $|f| + |g|$ is equivalent to $w$. Also, we obtain similar results in several complex variables for circular, strictly convex domains with smooth boundary.
Let $h(B_d)$ denote the space of real-valued harmonic functions on the unit ball $B_d$ of $\mathbb{R}^d$, $d\ge 2$. Given a radial weight $w$ on $B_d$, consider the following problem: construct a finite family $\{f_1, f_2, \dots, f_J\}$ in $h(B_d)$ such that the sum $|f_1| + |f_2|+\dots + |f_J|$ is equivalent to $w$. We solve the problem for weights $w$ with a doubling property. Moreover, if $d$ is even, then we characterize those $w$ for which the problem has a solution.
Let $\mathcal{B}^ω(B_d)$ denote the $ω$-weighted Bloch space in the unit ball $B_d$ of $\mathbb{C}^d$, $d\ge 1$. We show that the quadratic integral $$ \int_x^1 \frac{ω^2(t)}{t}\, dt,\quad 0<x<1, $$ governs the radial divergence and integral reverse estimates in $\mathcal{B}^ω(B_d)$.