arXiv subjects
Publications and source records attributed to Dmitry V. Rutsky.
Let $φ_j$, $j=1,2, \dots, N$, be holomorphic self-maps of the unit disk $\mathbb{D}$ of $\mathbb{C}$. We prove that the compactness of a linear combination of the composition operators $C_{φ_j}: f\mapsto f\circφ_j$ on the Hardy space $H^p(\mathbb{D})$ does not depend on $p$ for $0<p<\infty$. This answers a conjecture of Choe et al. about the compact differences $C_{φ_1} - C_{φ_2}$ on $H^p(\mathbb{D})$, $0<p<\infty$.
Let $(X, Y)$ be a couple of quasi-Banach lattices of measurable functions on $\mathbb T \times Ω$ satisfying some additional assumptions. The K-closedness of a couple of Hardy-type spaces $(X_A, Y_A)$ in $(X, Y)$ and the stability of the real interpolation $(X_A, Y_A)_{θ, p} = (X_A + Y_A) \cap (X, Y)_{θ, p}$ are shown to be equivalent to each other and to the BMO-regularity of the associated lattices $\left(L_1, \left(X^r\right)' Y^r\right)_{δ, q}$. The inclusion $\left(X^{1 - θ} Y^θ\right)_A \subset \left(X_A, Y_A \right)_{θ, \infty}$ is also characterized in these therms. New examples of couples $(X_A, Y_A)$ with this stability are given, proving that this property is strictly weaker than the BMO-regularity of $(X, Y)$.
Let $E$ be a Banach lattice on $\mathbb Z$ having order continuous norm. We show that for any function $f = \{f_j\}_{j \in \mathbb Z}$ from the Hardy space $H_\infty (E)$ such that $δ\leqslant \|f (z)\|_E \leqslant 1$ for all $z$ from the unit disk $\mathbb D$ there exists some solution $g = \{g_j\}_{j \in \mathbb Z} \in H_\infty (E')$, $\|g\|_{H_\infty (E')} \leqslant C_δ$ of the Bézout equation $\sum_j f_j g_j = 1$, also known as the vector-valued corona problem with data in $H_\infty (E)$.
A proof is given for the "only if" part of the result stated in the previous paper of the series that a suitably nondegenerate Calderón-Zygmund operator $T$ is bounded in a Banach lattice $X$ on $\mathbb R^n$ if and only if the Hardy-Littlewood maximal operator $M$ is bounded in both $X$ and $X'$, under the assumption that $X$ has the Fatou property and $X$ is $p$-convex and $q$-concave with some $1 < p, q < \infty$. We also get rid of a fixed point theorem in the proof of the main lemma and give an improved version of an earlier result concerning the divisibility of $\mathrm {BMO}$-regularity.
Let $(X, Y)$ be a suitable couple of quasi-Banach lattices of measurable functions on $\mathbb T \times Ω$, and let $(X_A, Y_A)$ be the couple of the corresponding Hardy-type spaces. It has long been suspected that the BMO-regularity property of $(X, Y)$ is not only sufficient for the $\mathrm K$-closedness of $(X_A, Y_A)$ in $(X, Y)$ but also necessary. We establish the equivalence of these two properties for a general couple of Banach lattices having the Fatou property when $Ω$ is a discrete measurable space, and also for couples $(X, Y)$ where $X$ is allowed to be quasi-Banach but $Y$ is assumed to be $p$-convex with some $p > 1$ (here $Ω$ is arbitrary). We show under certain mild restrictions that the "good interpolation" formula $$ \left(X_A, \mathrm H_q\right)_{θ, p} = \left[\left(X, \mathrm L_q\right)_{θ, p}\right]_A $$ holds true if and only if $X$ is BMO-regular.
The Coifman-Fefferman inequality implies quite easily that a Calderon-Zygmund operator $T$ acts boundedly in a Banach lattice $X$ on $\mathbb R^n$ if the Hardy-Littlewood maximal operator $M$ is bounded in both $X$ and $X'$. We discuss this phenomenon in some detail and establish a converse result under the assumption that $X$ is $p$-convex and $q$-concave with some $1 < p, q < \infty$ and satisfies the Fatou property: if a linear operator $T$ is bounded in $X$ and $T$ is nondegenerate in a certain sense (for example, if $T$ is a Riesz transform) then $M$ has to be bounded in both $X$ and $X'$.
The Riesz-Kantorovich formula expresses (under certain assumptions) the supremum of two operators $S, T : X \to Y$ where $X$ and $Y$ are ordered linear spaces as $$ S \vee T (x) = \sup_{0 \leqslant y \leqslant x} [S (y) + T (x - y)]. $$ We explore some conditions that are sufficient for its validity, which enables us to get extensions of known results characterizing lattice and decomposition properties of certain spaces of order bounded linear operators between $X$ and $Y$.
A. Ya. Zaslavskii's results on the existence of a linear (affine) selection for a linear (affine) or superlinear (convex) map $Φ: K \to 2^Y$ defined on a convex cone (convex set) $K$ having the interpolation property are extended. We prove that they hold true under more general conditions on the values of the mapping and study some other properties of the selections. This leads to a characterization of Choquet simplexes in terms of the existence of continuous affine selections for arbitrary continuous convex maps. A few applications to analysis are given, including a construction that leads to the existence of a (not necessarily bounded) solution for the corona problem in polydisk $\mathbb D^n$ with radial boundary values that are bounded almost everywhere on $\mathbb T^n$.