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Sergey V. Astashkin

Publications and source records attributed to Sergey V. Astashkin.

23 records · Page 2Linked to original sources
arXiv2018

Limiting interpolation spaces via extrapolation

We give a complete characterization of limiting interpolation spa\-ces for the real method of interpolation using extrapolation theory. For this purpose the usual tools (e.g., Boyd indices or the boundedness of Hardy type operators) are not appropriate. Instead, our characterization hinges upon the boundedness of some simple operators (e.g. $f\mapsto f(t^{2})/t$, or $f\mapsto f(t^{1/2}% )$) acting on the underlying lattices that are used to control the $K$- and $J$-functionals. Reiteration formulae, extending Holmstedt's classical reiteration theorem to limiting spaces, are also proved and characterized in this fashion. The resulting theory gives a unified roof to a large body of literature that, using ad-hoc methods, had covered only special cases of the results obtained here. Applications to Matsaev ideals, Grand Lebesgue spaces, Bourgain-Brezis-Mironescu-Maz'ya-Shaposhnikova limits, as well as a new vector valued extrapolation theorems, are provided.

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arXiv2015

Isomorphic structure of Cesàro and Tandori spaces

We investigate the isomorphic structure of the Cesàro spaces and their duals, the Tandori spaces. The main result states that the Cesàro function space $Ces_{\infty}$ and its sequence counterpart $ces_{\infty}$ are isomorphic, which answers to the question posted in \cite{AM09}. This is rather surprising since $Ces_{\infty}$ has no natural lattice predual similarly as the known Talagrand's example \cite{Ta81}. We prove that neither $ces_{\infty}$ is isomorphic to $l_{\infty}$ nor $Ces_{\infty}$ is isomorphic to the Tandori space $\widetilde{L_1}$ with the norm $\|f\|_{\widetilde{L_1}}= \|\widetilde{f}\|_{L_1},$ where $\widetilde{f}(t):= \esssup_{s \geq t} |f(s)|.$ Our investigation involves also an examination of the Schur and Dunford-Pettis properties of Cesàro and Tandori spaces. In particular, using Bourgain's results we show that a wide class of Ces{à}ro-Marcinkiewicz and Ces{à}ro-Lorentz spaces have the latter property.

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arXiv2012

Interpolation of Ces{à}ro sequence and function spaces

The interpolation property of Ces{à}ro sequence and function spaces is investigated. It is shown that $Ces_p(I)$ is an interpolation space between $Ces_{p_0}(I)$ and $Ces_{p_1}(I)$ for $1 < p_0 < p_1 \leq \infty$ and $1/p = (1 - θ)/p_0 + θ/p_1$ with $0 < θ< 1$, where $I = [0, \infty)$ or $[0, 1]$. The same result is true for Ces{à}ro sequence spaces. On the other hand, $Ces_p[0, 1]$ is not an interpolation space between $Ces_1[0, 1]$ and $Ces_{\infty}[0, 1]$.

math.FA↗
arXiv2012

New examples of K-monotone weighted Banach couples

Some new examples of K-monotone couples of the type (X, X(w)), where X is a symmetric space on [0, 1] and w is a weight on [0, 1], are presented. Based on the property of the w-decomposability of a symmetric space we show that, if a weight w changes sufficiently fast, all symmetric spaces X with non-trivial Boyd indices such that the Banach couple (X, X(w)) is K-monotone belong to the class of ultrasymmetric Orlicz spaces. If, in addition, the fundamental function of X is t^{1/p} for some p \in [1, \infty], then X = L_p. At the same time a Banach couple (X, X(w)) may be K-monotone for some non-trivial w in the case when X is not ultrasymmetric. In each of the cases where X is a Lorentz, Marcinkiewicz or Orlicz space we have found conditions which guarantee that (X, X(w)) is K-monotone.

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