On the Hardy-Littlewood-P\'olya and Taikov type inequalities for multiple operators in Hilbert spaces

TL;DR

This paper develops a unified framework for sharp inequalities involving multiple operators in Hilbert spaces, with applications to differential operators on manifolds and approximation theory.

Contribution

It introduces a novel unified approach to derive sharp Hardy-Littlewood-Pólya and Taikov inequalities for multiple operators, extending to unbounded operators and approximation problems.

Findings

Derived new sharp inequalities for powers of Laplace-Beltrami operators.

Established limit case inequalities for functions in Euclidean space.

Provided bounds for approximation of unbounded operators by bounded ones.

Abstract

We present unified approach to obtain sharp mean-squared and multiplicative inequalities of Hardy-Littlewood-Poly\'a and Taikov types for multiple closed operators acting on Hilbert space. We apply our results to establish new sharp inequalities for the norms of powers of the Laplace-Beltrami operators on compact Riemmanian manifolds and derive the well-known Taikov and Hardy-Littlewood-Poly\'a inequalities for functions defined on $d$-dimensional space in the limit case. Other applications include the best approximation of unbounded operators by linear bounded ones and the best approximation of one class by elements of other class. In addition, we establish sharp Solyar-type inequalities for unbounded closed operators with closed range.