Uniform Estimates of Remainders in Spectral Analysis of Linear Differential Systems

Abstract

We study the problem of estimating the expression Υ(λ) = sup{|∫₀^xf(t)e^iλω(t)dt|: x ∈ [0, 1]}, where the derivative of the function ω(t) is positive almost everywhere on [0, 1]. In particular, for f ∈ L_p[0, 1], p ∈ (1, 2], we prove the estimate ∥Υ(λ)∥ L_q(ℝ) ≤ C∥f∥L_p, where 1/p + 1/q = 1. The same estimate is obtained in the space Lq(dμ), where dμ is an arbitrary Carleson measure in the open upper half-plane ℂ+. In addition, we estimate more complicated expressions like Υ(λ) that arise when studying the asymptotics of fundamental solution systems for systems of the form y′ = λρ(x)By +A(x)y +C(x, λ)y as |λ| →∞in appropriate sectors of the complex plane.