The Calderon–Zygmund Operator and its Relation to Asymptotic Estimates of Ordinary Differential Operators

The problem to estimate expressions of the kind $$ Y\left(\lambda \right)=\underset{x\upepsilon \left[0,1\right]}{\sup}\left|\underset{0}{\overset{x}{\int }}f(t){e}^{i\lambda t} dt\right| $$ is considered. In particular, for the case f ∈ Lp[0, 1], p ∈ (1, 2], we prove the estimate $$ {\left\Vert Y\left(\lambda \right)\right\Vert}_{L_q\left(\mathbb{R}\right)}\le C{\left\Vert f\right\Vert}_{L_p} $$ for each q exceeding p', where 1/p+1/p' = 1. The same estimate is proved for the space Lq(dμ), where dμ is an arbitrary Carleson measure in the upper half-plane C+. Also, we estimate more complex expressions of the kind Υ(λ) arising in the study of asymptotical properties of the fundamental system of solutions for n-dimensional systems of the kind y'= By + A(x)y + C(x, λ)y as |λ| → ∞ in suitable sectors of the complex plane.