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 case f ∈ Lp[0, 1], p (1, 2], we prove {\left\Vert \right)\right\Vert}_{L_q\left(\mathbb{R}\right)}\le C{\left\Vert f\right\Vert}_{L_p} each q exceeding p', where 1/p+1/p' = 1. same proved space Lq(dμ), dμ an arbitrary Carleson measure in upper half-plane C+. Also, more complex Υ(λ) arising study asymptotical properties fundamental system solutions n-dimensional systems y'= By + A(x)y C(x, λ)y as |λ| → ∞ suitable sectors plane.