The Calderon--Zygmund operator and its relation to asymptotic estimates for ordinary differential operators

A. M. Savchuk

Contemporary Mathematics. Fundamental Directions, Differential and functional differential equations, Tome 63 (2017) no. 4, pp. 689-702

Résumé

We consider the problem of estimating of expressions of the kind $\Upsilon(\lambda)=\sup_{x\in[0,1]}\left|\int_0^xf(t)e^{i\lambda t}\,dt\right|$. In particular, for the case $f\in L_p[0,1]$, $p\in(1,2]$, we prove the estimate $\|\Upsilon(\lambda)\|_{L_q(\mathbb R)}\le C\|f\|_{L_p}$ for any $q>p'$, where $1/p+1/p'=1$. The same estimate is proved for the space $L_q(d\mu)$, where $d\mu$ is an arbitrary Carleson measure in the upper half-plane $\mathbb C_+$. Also, we estimate more complex expressions of the kind $\Upsilon(\lambda)$ arising in study of asymptotics of the fundamental system of solutions for systems of the kind $\mathbf y'=B\mathbf y+A(x)\mathbf y+C(x,\lambda)\mathbf y$ with dimension $n$ as $|\lambda|\to\infty$ in suitable sectors of the complex plane.