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

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.