The problem to estimate expressions of the kind Yλ=supx?01∫0xfteiλtdtdocumentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$ Yleft(lambda right)=underset{xupepsilon left0,1right}{sup}leftunderset{0}{overset{x}{int }}f(t){e}^{ilambda t} dtright $$end{document} is considered. In particular, for the case f ∈ Lp0, 1, p ∈ (1, 2, we prove the estimate YλLq?≤CfLpdocumentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$ {leftVert Yleft(lambda right)rightVert}_{L_qleft(mathbb{R}right)}le C{leftVert frightVert}_{L_p} $$end{document} 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.
展开▼
