Abstract
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.
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References
G. D. Birkhoff, “On the asymptotic character of the solutions of certain linear diferential equations containing a parameter,” Trans. Am. Math. Soc., 9, 21–231 (1908).
J. Garnett, Bounded Analytic Functions [Russian translation], Mir, Moscow (1984).
L. Grafakos, Classical Fourier Analysis, Springer Science+Business Media (2008).
L. Grafakos, Modern Fourier Analysis, Springer Science+Business Media (2009).
Y. Meyer, R. Coifman, Wavelets Calderon—Zygmund and Multilinear Operators, Cambridge Univ. Press (1997).
K. A. Mirzoev and A. A. Shkalikov, “Even-order differential operators with distributions as coefficients,” Mat. Zametki, 99, No. 5, 788–793 (2016).
V. S. Rykhlov, “Asymptotical formulas for solutions of linear differential systems of the first order,” Result. Math., 36, 342–353 (1999).
A. M. Savchuk and A. A. Shkalikov, “Sturm–Liouville operators with distributions as potentials,” Tr. Mosk. Mat. Ob-va, 64, 159–219 (2003).
A. M. Savchuk and A. A. Shkalikov, “The Dirac operator with complex-valued summable potential,” Math. Notes, 96, No. 5, 3–36 (2014).
A. M. Savchuk and A. A. Shkalikov, “Asymptotic formulas for fundamental system of solutions of high order ordinary differential equations with coefficients-distributions,” arXiv:1704.02736 (04/2017).
A. A. Shkalikov, “Perturbations of self-adjoint operators with discrete spectrum,” Usp. Mat. Nauk, 71, No. 5 (431), 113–174 (2016).
E. M. Stein, Singular Integrals and Differentiability Properties of Functions [Russian translation], Mir, Moscow (1973).
J. D. Tamarkin, On Some General Problems of the Theory of Ordinary Linear Differential Operators and on Expansion of Arbitrary Functions into Series, Petrograd (1917).
H. Triebel, Interpolation Theory, Function Spaces, Differential Operators [Russian translation], Mir, Moscow (1980).
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Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 63, No. 4, Differential and Functional Differential Equations, 2017.
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Savchuk, A.M. The Calderon–Zygmund Operator and its Relation to Asymptotic Estimates of Ordinary Differential Operators. J Math Sci 259, 908–921 (2021). https://doi.org/10.1007/s10958-021-05668-w
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DOI: https://doi.org/10.1007/s10958-021-05668-w