Measure of the level set for solutions of ordinary differential equations with constant coefficients(in Ukrainian)

V. S. Ilkiv
Lviv Polytechnic National University, Pidstryhach Institute for Applied Problems of Mechanics and Mathematics
For a real functions $f\in\mathcal{C}^n[a,b]$ such that $|L_nf(x)|\ge\delta$ on $[a,b]$, where $L_n$ is a differential expression, namely $L_n=(d/dx+\lambda_1)\dotsb(d/dx+\lambda_n)$ with real $\lambda_1,\dotsc,\lambda_n$, we find a Lebesgue measure estimate for the level set $G_{L_n}(\varepsilon,\delta;f)=\{x\in[a,b]\colon|f(x)|\leq\varepsilon\}$. In particular, we establish the inequality $\mathop{\rm meas} G_{L_n}(\varepsilon,\delta;f)\le\min\{b-a,n\sqrt{2^{n+1}}\sqrt[n]{q_n\varepsilon/\delta}\},$ where $q_n=\prod_{r=1}^n \frac{|\lambda_r|(b-a)/2}{\mathop{\rm th}|\lambda_r|(b-a)/2}$.
Lebesgue measure; level sets; small denominators
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