Measure of the level set for solutions of ordinary differential equations with constant
coefficient

V. S. Ilkiv

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 $q_n=\prod_{r=1}^n \frac{|\lambda_r|(b-a)/2}{\mathop{\rm th}|\lambda_r|(b-a)/2}$.

1. Ilkiv V.S., Maherovska T.V. Exact estimate for the measure of the level set of the modulus of a function with high-order constant-sign derivative// Mat. Stud. – 2010. – V.34, №1. – P. 57–64.

2. Ptashnyk B.Yo. Ill-posed boundary-value problems for partial differential equations. – Naukova dumka, Kiev, 1984. (in Russian)

3. Ptashnyk B.Yo., Ilkiv V.S., Kmit I.Ya., Polishchuk V.M. Nonlocal boundary-value problems for partial differential equations. – Naukova dumka, Kiev, 2002. (in Ukrainian)

4. Pyartli A.S. Diophantine approximation on submanifolds of euclidean space// Funkts. Anal. Priloz. – 1969. –V.3, №4. (in Russian)

5. Bernik V.I., Ptashnik B.I., Salyga B.O. An analog of a multipoint problem for a hyperbolic equation with constant coefficients// Differ. Equations. – 1977. – V.13, №4. – P. 637–645.

6. Ilkiv V.S. A generalization of a Pyartli lemma// In: Mater. 10th conf. mol. uchen. Inst. of appl. probl. mech. and math. AN USSR, part 2, Lviv, 1984. – P. 96–99.

7. Ilkiv V.S. Analogies of Piartly’s lemma with absolute constant// Mat. methods and fys.-mekh. polya. – 1999. – V.42, №4. – P. 68–74.

8. Il’kiv V.S., Maherovska T.V. On the constant in the Pyartli lemma// J. Lviv politech. nation. univ. “Phys. and math. sci.” – 2007. – V.601. – P. 12–17.

9. Symotyuk M.M. On the estimates of the measures of sets where the modulus of a smooth function is the upper bound// Mat. methods and fys.-mekh. polya. – 1999. – V.42, №4. – P. 90–95.

10. Beresnevich V.V. A Groshev type theorem for convergence on manifolds// Acta Math. Hungar. – 2002. – V.94, №1–2, P. 99–130.

11. Beresnevich V.V., Bernik V.I., Kleinbock D.Y., Margulis G.A. Metric Diophantine approximation: the Khintchine.Groshev theorem for nondegenerate manifolds// Moskow Math. Journ. - 2002. - V.2, №2. - P. 203-225.

12. Dani S.G., Margulis G.A. Limit distributions of orbits of unipotent flows and values of quadratic forms// Adv. in Soviet Math. - 1993. - V.16. - P. 91-137.

13. Kleinbock D., Margulis G.A. Flows on homogeneous spaces and diophantine approximation on manifold// Ann. Math. - 1998. - V.148. - P. 339-360.

14. Baker G.A.Jr., Graves-Morris P., Pade approximants. - Addison.Wesley, London, 1986.

15. Kondratiuk A.A. The Fourier series method for entire and meromorphic functions of completely regular growth. III// Math. sbornik. - 1983. - V.120(162). - P. 331-343.

16. Skaskiv O.B. Random gap series and Wimans inequality// Mat. Stud. - 2008. - V.30, №1. - P. 101-106.

17. Benvenuti P., Mesiar R., Vivona D. Monotone set functions-based integrals. . Handbook of measure theory, Elsevier, Amsterdam, 2002.

18. Cartan H. Sur les syst`emes de founctions holomorphes `a varietes lineaires et leurs applications// Ann. Sci. Ecole Norm. Sup. - 1928. - V.45, №3.

19. Levin B.Ya. Distribution of zeros of entire functions. - GITTL, 1956. (in Russian)

20. Levin B.Ya. Lectures on entire functions. - Amer. Math. Society, 1966, V.150.

21. Ilkiv V.S., Maherovska T.V. On inequalities between norms of derivatives of functions and area measure// Int. Conf. Functional methods in approximation theory and operator theory III, dedicated to the memory of V.K. Dzyadyk, 2009, Volyn, Ukraine. - P. 48-49.

22. Bernik V.I., Melnichuk Yu.V. Diophantine approximations and Hausdorff dimension. - Nauka i technika, Minsk, 1988. (in Russian)

	Measure of the level set for solutions of ordinary differential equations with constant coefficients(in Ukrainian)
Author	V. S. Ilkiv ilkivv@i.ua Lviv Polytechnic National University, Pidstryhach Institute for Applied Problems of Mechanics and Mathematics
Abstract	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 $q_n=\prod_{r=1}^n \frac{\|\lambda_r\|(b-a)/2}{\mathop{\rm th}\|\lambda_r\|(b-a)/2}$.
Keywords	Lebesgue measure; level sets; small denominators
Reference	1. Ilkiv V.S., Maherovska T.V. Exact estimate for the measure of the level set of the modulus of a function with high-order constant-sign derivative// Mat. Stud. – 2010. – V.34, №1. – P. 57–64. 2. Ptashnyk B.Yo. Ill-posed boundary-value problems for partial differential equations. – Naukova dumka, Kiev, 1984. (in Russian) 3. Ptashnyk B.Yo., Ilkiv V.S., Kmit I.Ya., Polishchuk V.M. Nonlocal boundary-value problems for partial differential equations. – Naukova dumka, Kiev, 2002. (in Ukrainian) 4. Pyartli A.S. Diophantine approximation on submanifolds of euclidean space// Funkts. Anal. Priloz. – 1969. –V.3, №4. (in Russian) 5. Bernik V.I., Ptashnik B.I., Salyga B.O. An analog of a multipoint problem for a hyperbolic equation with constant coefficients// Differ. Equations. – 1977. – V.13, №4. – P. 637–645. 6. Ilkiv V.S. A generalization of a Pyartli lemma// In: Mater. 10th conf. mol. uchen. Inst. of appl. probl. mech. and math. AN USSR, part 2, Lviv, 1984. – P. 96–99. 7. Ilkiv V.S. Analogies of Piartly’s lemma with absolute constant// Mat. methods and fys.-mekh. polya. – 1999. – V.42, №4. – P. 68–74. 8. Il’kiv V.S., Maherovska T.V. On the constant in the Pyartli lemma// J. Lviv politech. nation. univ. “Phys. and math. sci.” – 2007. – V.601. – P. 12–17. 9. Symotyuk M.M. On the estimates of the measures of sets where the modulus of a smooth function is the upper bound// Mat. methods and fys.-mekh. polya. – 1999. – V.42, №4. – P. 90–95. 10. Beresnevich V.V. A Groshev type theorem for convergence on manifolds// Acta Math. Hungar. – 2002. – V.94, №1–2, P. 99–130. 11. Beresnevich V.V., Bernik V.I., Kleinbock D.Y., Margulis G.A. Metric Diophantine approximation: the Khintchine.Groshev theorem for nondegenerate manifolds// Moskow Math. Journ. - 2002. - V.2, №2. - P. 203-225. 12. Dani S.G., Margulis G.A. Limit distributions of orbits of unipotent flows and values of quadratic forms// Adv. in Soviet Math. - 1993. - V.16. - P. 91-137. 13. Kleinbock D., Margulis G.A. Flows on homogeneous spaces and diophantine approximation on manifold// Ann. Math. - 1998. - V.148. - P. 339-360. 14. Baker G.A.Jr., Graves-Morris P., Pade approximants. - Addison.Wesley, London, 1986. 15. Kondratiuk A.A. The Fourier series method for entire and meromorphic functions of completely regular growth. III// Math. sbornik. - 1983. - V.120(162). - P. 331-343. 16. Skaskiv O.B. Random gap series and Wimans inequality// Mat. Stud. - 2008. - V.30, №1. - P. 101-106. 17. Benvenuti P., Mesiar R., Vivona D. Monotone set functions-based integrals. . Handbook of measure theory, Elsevier, Amsterdam, 2002. 18. Cartan H. Sur les syst`emes de founctions holomorphes `a varietes lineaires et leurs applications// Ann. Sci. Ecole Norm. Sup. - 1928. - V.45, №3. 19. Levin B.Ya. Distribution of zeros of entire functions. - GITTL, 1956. (in Russian) 20. Levin B.Ya. Lectures on entire functions. - Amer. Math. Society, 1966, V.150. 21. Ilkiv V.S., Maherovska T.V. On inequalities between norms of derivatives of functions and area measure// Int. Conf. Functional methods in approximation theory and operator theory III, dedicated to the memory of V.K. Dzyadyk, 2009, Volyn, Ukraine. - P. 48-49. 22. Bernik V.I., Melnichuk Yu.V. Diophantine approximations and Hausdorff dimension. - Nauka i technika, Minsk, 1988. (in Russian)
Pages	146-156
Volume	41
Issue	2
Year	2014
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

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