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 Mat. Zametki, 1978, Volume 23, Issue 1, Pages 67–78 (Mi mz8120)

Precise inequalities for norms of functions, third partial, second mixed, or directional derivatives

V. N. Konovalov

Mathematics Institute, Academy of Sciences of the Ukrainian SSR

Abstract: For functions $f$ which are bounded throughout the plane $R^2$ together with the partial derivatives $f^{(3,0)}$, $f^{(0,3)}$, inequalities
\begin{gather*} \|f^{(1,1)}\|\le\sqrt[3]3\|f\|^{1/3}\|f^{(3,0)}\|^{1/3}\|f^{(0,3)}\|^{1/3},
\|f_e^{(2)}\|\le\sqrt[3]3\|f\|^{1/3}(\|f^{(3,0)}\|^{1/3}|e_1|+\|f^{(0,3)}\|^{1/3}|e_2|)^2, \end{gather*}
are established, where $\|\cdot\|$ the upper bound on $R^2$ of the absolute values of the corresponding function, andf $f_e^{(2)}$ is the second derivative in the direction of the unit vector $e=(e_1,e_2)$. Functions are exhibited for which these inequalities become equalities.

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English version:
Mathematical Notes, 1978, 23:1, 38–44

Bibliographic databases:

UDC: 517.5

Citation: V. N. Konovalov, “Precise inequalities for norms of functions, third partial, second mixed, or directional derivatives”, Mat. Zametki, 23:1 (1978), 67–78; Math. Notes, 23:1 (1978), 38–44

Citation in format AMSBIB
\Bibitem{Kon78} \by V.~N.~Konovalov \paper Precise inequalities for norms of functions, third partial, second mixed, or directional derivatives \jour Mat. Zametki \yr 1978 \vol 23 \issue 1 \pages 67--78 \mathnet{http://mi.mathnet.ru/mz8120} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=480912} \zmath{https://zbmath.org/?q=an:0403.26008|0379.26008} \transl \jour Math. Notes \yr 1978 \vol 23 \issue 1 \pages 38--44 \crossref{https://doi.org/10.1007/BF01104884} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. S. M. Nikol'skii, “Aleksandrov and Kolmogorov in Dnepropetrovsk”, Russian Math. Surveys, 38:4 (1983), 41–55
2. V. V. Arestov, “Approximation of unbounded operators by bounded operators and related extremal problems”, Russian Math. Surveys, 51:6 (1996), 1093–1126
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4. V. F. Babenko, S. A. Pichugov, “Sharp Estimates of the Norms of Fractional Derivatives of Functions of Several Variables Satisfying Hölder Conditions”, Math. Notes, 87:1 (2010), 23–30
5. Babenko V.F., Parfinovych N.V., Pichugov S.A., “SHARP KOLMOGOROV-TYPE INEQUALITIES FOR NORMS OF FRACTIONAL DERIVATIVES OF MULTIVARIATE FUNCTIONS”, Ukrainian Math J, 62:3 (2010), 343–357
6. Trigub R.M., “ON Fourier MULTIPLIERS AND ABSOLUTE CONVERGENCE OF Fourier INTEGRALS OF RADIAL FUNCTIONS”, Ukrainian Math J, 62:9 (2011), 1487–1501
7. V. F. Babenko, N. V. Parfinovich, “Kolmogorov-type inequalities for the norms of Riesz derivatives of multivariable functions and some applications”, Proc. Steklov Inst. Math. (Suppl.), 277, suppl. 1 (2012), 9–20
8. V. F. Babenko, N. V. Parfinovich, S. A. Pichugov, “Kolmogorov-Type Inequalities for Norms of Riesz Derivatives of Functions of Several Variables with Laplacian Bounded in $L_\infty$ and Related Problems”, Math. Notes, 95:1 (2014), 3–14
9. S. B. Vakarchuk, A. V. Shvachko, “Inequalities of Kolmogorov's type for derived functions in two variables and application to approximation by an “angle””, Russian Math. (Iz. VUZ), 59:11 (2015), 1–18
10. A. A. Koshelev, “The Landau–Kolmogorov problem for the Laplace operator on a ball”, Russian Math. (Iz. VUZ), 60:2 (2016), 25–32
11. Babenko V.F. Parfinovich N.V., “Estimation of the Uniform Norm of One-Dimensional Riesz Potential of the Partial Derivative of a Function with Bounded Laplacian”, Ukr. Math. J., 68:7 (2016), 987–999
12. M. Sh. Shabozov, M. O. Akobirshoev, “O neravenstvakh tipa Kolmogorova dlya periodicheskikh funktsii dvukh peremennykh v $L_2$”, Chebyshevskii sb., 20:2 (2019), 348–365
13. V. V. Arestov, R. R. Akopyan, “Zadacha Stechkina o nailuchshem priblizhenii neogranichennogo operatora ogranichennymi i rodstvennye ei zadachi”, Tr. IMM UrO RAN, 26, no. 4, 2020, 7–31
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