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

This article is cited in 13 scientific papers (total in 13 papers)

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
Received: 29.11.1976

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
\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
\jour Math. Notes
\yr 1978
\vol 23
\issue 1
\pages 38--44

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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  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    2. V. V. Arestov, “Approximation of unbounded operators by bounded operators and related extremal problems”, Russian Math. Surveys, 51:6 (1996), 1093–1126  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    3. O. A. Timoshin, “The best approximation to the operator of the second mixed derivative”, Izv. Math., 62:1 (1998), 191–200  mathnet  crossref  crossref  mathscinet  zmath  isi
    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  mathnet  crossref  crossref  mathscinet  zmath  isi
    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  crossref  isi
    6. Trigub R.M., “ON Fourier MULTIPLIERS AND ABSOLUTE CONVERGENCE OF Fourier INTEGRALS OF RADIAL FUNCTIONS”, Ukrainian Math J, 62:9 (2011), 1487–1501  isi
    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  mathnet  crossref  isi  elib
    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  mathnet  crossref  crossref  mathscinet  isi  elib
    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  mathnet  crossref
    10. A. A. Koshelev, “The Landau–Kolmogorov problem for the Laplace operator on a ball”, Russian Math. (Iz. VUZ), 60:2 (2016), 25–32  mathnet  crossref  isi
    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  crossref  mathscinet  isi  scopus
    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  mathnet  crossref
    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  mathnet  crossref  elib
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