Some sharp Landau-Kolmogorov–Nagy-type inequalities in Sobolev spaces of multivariate functions

  • V. Babenko Dnipro National University named after Oles Honchar
  • V. Babenko Drake University, Des Moines, USA
  • O. Kovalenko Dnipro National University named after Oles Honchar
  • N. Parfinovych Dnipro National University named after Oles Honchar
Keywords: Nagy and Landau -- Kolmogorov type inequality, charge, gradient, mixed derivative


UDC 517.5

For a function $f$ from the Sobolev space $W^{1,p}(C),$ where $C\subset R^d$ is an open convex cone, we establish a sharp inequality  estimating $\| f\|_{L_{\infty}}$ via the $L_{p}$-norm of its gradient and a seminorm of the function. With the help of this inequality, we prove a sharp inequality estimating the ${L_{\infty}}$-norm of the Radon-Nikodym derivative of a charge defined on Lebesgue measurable subsets of  $C$ via the $L_p$-norm of the gradient of this derivative and the seminorm of the charge.  In the case where $C=R_+^m\times R^{d-m},$ $0\le m\le d,$ we obtain inequalities estimating the ${L_{\infty}}$-norm of a mixed derivative of the function $f\colon C\to R$ via its ${L_{\infty}}$-norm and the $L_p$-norm of the gradient of mixed derivative of this function. 


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How to Cite
Babenko, V., V. Babenko, O. Kovalenko, and N. Parfinovych. “Some Sharp Landau-Kolmogorov–Nagy-Type Inequalities in Sobolev Spaces of Multivariate Functions”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 75, no. 10, Oct. 2023, pp. 1347 -53, doi:10.3842/umzh.v75i10.7680.
Research articles