Nagy type inequalities in metric measure spaces and some applications

Authors

  • V.F. Babenko Oles Honchar Dnipro National University, 72 Gagarin avenue, 49010, Dnipro, Ukraine
  • V.V. Babenko Drake University, 2507 University avenue, Des Moines, USA
  • O.V. Kovalenko Oles Honchar Dnipro National University, 72 Gagarin avenue, 49010, Dnipro, Ukraine
  • N.V. Parfinovych Oles Honchar Dnipro National University, 72 Gagarin avenue, 49010, Dnipro, Ukraine
https://doi.org/10.15330/cmp.15.2.563-575

Keywords:

Nagy type inequality, Landau-Kolmogorov type inequality, Stechkin's problem, charge, modulus of continuity, mixed derivative
Published online: 2023-12-28

Abstract

We obtain a sharp Nagy type inequality in a metric space (X,ρ) with measure μ that estimates the uniform norm of a function using its Hω-norm determined by a modulus of continuity ω, and a seminorm that is defined on a space of locally integrable functions. We consider charges ν that are defined on the set of μ-measurable subsets of X and are absolutely continuous with respect to μ. Using the obtained Nagy type inequality, we prove a sharp Landau-Kolmogorov type inequality that estimates the uniform norm of a Radon-Nikodym derivative of a charge via a Hω-norm of this derivative, and a seminorm defined on the space of such charges. We also prove a sharp inequality for a hypersingular integral operator. In the case X=Rm+×Rdm, 0md, we obtain inequalities that estimate the uniform norm of a mixed derivative of a function using the uniform norm of the function and the Hω-norm of its mixed derivative.

How to Cite
(1)
Babenko, V.; Babenko, V.; Kovalenko, O.; Parfinovych, N. Nagy Type Inequalities in Metric Measure Spaces and Some Applications. Carpathian Math. Publ. 2023, 15, 563-575.