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A sharp Trudinger type inequality for harmonic functions and its applications.

Yili Tan1, Yongli An2, Hong Wang1

  • 1College of Science, North China University of Science and Technology, Tangshan, 063210 China.

Journal of Inequalities and Applications
|October 27, 2017
PubMed
Summary

This study presents a sharp Trudinger type inequality for harmonic functions using the Cauchy-Riesz kernel. Applications include Morrey representations for linear and set-valued maps in Banach spaces.

Keywords:
Cauchy-Riesz kernel functionMorrey representationTrudinger type inequalitymodified Poisson type kernel

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Area of Science:

  • Harmonic analysis
  • Functional analysis
  • Partial differential equations

Background:

  • Harmonic functions are solutions to Laplace's equation with significant applications in physics and mathematics.
  • Trudinger type inequalities are crucial in establishing embedding theorems and understanding function spaces.
  • The Cauchy-Riesz kernel and modified Poisson kernels are essential tools in potential theory and related fields.

Purpose of the Study:

  • To establish a sharp Trudinger type inequality for harmonic functions.
  • To explore the applications of this inequality in representing linear and set-valued maps.
  • To extend existing representation theorems to more general settings.

Main Methods:

  • Utilizing the Cauchy-Riesz kernel function and modified Poisson type kernels.
  • Applying techniques from harmonic analysis and the theory of partial differential equations.
  • Leveraging properties of harmonic functions in Banach spaces and convex sets.

Main Results:

  • A novel sharp Trudinger type inequality for harmonic functions is derived.
  • Morrey representations are obtained for continuous linear maps acting on harmonic functions within convex subsets of Banach spaces.
  • Representations are deduced for set-valued maps and scalar-valued maps of Dunford-Schwartz.

Conclusions:

  • The derived inequality provides a new sharp bound for harmonic functions.
  • The results offer valuable insights into the representation of various types of maps in functional analysis.
  • This work contributes to the understanding of harmonic functions and their associated representations in abstract spaces.