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Hölder norm estimate for the fractal Hilbert transform in Douglis analysis.

Yudier Peña Pérez1, Martín P Árciga Alejandre1, Ricardo Abreu Blaya2

  • 1Facultad de Matemática, Universidad Autónoma de Guerrero, Avenida Lázaro Cárdenas sn, Chilpancingo, Guerrero 39087 México.

Journal of Inequalities and Applications
|September 29, 2017
PubMed
Summary

This study estimates the Hölder norm for a fractal Hilbert transform within Douglis analysis. The research focuses on functions defined on h-summable curves, advancing fractal analysis and operator theory.

Keywords:
Hilbert transformHölder functionsWhitney decompositionfractal dimensionsh-summables curveshyperanalytic functions

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Area of Science:

  • Harmonic Analysis
  • Fractal Geometry
  • Functional Analysis

Background:

  • The Hilbert transform is a fundamental operator in harmonic analysis.
  • Fractal analysis extends classical concepts to irregular geometric objects.
  • Douglis analysis provides a framework for studying operators on specific function spaces.

Purpose of the Study:

  • To estimate the Hölder norm of a fractal Hilbert transform.
  • To analyze the transform's action on Hölder spaces of functions with values in a Douglis algebra.
  • To investigate the behavior of this transform on h-summable curves.

Main Methods:

  • Utilizing techniques from harmonic analysis and fractal geometry.
  • Applying concepts from Douglis analysis.
  • Estimating norms of operators acting on function spaces.

Main Results:

  • Successful estimation of the Hölder norm for the fractal Hilbert transform.
  • Characterization of the transform's boundedness properties in the specified context.

Conclusions:

  • The study provides a rigorous analysis of a fractal version of the Hilbert transform.
  • Findings contribute to the understanding of operator theory in fractal settings.
  • The results have implications for the analysis of functions on complex, irregular domains.