Some weighted inequalities for Hausdorff operators and commutators

Area of Science:

• Harmonic Analysis
• Functional Analysis
• Operator Theory

Background:

• The Hausdorff operator is a fundamental object in harmonic analysis, with its boundedness properties extensively studied on various function spaces.
• Weighted central Morrey spaces and weighted central BMO (Bounded Mean Oscillation) spaces are crucial settings for analyzing the behavior of operators, particularly concerning regularity and integrability conditions.

Purpose of the Study:

• To determine the boundedness of the Hausdorff operator on power-weighted central Morrey spaces.
• To establish sharp bounds for the Hausdorff operator in these specific weighted spaces.
• To investigate the boundedness of commutators of the Hausdorff operator with symbol functions in weighted central BMO spaces.

Main Methods:

• Utilizing techniques from harmonic analysis, including pointwise estimates and integral inequalities.
• Applying properties of weighted function spaces, such as Morrey and BMO spaces, to analyze operator norms.
• Developing specialized methods to handle the power weights and the commutator structure.

Main Results:

• Sharp boundedness criteria for the Hausdorff operator on power-weighted central Morrey spaces are established.
• The study provides precise upper bounds for the Hausdorff operator in this context.
• Analogous sharp results are obtained for the commutators of the Hausdorff operator when symbol functions are in weighted central BMO spaces.

Conclusions:

• The findings contribute to a deeper understanding of the spectral properties of Hausdorff operators in weighted function spaces.
• The established sharp bounds offer valuable insights into the regularity and behavior of these operators and their commutators.
• This research extends existing theories on operator boundedness to more generalized weighted function spaces.