Littlewood-Paley operators on Morrey spaces with variable exponent.

Area of Science:

• Harmonic Analysis
• Functional Analysis
• Real Analysis

Background:

• Littlewood-Paley theory is fundamental in harmonic analysis, analyzing functions via their derivatives.
• Lebesgue and Morrey spaces with variable exponents are crucial for studying function spaces with non-uniform smoothness.
• Commutators of operators with BMO (Bounded Mean Oscillation) functions probe regularity properties.

Purpose of the Study:

• To investigate the boundedness of Littlewood-Paley operators and their commutators.
• To extend these results to the setting of Morrey spaces with variable exponents.
• To utilize vector-valued inequalities for achieving these extensions.

Main Methods:

• Application of vector-valued inequalities for Littlewood-Paley operators.
• Analysis on Lebesgue spaces with variable exponents.
• Extension of boundedness results to variable exponent Morrey spaces.

Main Results:

• Established the boundedness of key Littlewood-Paley operators (Lusin area integrals, g-functions, g μ *-functions) on variable exponent Morrey spaces.
• Demonstrated the boundedness of their commutators generated by BMO functions on these spaces.
• Provided a comprehensive analysis of operator behavior in these generalized function spaces.

Conclusions:

• The boundedness of Littlewood-Paley operators and their commutators is confirmed on variable exponent Morrey spaces.
• This work extends existing theories to more complex function spaces.
• The results contribute to a deeper understanding of harmonic analysis in variable exponent settings.