A note on Marcinkiewicz integrals supported by submanifolds

Area of Science:

• Harmonic Analysis
• Real Analysis

Background:

• Parametric Marcinkiewicz integrals are crucial in harmonic analysis.
• Rough kernels and polynomial mappings present unique analytical challenges.
• Triebel-Lizorkin and Besov spaces are important function spaces for studying PDEs and Fourier analysis.

Purpose of the Study:

• To establish the boundedness and continuity of parametric Marcinkiewicz integrals with rough kernels.
• To analyze these integrals associated with polynomial mappings and compound submanifolds.
• To investigate their properties within Triebel-Lizorkin and Besov spaces.

Main Methods:

• Utilizing techniques from harmonic analysis.
• Employing methods for analyzing integrals with rough kernels.
• Leveraging the properties of polynomial mappings and submanifolds.
• Applying tools specific to Triebel-Lizorkin and Besov spaces.

Main Results:

• The boundedness and continuity of the parametric Marcinkiewicz integrals were established.
• The results hold for specific conditions on parameters $$ and $$.
• The study covers integrals associated with polynomial mappings and compound submanifolds.

Conclusions:

• The established results represent significant improvements and natural extensions of existing literature.
• This work contributes to a deeper understanding of Marcinkiewicz integrals in function spaces.
• The findings have potential implications for the study of partial differential equations and Fourier analysis.