Optimal Functional Inequalities for Fractional Operators on the Sphere and Applications

Area of Science:

• Analysis
• Geometric Analysis
• Partial Differential Equations

Background:

• Functional inequalities are crucial in analysis and have applications in various scientific fields.
• Fractional calculus extends classical calculus to non-integer orders, leading to new mathematical tools and problems.
• The n-dimensional sphere is a fundamental geometric object with rich mathematical properties.

Purpose of the Study:

• To establish a family of optimal functional inequalities on the n-dimensional sphere involving the fractional Laplace operator.
• To determine the optimal constants for these inequalities using spectral properties.
• To explore the implications of these inequalities for fractional heat flows and related inequalities.

Main Methods:

• Utilizing spectral properties of fractional operators on the n-dimensional sphere.
• Applying techniques from the theory of functional inequalities.
• Employing the stereographic projection to extend results to Euclidean space.

Main Results:

• Optimal functional inequalities of the form are established on $$ with optimal constants.
• These inequalities interpolate between fractional Sobolev, logarithmic Sobolev, and Poincaré inequalities for different ranges of the order $$.
• Remainder terms are provided for the subcritical range $$, yielding improved inequalities.
• Weighted inequalities involving the fractional Laplacian are derived in Euclidean space.

Conclusions:

• The established inequalities provide a comprehensive framework for understanding fractional calculus on spheres.
• The results offer new insights into the behavior of fractional heat flows.
• The methods and findings have potential applications in various areas of mathematics and physics.