Work

A generalized radial integration by parts formula and its applications to Caffarelli–Kohn–Nirenberg inequalities

Year: 2025

Type: article

Abstract: This paper builds upon the Caffarelli–Kohn–Nirenberg (CKN) weighted interpolation inequalities, which are fundamental tools in partial differential equations and geometric analysis for establishing relationships between functions and their gradients when power weights are involved. Our work broadens the scope of these inequalities by generalizing them to encompass a broader class of radial weights and exponents. Additionally, we extend the application of these inequalities to the class $$C^{\infty } ( {\overline{\Omega }})$$ C ∞ ( Ω ¯ ) of smooth functions defined on bounded domains with Lipschitz boundaries. To achieve this generalization, we formulate a new integration by parts formula that accounts for more general weights, a wider range of exponents, and $$C^{\infty }({\overline{\Omega }})$$ C ∞ ( Ω ¯ ) functions. The resulting generalized CKN-type inequalities offer explicit upper bounds on the optimal constants, independent of the domain’s geometry, consistent with the scaling invariant nature of the inequalities.

Source: Analysis and Mathematical Physics

DOI: 10.1007/s13324-025-01060-y

Authors: Giovanni Di Fratta, Alberto Fiorenza

Institutions: University of Naples Federico II, Istituto per le Applicazioni del Calcolo Mauro Picone, National Research Council

Language: English

Topic: Nonlinear Partial Differential Equations

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,290