Weighted Discrete Hardy Inequalities on Trees and Applications

• Published in: Potential analysis Vol. 59; no. 2; pp. 675 - 703
• Main Authors: López-García, Fernando, Ojea, Ignacio
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.08.2023; Springer Nature B.V
• Subjects: Atomic properties; Divergence; Functional Analysis; Geometry; Inequalities; Mathematics; Mathematics and Statistics; Potential Theory; Probability Theory and Stochastic Processes; Sobolev space; Trees; Poincaré-type inequalities; Secondary: 35A23; Primary: 26D10; Hölder; domains; Decomposition of functions; 46E35; Weights; Divergence equation; Korn’s inequality; Discrete Hardy inequality; Distance
• ISSN: 0926-2601; 1572-929X
• DOI: 10.1007/s11118-021-09982-5

In this paper, we study certain inequalities and a related result for weighted Sobolev spaces on Hölder- α domains, where the weights are powers of the distance to the boundary. We obtain results regarding the divergence equation’s solvability, and the improved Poincaré, the fractional Poincaré, and the Korn inequalities. The proofs are based on a local-to-global argument that involves a kind of atomic decomposition of functions and the validity of a weighted discrete Hardy-type inequality on trees. The novelty of our approach lies in the use of this weighted discrete Hardy inequality and a sufficient condition that allows us to study the weights of our interest. As a consequence, the assumptions on the weight exponents that appear in our results are weaker than those in the literature.