Geometry of Sobolev Spaces on Regular Trees and the Hardy Inequalities

Regular metric trees (see Definition 2.1) possess a rich group of symmetries. As a consequence, the Sobolev space H¹ (Γ) on such a tree Γ admits a direct decomposition that is simultaneously orthogonal with respect to many inner products, including those defined by (latin small letter esh)_Γp uv̄ dx and (latin small letter esh)_Γ u′v′̄ dx. This decomposition was discovered by the authors in [5]; here we describe the construction in detail. Applications to spectral analysis of Schrödinger operators and to the Hardy inequalities on trees are given.