Extensions of the theory of tent spaces and applications to boundary value problems

Abstract

We extend the theory of tent spaces from Euclidean spaces to various types of metric measure spaces. For doubling spaces we show that the usual ‘global’ theory remains valid, and for ‘non-uniformly locally doubling’ spaces (including R with the Gaussian measure) we establish a satisfactory local theory. In the doubling context we show that Hardy–Littlewood–Sobolev-type embeddings hold in the scale of weighted tent spaces, and in the special case of unbounded ADregular metric measure spaces we identify the real interpolants (the ‘Z-spaces’) of weighted tent spaces. Weighted tent spaces and Z-spaces on R are used to construct Hardy–Sobolev and Besov spaces adapted to perturbed Dirac operators. These spaces play a key role in the classification of solutions to first-order Cauchy–Riemann systems (or equivalently, the classification of conormal gradients of solutions to second-order elliptic systems) within weighted tent spaces and Z-spaces. We establish this classification, and as a corollary we obtain a useful characterisation of well-posedness of Regularity and Neumann problems for second-order complex-coefficient elliptic systems with boundary data in Hardy–Sobolev and Besov spaces of order s ∈ (−1, 0).