Sobolev spaces arising from the Stockwell transform on Gelfand pairs

Abstract

We introduce a family of Sobolev-type spaces on Gelfand pairs, built from the Stockwell transform: a function belongs to the space of a given smoothness when its Stockwell transform, multiplied by a weight that grows with the smoothness, is square integrable. We prove that these spaces are Hilbert spaces, that they form a decreasing scale under continuous embeddings, and that an interpolation inequality controls the norm at an intermediate smoothness by the norms at the two endpoints. Under integrability conditions on the weight and on the family of analysing windows, they embed continuously into Lebesgue spaces. Under a local domination condition on the window, we show that their elements have bounded continuous representatives. On the Euclidean motion group, where the Stockwell transform reduces to a windowed Hankel transform, this gives boundedness and continuity as soon as the smoothness exceeds half the dimension, which is the classical threshold for Euclidean spaces.