Carleson Measures for Slice Regular Hardy and Bergman Spaces in Quaternions

Abstract

We study the quaternionic Carleson measure, which provides an embedding of the slice regular Hardy space \({\mathcal {H}}^p({\mathbb {B}})\) into \(L^s({\mathbb {B}}, \text {d}\mu )\) with \(s>p.\) A new criterion is needed for a finite positive Borel measure to be an \(({\mathcal {H}}^p({\mathbb {B}}),s)\)-Carleson measure, given by the uniform integrability of slice Cauchy kernels. It turns out that the symmetric box and the symmetric pseudo-hyperbolic disc are equivalent in the characterization of \(({\mathcal {H}}^p({\mathbb {B}}),s)\)-Carleson measures, while they are not when \(s=p.\) We further study the s-Carleson measure for slice regular Bergman spaces \({{\mathcal {A}}}^p({\mathbb {B}})\) for all indices s, p. When \(s\ge p,\) our characterization relies on a close relation between the Carleson measure for Hardy and Bergman spaces and is primarily based on the slice Cauchy kernel, rather than the slice Bergman kernel. The advantage of the slice Cauchy kernel over the slice Bergman kernel is that, when restricted to any slice plane, the former, as a sum of two terms, transforms into a fractional linear transform, whereas the latter does not. This enables a locally uniform lower bound estimate for the slice Cauchy kernel, which is crucial in applications. In the case where \(s<p,\) we need to apply Khinchine’s inequality and a point-wise estimate for atoms in slice Bergman spaces based on the convex combination identity.