p-Carleson Measures in the Quaternionic Unit Ball with Applications to Slice Campanato and $Q_p$ Spaces

Abstract

The p-Carleson measure in the unit ball of quaternions is introduced in terms of the symmetric box. When $p=1$ or $p=2$, the p-Carleson measure becomes the Carleson measure for the Hardy or Bergman spaces, respectively. A criterion for a measure to be a p-Carleson measure is provided in terms of slice Cauchy kernels. Bergman type integral operators are shown to preserve the p-Carleson measure in some sense. As applications, we provide a global characterization of the slice Campanato space and the slice $Q_p^\mathcal{S}\mathcal{R}$ space. We also establish an isomorphism between these spaces via fractional order derivatives and introduce slice Jones estimates, which measure distances between functions from the slice Bloch space to the slice $Q_p^\mathcal{S}\mathcal{R}$ space.