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

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)(SR) 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)(SR) space.