Embedding theorems and integration operators on Hardy--Carleson type tent spaces induced by doubling weights

This paper develops the function and operator theory of Hardy–Carleson–type analytic tent spaces $AT_q^\infty(ω)$ induced by radial weights $ω$ satisfying a two-sided doubling condition. We first characterize the positive Borel measures $μ$ for which the embedding from $AT_p^\infty(ω)$ into the tent space $T_q^\infty(μ)$ is bounded for all $0 < p, q < \infty$. A Littlewood–Paley formula for $AT_q^\infty(ω)$ is then established. Using these results, we give a complete characterization of the boundedness (compactness) of Volterra-type integration operators between $AT_p^\infty(ω)$ and $AT_q^\infty(ω)$.

💡 Research Summary

This paper develops a comprehensive function‑and‑operator theory for analytic Hardy–Carleson type tent spaces denoted by (AT^\infty_q(\omega)). The underlying weight (\omega) is assumed to be radial and to satisfy a two‑sided doubling condition; in the terminology of the authors this means (\omega) belongs to the class (\mathcal D=\widehat{\mathcal D}\cap\widetilde{\mathcal D}). The main contributions can be grouped into three themes: (i) sharp embedding theorems for the identity map from (AT^\infty_p(\omega)) into weighted tent spaces (T^\infty_q(\mu)); (ii) a Littlewood–Paley type equivalence that characterises the norm of (AT^\infty_q(\omega)) via higher order derivatives; and (iii) a complete description of the boundedness and compactness of Volterra‑type integration operators (J_g) acting between these spaces.

1. Embedding theorems.
For a positive Borel measure (\mu) on the unit disc (\mathbb D) and a fixed radius (r\in(0,\tfrac14)) the authors introduce the auxiliary function
\