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 $AT_q^\infty(ω)$.