A Full Characterization of the Dirichlet Carleson embedding $\operatorname{id}: \mathcal D_{p-1}^p \to L^p(μ)$ for $p>2$

In this paper, we obtain a full characterization of the finite positive Borel measures $μ$ on $\mathbb D$ for which the embedding $$ \operatorname{id}:\mathcal D_{p-1}^p\longrightarrow L^p(μ),\qquad p>2, $$ is bounded. More precisely, for any dyadic system $\mathcal D$ on $\mathbb T$, we prove that