Infinitely supported harmonically weighted Dirichlet spaces which are de Branges Rovnyak spaces

Harmonically weighted Dirichlet spaces $\mathcal{D}_\mu$ and de Branges_Rovnyak spaces $\mathcal{H}(b)$ are two fundamental structures in analytic function theory exhibiting rich and often complementary properties. The question of when these spaces coincide, first raised and solved in Sarason's groundbreaking work in 1997 when $\mu$ is a single Dirac mass, is thus of fundamental importance in operator theory and analytic function spaces. In this paper, we focus on spaces $\mathcal{H}(b)$ with symbol $b = (1+u)/2$, where $u$ is a one-component inner function. While previous results extended Sarason's work to finitely supported measures $\mu$, the symbols we consider here give a natural framework to go beyond finiteness of the support. In our setting, we provide a complete characterization of measures $\mu$ for which $\mathcal{H}(b) = \mathcal{D}_\mu$, thereby resolving the long-standing open problem of constructing harmonically weighted Dirichlet spaces $\mathcal{D}_\mu$ associated with measures $\mu$ of infinite support that are also $\mathcal{H}(b)$ spaces. As a central ingredient to prove this result and which is of independent interest, we establish a $T(1)$-type result for the Cauchy transform on $L^2(\sigma)$, where $\sigma$ denotes the Clark measure associated with a one-component inner function $u$. Another notable result is a perturbation theorem for one-component inner functions that allows us to present a large class of function spaces satisfying $\mathcal{H}(b)=\mathcal{D}_\mu$. Furthermore, we settle the Brown--Shields conjecture within this setting.