I have a question about singularities of conformal mappings.

Let $\mathbb{H} \subset \mathbb{C} \cong \mathbb{R}^2$ be the upper half-place and let $D$ be a Jordan domain. Let $\varphi:\mathbb{H} \to D$ denote a conformal map.

I am concerned with the following quantity: \begin{align*} I(z,r)=\int_{\mathbb{H} \cap B(z,r)}\log(|z-w|^{-1})|\varphi'(w)|^2\,dm(w),\quad z \in \bar{\mathbb{H}},\ r>0. \end{align*} Here $m$ denotes the two-dimensional Lebesgue measure and $B(z,r)$ denotes the open ball centered at $z$ with radius $r>0$. Of course, $I(z,r)$ is a variant of logarithmic potential. $I(z,r)$ roughly represents a singularity of $\varphi$ around the point $z$.

In fact, the quantity $I(z,r)$ naturally appears in the context of "random time-change" in probability theory. This controls local behaviors of the reflected Brownian motion in $D$ in some sense.

I am interested when $\lim_{r \to 0}I(z,r)=0$ uniformly in $z$ over each compact subset of $\bar{\mathbb{H}}$ (of course, when $\partial D$ is smooth, this question is not interesting).

Has such a thing been studied in the context of conformal mappings and logarithmic potential theory?

I have a question about singularities of conformal mappings.

Let $\mathbb{H} \subset \mathbb{C} \cong \mathbb{R}^2$ be the upper half-place and let $D$ be a Jordan domain. Let $\varphi:\mathbb{H} \to D$ denote a conformal map.

I am concerned with the following quantity: \begin{align*} I(z,r)=\int_{\mathbb{H} \cap B(z,r)}\log(|z-w|^{-1})|\varphi'(w)|^2\,dm(w),\quad z \in \bar{\mathbb{H}},\ r>0. \end{align*} Here $m$ denotes the two-dimensional Lebesgue measure and $B(z,r)$ denotes the open ball centered at $z$ with radius $r>0$. Of course, $I(z,r)$ is a variant of logarithmic potential. $I(z,r)$ roughly represents a singularity of $\varphi$ around the point $z$.

In fact, the quantity $I(z,r)$ naturally appears in the context of "random time-change" in probability theory. This controls local behaviors of the reflected Brownian motion in $D$ in some sense.

I am interested when $$\text{(A)}\quad \lim_{r \to 0}I(z,r)=0 \text{ uniformly in $z$ over each compact subset of $\bar{\mathbb{H}}$}$$ (of course, when $\partial D$ is smooth, this question is not interesting).

Has such a thing been studied in the context of conformal mappings and logarithmic potential theory?