‎Difference of weighted composition operator from Cauchy transform space into Dirichlet ‎space

‎‎‎‎‎Let $H(\mathbb{D})$ be the space of all analytic functions on $\mathbb{D}$‎, ‎$u,v\in H(\mathbb{D})$ and $\varphi,\psi$ be self-map $(\varphi,\psi:\mathbb{D}\rightarrow \mathbb{D})$‎. Difference of weighted composition operator is denoted by $uC_\varphi‎ -‎vC_\psi$ and defined as follows‎ ‎\begin{align*}‎ (uC_\varphi‎ -‎vC_\psi)f(z) = u(z) f{(\varphi(z))}‎- ‎v(z) f(\psi(z))‎ ,‎\quad f\in H(\mathbb{D} )‎, ‎\quad z\in \mathbb{D}‎. ‎\end{align*}‎ ‎In this paper‎, ‎boundedness of difference of weighted composition operator from Cauchy transform into Dirichlet space will be considered and ‎an ‎equivalence condition for boundedness of such operator will be given‎.‎‎ Then the norm of composition operator between the mentioned spaces will be studied and it will be shown ‎that‎‎ $\|C_\varphi\|\geq 1$ ‎and‎ there is no composition isometry from Cauchy transform into Dirichlet ‎space‎.‎