\Delta_{\nu,\mu} f (x)\nu(\{ x^+ \} ) (1 - \mathds{1}_{\{x^+\}}(x) )
    = \lim_{ \substack{ y \uparrow x^-}} \dfrac{f(x)-f(y)}{F_{\mu}(x)-F_{\mu}(y)}   +  \Delta_{\nu,\mu} f (x)\nu(\{ x^- \} ).
    \end{aligned}$$ Here, $x^+ \coloneqq \inf\{ y \in [0,1] \colon F_{\mu}(x) < F_{\mu}(y) \}$ and $x^- \coloneqq \sup\{ y \in [0,1] \colon F_{\mu}(x)>F_{\mu}(y) \}$. Observe, if $\nu$ is atomless, then $\nabla_{\mu}(f)$ is independent of $\nu$ and given by an ‘ordinary’ differential quotient.

For $ \gamma =(\alpha,\beta) \in [0,{\pi/2}]^{2}$ we consider the following eigenvalue problem for $\Delta_{\nu,\mu}$, see [@Fu87], $$\begin{aligned}
\label{EWC}
    \Delta_{\nu,\mu}f= \lambda f  
    \end{aligned}$$ with [*Robin boundary conditions*]{} $$\begin{aligned}
\label{BC}
    f(0)\cos(\alpha)-\nabla_{\mu}f(0)\sin(\alpha)=0
    \quad \text{and} \quad
    f(1)\cos(\beta)+\nabla_{\mu}f(1)\sin(\beta)=0.
    \end{aligned}$$ We refer to the particular case $\gamma=(\pi/2,\pi/2)$ as the [*Neumann case*]{} and the case $\gamma=(0,0)$ as the [*Dirichlet case*]{}. We denote by $\mathcal{D}^{*}_{\gamma}( \Delta_{\nu,\mu} )$ the set of all $f \in \mathcal{D}^{*}( \Delta_{\nu,\mu} )$ which satisfy and . Combining and with Fubini’s theorem and the assumptions on $\nu,\mu$ one obtains a [*Gauss-Green formula*]{}; namely, for $f,g \in \mathcal{D}_{\gamma}^s( \Delta_{\nu,\mu} )$, $$\begin{aligned}
    \int (\Delta_{\nu,\mu}f )g\mathrm{d} \nu
        &= \left(
        \nabla_{\mu}f (1)-\nabla_{\mu}f (0)\right)g(0)+ \int\nabla_{\mu}g(y)\left(
\nabla_{\mu}f (1)+\Delta_{\nu,\mu}f (y)\nu \left(\{y\} \right)-\nabla_{\mu}f (y)        
         \right)
{\mathrm{d}}\mu(y)
    \\
    &= \nabla_{\mu}f (1) g(1) - \nabla_{\mu}f (0) g(0) - \int \nabla_{\mu}f \nabla_{\mu} g \;\mathrm{d} \mu\end{aligned}$$ where $\int  \nabla_{\nu,\mu}g(y)\Delta_{\nu,\mu}f (y)\nu \left(\{y\} \right) 