This is a lemma for: Helffer-Sjöstrand
Given the complex plane.
Consider a smooth function: $$f_E\in\mathcal{C}^\infty_c(\mathbb{R}^2):\quad\bar{\partial}f_E\restriction_\mathbb{R}=0$$
How to apply Stone's theorem here? $$\int_{|\Im z|\geq\varepsilon}\frac{\bar{\partial}f_E(z)}{z-\lambda}\mathrm{d}z\wedge\mathrm{d}\bar{z}=\int_{-\infty}^\infty\left(\frac{f_E(x-i\varepsilon)}{x-i\varepsilon-\lambda}-\frac{f_E(x+i\varepsilon)}{x+i\varepsilon-\lambda}\right)\mathrm{d}x$$ (I'm not so familiar with complex manifolds yet.)