working on the exercises of section F. Constant-Coefficient Operators: Fundamental Solutions but I'm stuck in the following problem:
I'm not sure how to use Green's theorem to conclude the second equality since the classical version works in a domain bounded by picewise smooth curve, but the integration domain is basically the complement of the closed ball $B(0,\varepsilon)$.
Any suggestion would be appreciated.
Thanks for the help, this is a sketch of my attempt, any suggestion would be appreciated.
Using the hint, we know that $$ \langle LK,\phi\rangle = - \langle K,L\phi\rangle = -\frac{1}{2\pi i}\iint_{x^2+y^2>\varepsilon} \frac{\partial_x \phi + i \partial_y \phi}{x+iy}\, dx\,dy \quad \forall \phi\in C^\infty_c $$ Since $\phi$ has compact support $$ \iint_{x^2+y^2>\varepsilon} \frac{\partial_x \phi + i \partial_y \phi}{x+iy}\, dx\,dy = \iint_{A_\varepsilon} \frac{\partial_x \phi + i \partial_y \phi}{x+iy}\, dx\,dy $$ where $A_\varepsilon = B(0,R)\setminus B(0,\varepsilon)$. Using Green's theorem for two regions (see: Taking the line integral of a region with holes with Green's Theorem) and since $\text{supp}(\phi)\subseteq B(0,R)$
\begin{align} \iint_{A_\varepsilon} \frac{\partial_x \phi + i \partial_y \phi}{x+iy}\, dx\,dy & = \int_{\partial B(0,R)} \phi(x,y)\,\frac{dx+ idy}{x+iy} + \int_{\partial B(0,\varepsilon)} \phi(x,y)\,\frac{dx+ idy}{x+iy} \\ & = \int_{\partial B(0,\varepsilon)} \phi(x,y)\,\frac{dx+ idy}{x+iy} \end{align} By the integration over polar coordinates $$ \int_{\partial B(0,\varepsilon)} \phi(x,y)\,\frac{dx+ idy}{x+iy} = \int_0^{2\pi} \phi(\varepsilon \cos(\theta), \varepsilon \sin\theta)\, d\theta $$ Finally, let $\varepsilon \to 0$ and by dominated convergence $$ \langle LK,\phi\rangle = \phi(0,0) $$ with this, $K$ is a fundamental solution of $L$.