Every non-constant holomorphic function in C^n generates a counterexample to L^1 regularity of the Poisson equation via sharp level-set estimates from resolution of singularities and the Lojasiewicz inequality.
Harvey and J.Polking,Removable singularities of solutions of linear partial differential equations, Acta Math.125 (1970) 39-56
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Energy estimates for level sets of holomorphic functions and universal counterexamples to Calder\'on-Zygmund theory
Every non-constant holomorphic function in C^n generates a counterexample to L^1 regularity of the Poisson equation via sharp level-set estimates from resolution of singularities and the Lojasiewicz inequality.