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Let assume that $u$ is smooth enough and $ -\Delta (u \phi) \in L^1(\Omega)$ for any $\phi \in C_c^{\infty}(\Omega)$. Then it easily follows that $ -\Delta u \in L^1_{\mathrm{loc}}(\Omega)$ by ...

Suppose we have a pseudo differential operator of principal type with a complex symbol and such that the poisson bracket of the real and imaginary parts on the characteristic set is non-negative.I ...