On the LpHelmholtz decomposition: A review of a result due to Solonnikov*

We consider the Helmholtz decomposition of the Lebesgue space Lp(Ω). We essentially reproduce a proof given by Solonnikov in [V.A. Solonnikov, Estimates of the solutions of the nonstationary Navier–Stokes system, Boundary Value Problems of Mathematical Physics and Related Questions in the Theory of Functions. Part 7, Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova, Vol. 38, Nauka, Leningrad, 1973, pp. 153–231 (in Russian)] and [V.A. Solonnikov, Estimates for solutions of nonstationary Navier–Stokes equations, J. Sov. Math., 8(4):467–529, 1977]