Maximal regularity theory and its application

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We develop the method to prove maximal regularity by proving R-bounded of an solution operator in view of operator-valued Fourier-multiplier theorem. As an application of the maximal regularity, we prove local solvability of free boundary problems for the Navier-Stokes equations with surface tension in a scale invariant Sobolev space. Moreover we prove maximal regularity of the Cauchy problem for the heat equation in homogeneous Besov space that is not a UMD (unconditional martingale differences) Banach space.