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This is an improvement of the so-called Matsumura estimates. Moreover, as its application, we consider the nonlinear problem with initial data in $(H^s\\cap H_r^{\\beta}) \\times (H^{s-1} \\cap L^r)$ with $r \\in (1,2]$, $s\\ge 0$, and $\\beta = (n-1)|\\frac{1}{2}-\\frac{1}{r}|$, and prove the local and global existence of solutions. 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