We consider the series expansion of the $L^p$-Hardy inequality of [G. Barbatis, S. Filippas and A. Tertikas, Series expansion for $L^p$ Hardy inequalities, Indiana Univ. Math. J. 52 (2003) 171–190], in the particular case where the distance is taken from an interior point of a bounded domain in ${ℝ}^n$ and $1<p\ne n$. For $p<n$ we improve it by adding as a remainder term an optimally weighted critical Sobolev norm, generalizing the $p$ = 2 result of [S. Filippas and A. Tertikas, Optimizing improved Hardy inequalities, J. Funct. Anal. 192 (2002) 186–233] and settling the open question raised in [G. Barbatis, S. Filippas and A. Tertikas, A unified approach to improved $L^p$ Hardy inequalities with best constants, Trans. Amer. Math. Soc. 356 (2004) 2169–2196]. For $p>n$ we improve it by adding as a remainder term the optimally weighted Hölder seminorm, extending the Hardy–Morrey inequality of [G. Psaradakis, An optimal Hardy–Morrey inequality, Calc. Var. Partial Differential Equations 45 (2012) 421–441] to the series case.

我们考虑级数展开${\mathrm{大号}}^p$- [G. 的 Hardy 不等式。Barbatis、S. Filippas 和 A. Tertikas，系列扩展${\mathrm{大号}}^p$哈代不平等，印第安纳大学。数学。J.  52 (2003) 171–190]，在特定情况下，距离取自有界域的内部点${ℝ}^n$和$1<p\ne n$. 为了$p<n$我们通过添加一个最优加权的关键 Sobolev 范数作为余项来改进它，推广$p$= 2 [S. Filippas 和 A. Tertikas，优化改进的 Hardy 不等式，J. Funct。肛门。 192 (2002) 186–233] 并解决 [G. Barbatis、S. Filippas 和 A. Tertikas，改进的统一方法${\mathrm{大号}}^p$具有最佳常数的 Hardy 不等式，Trans。阿米尔。数学。社会党。 356 (2004) 2169–2196]。为了$p>n$我们通过添加最优加权 Hölder 半范式作为余项来改进它，扩展了 [G. Psaradakis，最优的 Hardy-Morrey 不等式，Calc。变量。偏微分方程 45 (2012) 421–441] 到系列案例。