Optimal non-homogeneous improvements for the series expansion of Hardy’s inequality

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