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

Abstract

We consider the series expansion of the [Formula: see text]-Hardy inequality of [G. Barbatis, S. Filippas and A. Tertikas, Series expansion for [Formula: see text] 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 [Formula: see text] and [Formula: see text]. For [Formula: see text] we improve it by adding as a remainder term an optimally weighted critical Sobolev norm, generalizing the [Formula: see text] = 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 [Formula: see text] Hardy inequalities with best constants, Trans. Amer. Math. Soc. 356 (2004) 2169–2196]. For [Formula: see text] 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.