Sharp Weight Range for Critical Hardy–Rellich Inequalities

Abstract

We revisit a recently established family of critical Hardy–Rellich inequalities. Combining the Emden–Fowler transformation with the classical one-dimensional weighted Hardy inequality, we show that the weighted Hardy–Rellich inequality holds precisely when the weight exponent differs from the dimension, so that this single value is the unique critical one. We obtain explicit constants in every dimension greater than one, determine the sharp constant in the one-dimensional case, and extend the formulation involving the Laplacian to the full Muckenhoupt range. These results sharpen and unify the existing theory and answer in part an open problem raised in the recent literature.