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This paper is a continuation of [N. Ghoussoub and A. Moradifam, ‘On the best possible remaining term in the improved Hardy inequality’,  Proc. Natl. Acad. Sci. USA  105 (2008) 13746–13751] and [N. Ghoussoub and A. Moradifam, ‘Bessel pairs and optimal Hardy and Hardy–Rellich inequalities’,  Math. Ann.  349 (2011) 1–57], where the authors introduced a general approach for improved Hardy and Hardy–Rellich‐type inequalities. In this paper, we present equivalent conditions on a pair of positive radial functions  V  and  W  on a ball  B  in  R   n  ,  n ⩾1, and  b ∈ℝ so that the following inequalities hold∫   B    V  (  x  )   |   ∇  u    |    2    d  x  ⩾   ∫   B    W  (  x  )   u   2    d  x  +  b   ∫   ∂  Bu   2            d  s     f  o  r     a  l  l       u  ∈   H   1    (  B  )and∫   B    V  (  x  )   |   △  u    |    2    d  x  ⩾   ∫   B    W  (  x  )   |   ∇  u    |    2    d  x  +  b   ∫   ∂  B|   ∇  u    |    2            d  s     f  o  r     a  l  l       u  ∈   H   2    (  B  )Then we present various classes of optimal weighted Hardy–Rellich inequalities on  H  2  ∩  H  0  1 . The proofs are based on decomposition into spherical harmonics. This type of inequalities is important in the study of the systems of second‐order elliptic equations as well as fourth‐order elliptic equations with Navier boundary condition.

Optimal weighted Hardy–Rellich inequalities on H 2 ∩ H 0 1