On Opial-type inequalities in two varaiables.
Necessary and sufficient condition on the weights will be derived under which a -th order Hardy inequality holds on classes of functions satisfying more than “boundary” conditions.
Pointwise interpolation inequalities, in particular, ku(x)c(Mu(x)) 1-k/m (Mmu(x))k/m, k<m, and |Izf(x)|c (MIf(x))Re z/Re (Mf(x))1-Re z/Re , 0<Re z<Re<n, where is the gradient of order , is the Hardy-Littlewood maximal operator, and is the Riesz potential of order , are proved. Applications to the theory of multipliers in pairs of Sobolev spaces are given. In particular, the maximal algebra in the multiplier space is described.