Hardy–Sobolev inequalities for the biharmonic operator with remainder terms

We first review improvements of (first-order) Sobolev and Hardy inequalities by the addition of suitable lower-order terms (these improved inequalities have been pioneered by Brezis and Nirenberg (1983) and Brezis and Vázquez (1997)). Recently, corresponding results concerning first-order Hardy–Sobolev and higher-order Sobolev and Hardy inequalities have been proved.We then consider second-order inequalities of Hardy–Sobolev type, which are an interpolation between the second-order Sobolev and the second-order Hardy inequalities. We prove that the corresponding critical Hardy–Sobolev constants (Formula presented.) ((Formula presented.), where (Formula presented.) corresponds to the Sobolev case and (Formula presented.) to the Hardy case) do not depend on all traces of the space (Formula presented.); i.e., we prove that the critical Hardy– Sobolev constant with Navier conditions coincides with the constant with Dirichlet conditions. Moreover, under the same assumptions and with both Navier and Dirichlet boundary conditions, we derive lowerorder improvements of these second-order Hardy–Sobolev inequalities.