In the previous note, I showed a Rayleigh-type quotient for the conformal Killing operator on manifolds with boundary , i.e. the following result holds:

Whenever admits no non-zero conformal Killing vector fields, the following holds

where the infimum is taken over all smooth vector fields on with .

Today, I am going to prove a slightly stronger version of the above inequality, namely, when some terms on the boundary take part in. Precisely, we shall prove

Whenever admits no non-zero conformal Killing vector fields, the following holds

where the infimum is taken over all smooth vector fields on with .

However, a proof for this new inequality remains the same. To do so, we first make use of some Sobolev embeddings as follows: