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:

First, we denote . In view of the Sobolev embedding and Sobolev trace embedding , we obtain the following inequalities

and

for some positive constants and independent of . Therefore,

By simply rewriting as

it suffices to prove that

We are now able to repeat all arguments in the previous note to conclude the result. Apparently, one can easly verify that

I take this chance to explain that all arguments for proving work and the fact holds true since the numerator involves which is of fourth order. Thefore, one can expect that the inequalities are actually stronger than usual Sobolev inequalities. One main difficulty is how to connect these fourth order operators acting on , , and .

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