A remark on the kernel of the CR Paneitz operator

Abstract

Abstract. For CR structures in dimension three, the CR pluriharmonic functions are characterized by the vanishing of a third order operator. This third order operator, after composition with the divergence operator, gives the fourth order analogue of the Paneitz operator. In this short note, we give criteria under which the kernel of the CR Paneitz operator contains a supplementary space to the CR pluriharmonic functions.

The CR Paneitz operator and the stability of CR pluriharmonic functions

“…For an elementary proof, see Lemma 2.2. Generic results about the existence of W may be found in [1]. We need one more definition before we can state our main result:…”

“…The Reeb vector field T is the unique vector field such that θ(T ) = 1 and dθ(T, •) = 0. Let Z 1 be a local frame of T 1,0 and consider the frame {T, Z 1 , Z1} of T M ⊗ C. Then θ, θ 1 , θ 1 , the coframe dual to {T, Z 1 , Z1}, satisfies (1.2) dθ = ih 1 1θ 1 ∧ θ 1 for some positive function h 1 1. We can always choose Z 1 such that h 1 1 = 1; hence, throughout this paper, we assume h 1 1 = 1…”

Logarithmic Hardy-Littlewood-Sobolev Inequality on Pseudo-Einstein 3-manifolds and the Logarithmic Robin Mass

“…The equality in (7), follows then from an easy computation starting from the LHLS inequality in H, proved in [4,15] and stated in the theorem below.…”

“…Hence, ker P θ is infinite dimensional. For a thorough study of the analytical properties of P θ and its kernel, we refer the reader to [19,7,8]. The main property of the Paneitz operator P θ is that it is CR covariant [17].…”

The Mathematical Work of S.-Y. Alice Chang and Paul C. Yang

“…In subsequent work of Paul with J. Case and Chanillo [7] they studied conditions under which positivity of the CR Paneitz operator holds, and used these conditions to give concrete examples (for example, real ellipsoids in C 2 ).…”