Hypoellipticity of the $\overline{\partial}$-Neumann problem at a point of infinite type

Abstract

We prove local hypoellipticity of the complex Laplacian in a domain which has superlogarithmic estimates outside a curve transversal to the CR directions and for which the holomorphic tangential derivatives of a defining function are superlogarithmic multipliers in the sense of [10].

Regularity of the $\bar\partial$-Neumann problem by means of superlogarithmic multipliers

“…In [16] Kohn provided an example of a domain in which not even superlogarithmic estimates hold, but we have local hypoellipticity for the ∂-Neumann problem. Kohn proved in fact this result for the tangential problem, but it was generalized to the ∂-Neumann by Baracco, Khanh and Zampieri in [2]. Moreover, as we have already stated in the introduction, Baracco, Pinton and Zampieri in [3] gave a fully geometrical explanation of the phenomenon, relating local hypoellipticity to the presence of a sequence of cut-off {η} such that the gradient ∂η and the Levi form ∂ ∂η are are subelliptic multipliers.…”

“…In [16] Kohn showed that if the estimates fail but the points of failure are confined to a real curve transversal to the CR directions, local regularity for the tangential problem still holds (cf. [16] and [2]). This is an exquisitely geometric conclusion.…”

“…Note that Kohn's work in [16] only deals with the tangential problem, but there is some literature for transferring regularity of b to regularity of (cf. [17], [2], [14]). This consists in exploiting the decomposition Q = Q τ ⊕ Ln , and requires the heavy technicalities of the harmonic extension.…”

Local regularity of the Green operator in a CR manifold of general "type"

“…Under the assumption that [∂ b , η] and [∂ b , [ ∂b , η]] are superlogarithmic multipliers in a sense inspired to Kohn, we get the local regularity of the Green operator G = −1 b . In particular, if M has "infraexponential type" along S \ Γ where S is a manifold of CR dimension 0 and Γ a curve transversal to T C M , then we have local regularity of G. This gives an immediate proof of [1] in tangential version and of [14]. The conclusion extends to "block decomposed" domains for whose blocks the above hypotheses hold separately.…”

The Kohn-Hörmander-Morrey formula twisted by a pseudodifferential operator

“…For a gain such a subelliptic [13] or superlogarithmic [12] the regularity is stronger, that is, local: u is regular precisely in the portion of bΩ where u is. When these latter gains do not occur, but the points of failure are confined to a real curve transversal to the CR directions, local regularity still holds [11] and [1]. This is an exquisitely geometric conclusion.…”