The Diederich–Fornaess index and the global regularity of the $$\bar{\partial }$$ ∂ ¯ -Neumann problem

Abstract

We describe along the guidelines of Kohn [11], the constant E s which is needed to control the commutator of a totally real vector field T E with∂ * in order to have H s a-priori estimates for the Bergman projection B k , k ≥ q − 1, on a smooth qpseudoconvex domain D ⊂⊂ C n . This statement, not explicit in [11], yields regularity results for B k in specific Sobolev degree s. Next, we refine the pseudodifferential calculus at the boundary in order to relate, for a defining function r of D, the operators (T + ) − δ 2 and (−r) δ 2 . We are thus able to extend to general degree k ≥ 0 of B k , the conclusion of [11] which only holds for q = 1 and k = 0: if for the Diederich-Fornaess index δ of D, we have (1 − δ) 1 2 ≤ E s , then B k is H s -regular. MSC: 32F10, 32F20, 32N15, 32T25

The structure of the fractional powers of the noncommutative Fourier law

“…Since |s| 2 ||u w || L 2 ≤ ||w|| L 2 , from the last inequality above, we get the second estimate in (21):…”

“…The pseudo-resolvent operator is the inverse of T 2 − 2Re(s)T + |s| 2 . We look for conditions on the coefficients a, b, c such that purely imaginary quaternions (s ∈ H with Re(s) = 0) are in the S-resolvent set of T. Since we will prove our results using the Lax-Milgram lemma, we have to define the sesquilinear form associated with  s (T) (the technique related to the Lax-Milgram lemma to solve a differential equation is extensively used in thē-Neumann problem and, in this field, the third author used it in previous studies [16][17][18][19][20][21][22][23] ). So we consider, for…”

“…The pseudo‐resolvent operator is the inverse of We look for conditions on the coefficients a , b , c such that purely imaginary quaternions ( with Re( s )=0) are in the S ‐resolvent set of T . Since we will prove our results using the Lax‐Milgram lemma, we have to define the sesquilinear form associated with (the technique related to the Lax‐Milgram lemma to solve a differential equation is extensively used in the ‐Neumann problem and, in this field, the third author used it in previous studies). So we consider, for u , , that is, where dμ = dxdydz .…”

The Diederich-Fornaess index and good vector fields

“…(see also [22] for an alternative condition). Herbig and Fornaess have shown that the Diederich-Fornaess Index is equal to 1 whenever Ω has a defining function that is plurisubharmonic on the boundary in [14] and [15] (their construction also implies (1.1); see Remark 6.3 in [16]), so this is a true generalization of Boas and Straube's original condition.…”

Hartogs domains and the Diederich–Fornæss index

“…Given the connection between Condition R and the Diederich-Fornaess Index studied in [23], [20], and [28], this result can be seen as a parallel result to Zeytuncu's Theorem 8 (and Remark 6) in [33]. Zeytuncu shows that on the Hartogs domain Ω g = {(z, w) ∈ C 2 : |w| < 1 and |z| < |g(w)|}, where g is a bounded holomorphic function on the unit disk, Condition R implies that Ω g admits a Stein neighborhood basis.…”