Removable singularities of CR functions on singular boundaries

Holomorphic extension of CR functions from quadratic cones

“…However, if we assume further conditions regarding the growth of f near M sng , we may conclude that f is CR on M . The following result is also proved in [16]. Let M = {ρ(z) = 0} be a real analytic hypersurface, and suppose that σ = M ∩ {dρ = 0} has 2n − 1 measure zero.…”

“…It is important to note that according to this definition, a function f in L 1 loc (M ), which is CR on M reg , is not necessarily CR on M , even when M sng is a single point, see e.g., [16,Example 2.2]. In other words, the singularity of M is not in general CR-removable for CR functions on M reg .…”

“…Then by Lemma 4.1(b) there exists g ∈ SL(2, R) such that g t P g is the matrix of the form ± √ − det P diag(1, −1). Since det Q = det P − det S ≤ 0, and thanks to (16) and (17) above, it follows that the sum of the entries of g t P g cannot be zero, so by Lemma 4.2 there exists a matrix k ∈ SO(1, 1) such that for Q ′ = k t (g t Qg)k = (q ′ ij ), where either q ′ 11 or q ′ 22 vanishes. Since the subgroup SO(1, 1) ⊂ SL(2, R) preserves P ′ = g t P g, it follows from (17) that either q ′ 11 √ − det P = 0, or q ′…”

Holomorphic extension of CR functions from quadratic cones

“…In other words, the singularity of M is not in general CR-removable for CR functions on M reg . However, if we assume further conditions regarding the growth of f near M sng , we may conclude that f is CR on M. The following result is also proved in [14]. Let M = {ρ(z) = 0} be a real analytic hypersurface, and suppose that σ = M ∩ {dρ = 0} has 2n − 1 measure zero.…”

“…It is important to note that according to this definition, a function f in L 1 loc (M), which is CR on M reg , is not necessarily CR on M, even when M sng is a single point, see, e.g., [14,Example 2.2]. In other words, the singularity of M is not in general CR-removable for CR functions on M reg .…”