Tejinder S. Neelon

On solutions of real analytic equations

Analyticity of C∞ solutions yi = fi(x), 1 ≤ i ≤ m, of systems of real analytic equations pj(x, y) = 0, 1 ≤ j ≤ l, is studied. Sufficient conditions for C∞ and power series solutions to be real analytic are given in terms of iterative Jacobian ideals of the analytic ideal generated by p1, p2, . . . , pl. In a special case when the pi’s are independent of x, we prove that if a C∞ solution h satisfies the condition det ( ∂pi pyj ) (h(x)) 6≡ 0, then h is necessarily

A correction to “Ultradifferentiable functions on lines in ℝⁿ"

The proof of Theorem 1 in Proc. Amer. Math. Soc. 127 (1999), no. 7, 2099–2104, is revised. The proof of Theorem 1 in [1] uses Lemma 3(ii) which turns out to be valid only for the case n = 2. In this note the proof of Theorem 1 is revised by replacing Lemma 3(ii) by the proposition stated below. For x ∈ R, 0 6= ξ ∈ Sn−1, let Hxξ denote the hyperplane in R that passes through x with ξ as its normal vector. If φ : Rn−1 → R is a linear map such that Hxξ = φ(Rn−1), then the restriction of a function u ∈ C∞(Rn) to Hxξ is defined by uxξ(t) = u(x+ φ(t)), t ∈ Rn−1. Proposition. For any u ∈ C∞(Rn), the following inequality holds: (0.1) max |α|=k |∂u(x)| ≤ max β∈Zn−1 + ,|β|=k max |ξ|=1 ∣∣∂β (uxξ) (0)∣∣ , ∀k ≥ 0. Proof. Conforming to the notation in [1], let a1, e ∈ R, e > 0, x ∈ R, k ∈ Z, k ≥ 0, and (α1, α2, . . . , αn) ∈ Z+, |α| = k, be fixed. Put a1j = a1+ ej k , j = 0, ..., k. Consider the hyperplanes given by the images of the linear maps φj : Rn−1 → R, φj(t2, ..., tn) = (x1 + a1jt2, x2 + t2, ..., xn + tn) , ∀j, 0 ≤ j ≤ k. Then, ∂12 2 ∂ α′′ (u ◦ φj) (0) = α1+α2 ∑ l=0 a1 1j ( ∂ 1∂ α1+α2−l 2 ∂ αu ) (x) , where ∂ ′′ = ∂3 3 · · · ∂n n . By applying Lemma 3(i), we have for e = 8e and l = α1, |∂u(x)| ≤ max |β|=|α| max 0≤r≤k ∣∣∂β (u ◦ φr) (0)∣∣ . Let V be an n-dimensional vector space. By using a linear isomorphism between V and R, the classes C∞(V ) and C (V ) can be identified with C∞(Rn) and Received by the editors June 20, 2002. 2000 Mathematics Subject Classification. Primary 30D60, 46F05.

On reflection principle in several complex variables

Let J be a holomorphic mapping where is a domain with real analytic boundary ∂Ω such that . Suppose J maps ∂Ω into an algebraically normal real hypersurface ∑ By closely following the argument of Boauendi and Rotschild |2. 4| we prove the following results. Theorem 1: If is rigid and J is nondegenerate at point . then J extends holomorphically to a full neighbourhood of ρ. Theorem 2: If ∂Ω and ∑ are of infinite type and J not completely degenerate, then J extends holomorphically to a full neighborhood of ρ