CR Extension for L CR Functions on a Quadric Submanifold of C

We consider the space, CR(M), consisting of CR functions which also lie in L(M) on a quadric submanifold M of C of codimension at least one. For 1 ≤ p ≤ ∞, we prove that each element in CR(M) extends uniquely to an H function on the interior of the convex hull of M . As part of the proof, we establish a semi-global version of the CR approximation theorem of Baouendi and Treves for submanifolds which are graphs and whose graphing functions have polynomial growth. AMS Classification Numbers 32, 42 and 43. 1 Definitions and Main Results We will be working in C = C × C with coordinates (w = u+ iv, z = x+ iy) ∈ C × C. A bilinear form q : C × C 7→ C is said to be a quadric form if q(w1, w2) = q(w2, w1) for w1, w2 ∈ C. Note that this requirement implies that q(w,w) = q(w,w) ∈ R for all w ∈ C. A submanifold M ⊂ C is said to be a quadric submanifold if there exists a quadric form q such that M = {(w, z) ∈ C × C;Rez = q(w,w)}. The closed convex hull of M , denoted ch(M), can be identified with M + Γ where Γ = closed convex hull of {q(w,w); w ∈ C} ⊂ R. (1) The set Γ can be identified with the convex hull of the image of the Levi form of M at the origin (see [B1] or [BP] for details). We are interested in the case where the interior of ch(M) is non-empty. We say that F belongs to H(M + Γ) if F is holomorphic on the interior of M + Γ and ||F ||Hp(M+Γ) = sup x∈interior{Γ} ( ∫ m∈M |F (m+ x)| dσ(m) 1/p is finite