Local Peak Sets and Maximum Modulus Sets in Products of Strictly Pseudoconvex Domains

Abstract

Let D=D₁ x...x D_#x0393;⊂ℂⁿ, where D_i is a bounded strictly pseudoconvex domain in ℂ ⁿ _i, i = l,...,r, n=n₁ +...+n_r. A closed subset E of Γ=∂D₁x...x∂D_Γ is locally a peak set for A^∞(D) if and only if E is locally contained in n-1 dimensional interpolation submanifolds. A closed subset E of Γ which is locally a maximum modulus set for A^∞ (D) is locally contained in totally real n dimensional submanifolds which admit a foliation by n-1 dimensional submanifolds which verify the cone condition at every point of E. If the domains D_i have real analytic boundary, a real analytic n dimensional totally real submanifold of Γ is locally a maximum modulus set for the holomorphic functions in the neighborhood of d̄ if and only if it admits local foliations by real analytic n-1 dimensional interpolation submanifolds.