We derive a new homotopy formula for a strictly pseudoconvex domain of C 2 boundary in C n by using a method of Lieb and Range and obtain estimates in Lipschitz spaces for the homotopy operators. For r > 1 and q > 0, we obtain a r +1/2 solution u to ∂u = f for a ∂-closed (0, q)-form f of class r in the domain. We apply the estimates to obtain boundary regularities of D-solutions for a domain inthat admits a derivative estimate. Here ϕ is a (0, q)-form in D and ϕ, ∂ϕ are in C 1 (D). We will prove the following C r +1/2 estimate. Communicated by Ngaiming Mok.

Taking as a start point the recent article of Meziani [7], we present several results concerning the normalization of a class of complex vector fields in the plane which degenerate along a real curve. We mainly deal with operators with finite regularity and analyze both the local situation as well as the case of normalization near a circle. Some related questions (e.g., on semi-global solvability and on the normalization of a class of generalized Mizohata operators) are also discussed.

Let Γ(·, λ) be smooth, i.e. C ∞ , embeddings from Ω onto Ω λ , where Ω and Ω λ are bounded domains with smooth boundary in the complex plane and λ varies in I = [0, 1]. Suppose that Γ is smooth on Ω × I and f is a smooth function on ∂Ω × I. Let u(·, λ) be the harmonic functions on Ω λ with boundary values f (·, λ). We show that u(Γ(z, λ), λ) is smooth on Ω × I. Our main result is proved for suitable Hölder spaces for the Dirichlet and Neumann problems with parameter. By observing that the regularity of solutions of the two problems with parameter is not local, we show the existence of smooth embeddings Γ(·, λ) from D, the closure of the unit disc, onto Ω λ such that Γ is smooth on D × I and real analytic at ( √ −1, 0) ∈ D × I, but for every family of Riemann mappings R(·, λ) from Ω λ onto D, the function R(Γ(z, λ), λ) is not real analytic at ( √ −1, 0) ∈ D × I.2010 Mathematics Subject Classification. 31A10, 45B05, 30C35, 35B30. Key words and phrases. Dirichlet and Neumann problems, Kellogg's theorem with parameter, domains with parameter, integral equations with parameter.

The Newlander–Nirenberg theorem says that a formally integrable complex structure is locally equivalent to the complex structure in the complex Euclidean space. We will show two results about the Newlander–Nirenberg theorem with parameter. The first extends the Newlander–Nirenberg theorem to a parametric version, and its proof yields a sharp regularity result as Webster’s proof for the Newlander–Nirenberg theorem. The second concerns a version of Nirenberg’s complex Frobenius theorem and its proof yields a result with a mild loss of regularity.

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