SOBOLEV ESTIMATES FOR THE LOCAL EXTENSION OF BOUNDARY HOLOMORPHIC FORMS ON REAL HYPERSURFACES IN ℂⁿ

Abstract

Abstract. Let M be a smooth real hypersurface in complex space of dimension n, n ≥ 3, and assume that the Levi-form at z 0 on M has at least (q + 1)-positive eigenvalues, 1 ≤ q ≤ n − 2. We estimate solutions of the local∂-closed extension problem near z 0 for (p, q)-forms in Sobolev spaces. Using this result, we estimate the local solution of tangential Cauchy-Riemann equation near z 0 in Sobolev spaces.

Sobolev estimates for the local extension of ∂̄_{𝑏}-closed (0,1)-forms on real hypersurfaces in ℂⁿ with two positive eigenvalues

“…Also, for z ∈ C n , we write z = (z , z ) where z ∈ C 2 and z ∈ C n−2 . In [8], we filled a neighborhood of z 0 = (z 0 , z 0 ) ∈ M by strongly convex domains {Ω z } B σ (z 0 ) whose diameters converge to zero as z → z 0 . Therefore the estimates of the Neumann operators are not uniform in Sobolev spaces.…”

“…In [23], Shaw also constructed homotopy formulas for∂ b using integral kernel methods and showed the local solvability for∂ b in Hölder space when the Levi-form at z 0 ∈ M has at least (q + 2) positive and (q + 2) negative eigenvalues. In [10], the author obtained estimates for the local one-sided extension problem and proved the local solvability of∂ b equations for (p, q)-forms with estimates in the Sobolev spaces when the Leviform has at least (q + 2) positive eigenvalues (and hence n ≥ 4).…”

Hölder Estimates for the Cauchy-Riemann Equation on Parameters

“…As for the estimates in Sobolev space [3], we lose some regularity when we take derivatives of the solution. For each z, ζ ∈ Ω 0 , we set z τ = Ψ τ (z) and ζ τ = Ψ τ (ζ), and we let ρ τ (z) = ρ (Ψ τ (z)).…”

“…Let D ⊂ C n be a domain and bD is smooth near z 0 ∈ bD and the Levi-form of bD has k positive eigenvalues at z 0 . In this situation, the author constructed a family of smooth strongly convex family of domains of complex dimension k which are foliated inside of D, making a neighborhood of z 0 in D, and got estimates of ∂ and∂ b equation in Sobolev spaces near z 0 ∈ bD [3]. Let W be a set and k be a nonnegative integer and 0 < α ≤ 1.…”

“…We have the following derivative estimates for the ∂-equation in Hölder space. As for the estimates in Sobolev space [3], we lose some regularity.…”