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

Abstract

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

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

“…We note that the estimate (1.1) is comparable to the case when q = 1 in [10]. We also note that there are well-known non-solvability results of tangential Cauchy-Riemann equation for n = 2 [18] and for q = n − 1 [11].…”

“…In [10], the author proved the local extension problem, with estimates in Sobolev spaces, for∂ b -closed (0, 1)-forms on real hypersurfaces M in C n when the Leviform at z 0 ∈ M has two positive eigenvalues. Therefore, it is natural to ask the local extension problem, with estimates in Sobolev spaces, for (p, q)-forms when the Levi-form at z 0 ∈ M has at least (q + 1) positive eigenvalues (not mixed).…”