ERRATUM TO “PSEUDOLOCAL ESTIMATES FOR ¯∂

ON GENERAL PSEUDOCONVEX DOMAINS”

Miroslav Engliš

Indiana Univ. Math. J. 50 (2001), 1593–1607

The author thanks Nils Øvrelid for pointing out to him a gap in the above paper on page 1600, in the Remark at the end of Section 3: namely, while it is indeed true that the constant C in the estimate

∥ζMα∥_{kϵ}^{2} ≤ C(∥ζ_{1}α∥_{(k−2)ϵ}^{2} + ∥α∥^{2}) ∀α ∈ L^{2}

(or, equivalently, in the estimate (3.1) there) depends only on k, ζ, ζ_{1} and the coordinate chart U — and thus, in particular, can be chosen the same for all domains Ω_{δ} appearing above — this is no longer true if the operator M is replaced by the Neumann operator N. All one gets by the argument of Folland and Kohn ([FK], (3.1.2) and (3.1.3)) is that

(∗)  ∥ζNα∥_{kϵ}^{2} ≤ C(∥ζ_{1}α∥_{(k−2)ϵ}^{2} + ∥α + Nα∥^{2})

with C depending only on k, ζ, ζ_{1} and U. Unfortunately, this is not sufficient for the application in Proposition 4.1, and thus all the proofs from Section 4 (and, hence, Theorems 0.2 and 0.3, and Corollary 0.5) hold only under the additional hypothesis that the domain Ω be bounded (though possibly still with nonsmooth boundary): in that case, since all the domains Ω_{δ} are contained in Ω, the norms of the corresponding Neumann operators N_{δ} are jointly bounded by e diam(Ω)^{2} (by the well-known result of Hörmander), and thus one can replace ∥α + N_{δ}α∥ in (∗) by ∥α∥.

It would certainly be interesting to know whether the results just mentioned remain in force also in the unbounded situation.

Mathematics Institute, Žitná 25, 11567 Prague 1, Czech Republic

   E-mail address: englis@math.cas.cz