25.1.11 Weighted estimates and local extensionE796BF553819

For unbounded operators between Hilbert spaces, their adjoints and complexes of such operators compare [Chapter VIII, Section 1, pp. 363–365, DemBook]; the facts used here are proved in lemma 6.2.3, theorem 6.2.4. Hörmander's estimate on pseudoconvex domains, with the method of three weights, is presented in [Lemma 7.2, Proposition 7.3 and Theorem 7.1, pp. 35–39, Blo14SCV], and our proof of theorem 6.4.7 follows this presentation; in the density lemma (lemma 6.4.2) we verify that multiplication by cutoff functions preserves the domain of the adjoint before smoothing. Theorem 6.4.9 allows a plurisubharmonic weight together with a smooth weight whose complex Hessian is positive semidefinite; it is proved by approximation with weights for which the right-hand sides are bounded pointwise.

For Richberg's regularization and the boundary distance of pseudoconvex domains (lemma 6.3.3, theorem 6.3.4) compare [Chapter I, (5.18)–(5.21), pp. 43–44, and (7.1)–(7.2), pp. 53–55, DemBook]. These results give the invariant exhaustion needed in corollary 6.3.6.

Our proof of the Ohsawa–Takegoshi extension theorem, theorem 6.5.4, follows Chen's method with a bounded twisting function, as presented in [Theorems 9.1–9.3 and Lemma 9.4, pp. 46–48, Blo14SCV]. The auxiliary weight of lemma 6.5.3 is $-\log (s+\log s)$, where the function $s=-\log (|\zeta |^2+\varepsilon ^2)$ is positive. The constant in theorem 6.5.4 depends only on the radius in the normal direction, and iteration of the theorem gives the estimate with prescribed point value, corollary 6.5.8.