Sobolev regularity of the canonical solutions to \(\bar{\partial}\) on product domains. (English) Zbl 07816889
The Sobolev regularity of the canonical solution to \( \bar{ \partial } \) is studied on product domains of the form \( D_1 \times \cdots \times D_n \) with \( n \ge 2 \) where \( D_i \) is a smoothly bounded domain in \( \mathbb{C} \) for \( 1 \le i \le n \).
\( C^k \)-regularity of the canonical solution was established by M. Landucci (on the bi-disc) [Proc. Symp. Pure Math. 30, Part 1, 177–180 (1977; Zbl 0357.35063)] and J. Bertrams on poly-discs [Randregularität von Lösungen der \({\bar \partial}\)-Gleichung auf dem Polyzylinder und zweidimensionalen analytischen Polyedern. (Boundary regularity of solutions of the \({\bar \partial}\)-equation on the polycylinder and on two-dimensional analytic polyhedra). Bonn: Mathematisches Institut der Universität Bonn (1986; Zbl 0622.32001)]. The \( C^k \)-estimates can thus be carried over to more general product domains by means of bi-holomorphic mappings. The article under review establishes similar estimates to those of Bertrams, but in the context of Sobolev spaces. In particular, the following estimates are established: for non-negative integer \( k \), and \( 1 < p < \infty \), and for \( \bar{ \partial } \)-closed \( ( 0, 1 ) \)-forms, \( f \in W^{ k, p }_{ ( 0, 1 ) } ( \Omega ) \), \[ \| T f \|_{ W^{ k, p } ( \Omega ) } \lesssim \| f \|_{ W^{ k, p }_{ (0, 1 ) } ( \Omega ) } , \] where \( T \) is the canonical solution operator to \( \bar{ \partial } \) on \( \Omega \), a product domain.
The estimates above are especially interesting as the \( \bar{ \partial } \)-Neumann operator, \( N \), does not preserve Sobolev regularity, and in fact, exhibits singularities [the reviewer, Math. Ann. 337, No. 4, 797–816 (2007; Zbl 1130.32022)]. The canonical solution operator can be expressed in the form \[ T = \bar{ \partial }^{ \ast } N , \] and so the estimates suggest that there is a canceling of singularities from the application of the \( \bar{ \partial }^{ \ast } \) operator.
The following Sobolev regularity results of the Bergman projection (the orthogonal projection of \( L^2 \) functions onto \( L^2 \) holomorphic functions), \[ P = I - \bar{ \partial }^{ \ast } N \bar{ \partial } , \] are also presented: for \( f \in W^{ k, p } ( \Omega ) \), \[ \| P f \|_{ W^{ k, p } ( \Omega ) } \lesssim \| f \|_{ W^{ k, p } ( \Omega ) } . \] The regularity of the Bergman projection on product domains, in the case \( p = 2 \) can also be seen as a consequence of transversal symmetries (of the polydisc) [D. E. Barrett, Math. Ann. 258, 441–446 (1982; Zbl 0486.32015)].
The preservation of Sobolev regularity even on smooth pseudoconvex domains is not always guaranteed [D. E. Barrett, Acta Math. 168, No. 1–2, 1–10 (1992; Zbl 0779.32013)] so this fact combined with the above mentioned singular behavior of the \( \bar{ \partial } \)-Neumann operator makes the regularity of the Bergman projection here noteworthy.
\( C^k \)-regularity of the canonical solution was established by M. Landucci (on the bi-disc) [Proc. Symp. Pure Math. 30, Part 1, 177–180 (1977; Zbl 0357.35063)] and J. Bertrams on poly-discs [Randregularität von Lösungen der \({\bar \partial}\)-Gleichung auf dem Polyzylinder und zweidimensionalen analytischen Polyedern. (Boundary regularity of solutions of the \({\bar \partial}\)-equation on the polycylinder and on two-dimensional analytic polyhedra). Bonn: Mathematisches Institut der Universität Bonn (1986; Zbl 0622.32001)]. The \( C^k \)-estimates can thus be carried over to more general product domains by means of bi-holomorphic mappings. The article under review establishes similar estimates to those of Bertrams, but in the context of Sobolev spaces. In particular, the following estimates are established: for non-negative integer \( k \), and \( 1 < p < \infty \), and for \( \bar{ \partial } \)-closed \( ( 0, 1 ) \)-forms, \( f \in W^{ k, p }_{ ( 0, 1 ) } ( \Omega ) \), \[ \| T f \|_{ W^{ k, p } ( \Omega ) } \lesssim \| f \|_{ W^{ k, p }_{ (0, 1 ) } ( \Omega ) } , \] where \( T \) is the canonical solution operator to \( \bar{ \partial } \) on \( \Omega \), a product domain.
The estimates above are especially interesting as the \( \bar{ \partial } \)-Neumann operator, \( N \), does not preserve Sobolev regularity, and in fact, exhibits singularities [the reviewer, Math. Ann. 337, No. 4, 797–816 (2007; Zbl 1130.32022)]. The canonical solution operator can be expressed in the form \[ T = \bar{ \partial }^{ \ast } N , \] and so the estimates suggest that there is a canceling of singularities from the application of the \( \bar{ \partial }^{ \ast } \) operator.
The following Sobolev regularity results of the Bergman projection (the orthogonal projection of \( L^2 \) functions onto \( L^2 \) holomorphic functions), \[ P = I - \bar{ \partial }^{ \ast } N \bar{ \partial } , \] are also presented: for \( f \in W^{ k, p } ( \Omega ) \), \[ \| P f \|_{ W^{ k, p } ( \Omega ) } \lesssim \| f \|_{ W^{ k, p } ( \Omega ) } . \] The regularity of the Bergman projection on product domains, in the case \( p = 2 \) can also be seen as a consequence of transversal symmetries (of the polydisc) [D. E. Barrett, Math. Ann. 258, 441–446 (1982; Zbl 0486.32015)].
The preservation of Sobolev regularity even on smooth pseudoconvex domains is not always guaranteed [D. E. Barrett, Acta Math. 168, No. 1–2, 1–10 (1992; Zbl 0779.32013)] so this fact combined with the above mentioned singular behavior of the \( \bar{ \partial } \)-Neumann operator makes the regularity of the Bergman projection here noteworthy.
Reviewer: Dariush Ehsani (Berlin)
MSC:
| 32W05 | \(\overline\partial\) and \(\overline\partial\)-Neumann operators |
| 32A25 | Integral representations; canonical kernels (Szegő, Bergman, etc.) |
| 32A36 | Bergman spaces of functions in several complex variables |
Keywords:
canonical solution; \(\bar{\partial}\) equation; Bergman projection; product domains; Sobolev regularityReferences:
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