The Bergman space Ap(Ω), 1 ≤ p < ∞, on a domain Ω ⊂ 𝐂 is by definition the closed subspace of Lp(Ω) consisting of holomorphic functions; here, the Lp-space is defined via the real area measure of Ω. The Bergman projection PΩ is the orthogonal projection from L2(Ω) onto A2(Ω). Operator theory in Bergman spaces has been a subject of active research already for several decades, and particular questions concern the boundedness of PΩ with respect to the other Lp-norms, the properties of the integral kernel KΩ (called the Bergman kernel), and generalizations to higher dimensional domains.
We study Bergman kernels KΠ and projections PΠ in unbounded planar domains Π, which are periodic in one dimension. As a new innovation, we introduce and adapt to the Bergman space setting the Floquet transform technique, which is a modification of the Fourier-transform and also a standard tool for elliptic spectral problems in periodic domains. We investigate the boundedness properties of the Floquet transform operators in Bergman spaces and derive a general formula connecting PΠ to a projection on a bounded domain. In the case Π is simply connected we write the kernel KΠ in terms of a Riemann mapping φ related to the bounded periodic cell $ of the domain Π.
Reference
J. Taskinen: On the Bergman projection and kernel in periodic planar
domains, submitted.
