-regularity of the Bergman projection on quotient domains

Abstract

We obtain sharp ranges of L p -boundedness for domains in a wide class of Reinhardt domains representable as sub-level sets of monomials, by expressing them as quotients of simpler domains. We prove a general transformation law relating L pboundedness on a domain and its quotient by a finite group. The range of p for which the Bergman projection is L p -bounded on our class of Reinhardt domains is found to shrink as the complexity of the domain increases. This is a ``preproof'' accepted article for Canadian Journal of Mathematics This version may be subject to change during the production process.

$$L^p$$ regularity of Toeplitz operators on generalized Hartogs triangles

“…This domain has the same kind of boundary singularities as H γ , and it would be interesting to see how the L p mapping properties of Toeplitz operators depend on the exponents and their arithmetic properties. See [4] for similar results for the Bergman projection. 3.…”

Nonabelian ramified coverings and $L^p$-boundedness of Bergman projections in $\mathbb C^2$

“…In particular, it would be desirable to better understand how the presence, and nature, of singular points in the boundary of the domain D is reflected on I(P D ). The recent papers [CKY20,BCEM21] made progress on this question, investigating the L p -boundedness problem for certain classes of singular domains covered, in the sense of ramified coverings, by polydiscs. It is the goal of this paper to further develop this theme.…”

“…Example 1.1 (from [CKY20,BCEM21]). The generalized (rational) Hartogs triangle is the nonLipschitz domain…”

“…If P E and P D denote the Bergman projections on E and D respectively, a result of S. Bell (Theorem 1 of [Bel81]) implies the following two propositions, showing that the L p -boundedness problem may be "lifted along a ramified covering". Detailed proofs are given in Section 2 below (but see also Theorem 4.15 of [BCEM21], where a different terminology is employed and also [Gho21,GN22] for related work).…”

Toeplitz operators on the weighted Bergman spaces of quotient domains

“…• Rudin's domains are realized as B d /G (B d is the unit ball of C d with respect to ℓ 2 -norm) for a finite pseudoreflection group G [26] • In [9], Bender et al realized a monomial polyhedron as a quotient domain Ω/G for Ω ⊆ D d and a finite abelian group G. Under natural conditions we prove zero product theorems and commutative properties of Toeplitz operators on Bergman spaces over the domains described above. We emphasize that our method yields solutions to some problems of Toeplitz operators without making an appeal to the geometry of these domains.…”