$L^p$ mapping properties of the Bergman projection on the Hartogs triangle

Abstract

We prove optimal estimates for the mapping properties of the Bergman projection on the Hartogs triangle in weighted L p spaces when p > 4 3 , where the weight is a power of the distance to the singular boundary point. For 1 < p ≤ 4 3 we show that no such weighted estimates are possible.2010 Mathematics Subject Classification. 32A25, 32A07.

$L^p$ estimates for the Bergman projection on some Reinhardt domains

“…For other domains, the projection has only a finite range of mapping regularity. See [Zey13,CZ16,EM16,EM17,Che17]. There are also smooth bounded domains where the projection has limited L p range.…”

Special Toeplitz operators on a class of bounded Hartogs domains

“…Besides the power-generalized Hartogs triangles [4] and Edholm-McNeal [7,8], Beberok [2] also considered the L p boundedness of the Bergman projection on the following generalization of the Hartogs triangle…”

“…. We denote its inverse by G. Then the determinant of the complex Jacobian of G is given by 4) and the Bergman kernel on diagonal for H n {k j ,b} is…”

Sobolev Mapping of Some Holomorphic Projections

“…Proving Theorems 1.3 and 1.4 requires understanding how derivatives commute past the Bergman projection. An initial difficulty is that H γ is not smoothly bounded, so Stokes' theorem cannot be applied in the usual way, e.g., as in [28,Lemma 3], [ There are other papers showing Bergman irregularity on L p 0 (Ω), for specific pseudoconvex Ω: [1,11,13,34]. A unifying result, explaining irregularity in these cases and [19,20], is lacking.…”