Sobolev Mapping of Some Holomorphic Projections

Abstract

Sobolev irregularity of the Bergman projection on a family of domains containing the Hartogs triangle is shown. On the Hartogs triangle itself, a sub-Bergman projection is shown to satisfy better Sobolev norm estimates than its Bergman projection.

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

“…Now the proof of in [11,Lemma 4.1] shows (4) holds as well as the analogous statement when the roles of z and w are interchanged. Thus we can apply Lemma 2.9, so long as R is sufficiently large, as in [11,Lemma 4.1]. We wish to maximize r subject to the constraints tq λ and…”

Bergman projection on the symmetrized bidisk

“…For β ≥ 1, following the idea of the partial Bergman kernel in [EM2], one can define K β (w, η) = 1 (1−w η) 2 − β−1 j=0 (j + 1)(w η) j as Bergman kernel subtracting the first β terms in its Taylor series in w η. Then one obtains…”

“…For the Sobolev regularity, we follow the strategy of holomorphic integration by parts (cf. [Bo,St1,Che1,EM2]) to reduce the Sobolev regularity of B G to the boundness of B + D on the weighted L p space (see the definition B + D of in section 2) and study those weighted L p regularities by checking the Bekollé-Bonami constant of the corresponding weights.…”

$L^p$ boundedness of the Bergman Projection on generalizations of the Hartogs triangle in $\mathbb{C}^{n+1}$

“…and further Edholm, McNeal [4] and Chen [13] continue their study of the Bergman projection on H γ and obtain Sobolev estimates.…”