Sharp Estimates for the Szegő Projection on the Distinguished Boundary of Model Worm Domains

Abstract

In this paper we study the regularity of the Szegő projection on Lebesgue and Sobolev spaces on the distinguished boundary of the unbounded model worm domain D β .We denote by d b (D β ) the distinguished boundary of D β and define the corresponding Hardy space H 2 (D β ). This can be identified with a closed subspace ofThe orthogonal Hilbert space projection P :is called the Szegő projection on the distinguished boundary.We prove that P, initially defined on the dense subspace L2β−π , β > π. Furthermore, we also prove that P defines a bounded operator P :denotes the Sobolev space of order s and underlying L 2 -norm.Finally, we prove a necessary condition for the boundedness of P on W s,p (d b (D β ), dσ), p ∈ (1, ∞), the Sobolev space of order s and underlying L p -norm.2010 Mathematics Subject Classification. 32A25, 32A36, 30H20.

On Hardy spaces on worm domains

“…The results regarding D 0 are proved in [41] together with a detailed analysis of the space H 2 .D 0 /, whereas the results on Dǎ re new and greater details will appear in a forthcoming paper [42].…”

“…We remark that Barrett only proved the irregularity of the Bergman projection P Wˇ, whereas the easier geometry of Dˇallows also to prove the regularity of S Dˇ. The proofs of Theorem 4.6,Theorem 4.7 and Theorem 4.8 will appear in [42].…”

“…The details of the proofs will appear in a forthcoming paper [42] First, we need to find an explicit formula for the operator S Dˇ. We do it by showing that the biholomorphism (7) induces an isometry between the spaces H 2 .D 0 / and H 2 .Dˇ/.…”

Regularity of the Szegö projection on model worm domains

“…We mention that on the domains D ′ β and D β it is possible to introduce and study also Hardy spaces defined in terms of the induced surface measure on the distinguished boundary, that is, in the case of the domain D ′ β , the four vertices in Figure 1. This has been done by the first author [19] in the case of D ′ β , and in [20] in the case of D β . More precisely, denote by ∂D ′ β and ∂D β the distinguished boundaries of D ′ β and D β , resp., and by P ′ and P the corresponding projections, resp.…”

The structure of the fractional powers of the noncommutative Fourier law

“…The case of spectral operators has, for instance, been studied in Gantner, 42 and the theory of groups and semigroups of quaternionic linear operators has been studied in Alpay et al, 43,44 Colombo and Sabadini, 45 and Ghiloni and Recupero. 46 We are now interested in developing ideas from one and several complex variables, such as in previous studies, 16,[47][48][49][50][51][52][53][54][55] to the quaternionic setting.…”