Sharp Estimates for the Szegő Projection on the Distinguished Boundary of Model Worm Domains
Articolo
Data di Pubblicazione:
2017
Citazione:
Sharp Estimates for the Szegő Projection on the Distinguished Boundary of Model Worm Domains / A. Monguzzi, M.M. Peloso. - In: INTEGRAL EQUATIONS AND OPERATOR THEORY. - ISSN 0378-620X. - 89:3(2017 Nov), pp. 315-344.
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 DβDβ . We denote by db(Dβ)db(Dβ) the distinguished boundary of DβDβ and define the corresponding Hardy space ℋ2(Dβ)H2(Dβ) . This can be identified with a closed subspace of L2(db(Dβ),dσ)L2(db(Dβ),dσ) , that we denote by ℋ2(db(Dβ))H2(db(Dβ)) , where dσdσ is the naturally induced measure on db(Dβ)db(Dβ) . The orthogonal Hilbert space projection :L2(db(Dβ),dσ)→ℋ2(db(Dβ))P:L2(db(Dβ),dσ)→H2(db(Dβ)) is called the Szegő projection on the distinguished boundary. We prove that P , initially defined on the dense subspace L2∩Lp(db(Dβ),dσ)L2∩Lp(db(Dβ),dσ) extends to a bounded operator :Lp(db(Dβ),dσ)→Lp(db(Dβ),dσ)P:Lp(db(Dβ),dσ)→Lp(db(Dβ),dσ) if and only if 21+νβ
πνβ=π2β−π,β>π . Furthermore, we also prove that P defines a bounded operator :Ws,2(db(Dβ),dσ)→Ws,2(db(Dβ),dσ)P:Ws,2(db(Dβ),dσ)→Ws,2(db(Dβ),dσ) if and only if 0≤s<νβ20≤s<νβ2 where Ws.2(db(Dβ),dσ)Ws.2(db(Dβ),dσ) denotes the Sobolev space of order s and underlying L2L2 -norm. Finally, we prove a necessary condition for the boundedness of P on Ws,p(db(Dβ),dσ)Ws,p(db(Dβ),dσ) , p∈(1,∞)p∈(1,∞) , the Sobolev space of order s and underlying LpLp -norm.
Tipologia IRIS:
01 - Articolo su periodico
Keywords:
Hardy spaces; Szegő kernel; Szegő projection; Worm domain;
Elenco autori:
A. Monguzzi, M.M. Peloso
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