scholarly journals Boundedness of the Bergman Projection on Generalized Fock–Sobolev Spaces on $${{\mathbb {C}}}^n$$Cn

2020 ◽  
Vol 14 (2) ◽  
Author(s):  
Carme Cascante ◽  
Joan Fàbrega ◽  
Daniel Pascuas
Keyword(s):  
Sobolev Spaces ◽  
Bergman Projection

Mapping properties of the Bergman projections on elementary Reinhardt domains

2021 ◽  
Vol 71 (4) ◽  
pp. 831-844
Author(s):  
Shuo Zhang
Keyword(s):  
Sobolev Spaces ◽  
Bergman Projection ◽  
Reinhardt Domain ◽  
Reinhardt Domains ◽  
Multi Index ◽  
Bergman Projections

Abstract The elementary Reinhardt domain associated to multi-index k = (k 1, …, k n ) ∈ ℤ n is defined by ℋ ( k ) : = { z ∈ D n : z k   is defined and   | z k | < 1 } . $$\mathcal{H}(\mathbf{k}):=\{z\in\mathbb{D}^n: z^{\mathbf{k}}\ \text{is defined and}\ |z^{\mathbf{k}}|<1\}.$$ In this paper, we study the mapping properties of the associated Bergman projection on L p spaces and L p Sobolev spaces of order ≥ 1.

scholarly journals On Hardy spaces on worm domains

2016 ◽  
Vol 3 (1) ◽  
Author(s):  
Alessandro Monguzzi
Keyword(s):  
Sobolev Spaces ◽  
Hardy Spaces ◽  
Pseudoconvex Domain ◽  
Bergman Projection ◽  
Review Article ◽  
High Order ◽  
Worm Domains

AbstractIn this review article we present the problem of studying Hardy spaces and the related Szeg˝o projection on worm domains. We review the importance of the Diederich–Fornæss worm domain as a smooth bounded pseudoconvex domain whose Bergman projection does not preserve Sobolev spaces of sufficiently high order and we highlight which difficulties arise in studying the same problem for the Szeg˝o projection. Finally, we announce and discuss the results we have obtained so far in the setting of non-smooth worm domains.

Sobolev Spaces

2018 ◽  
Author(s):  
D. E. Edmunds ◽  
W. D. Evans
Keyword(s):  
Sobolev Spaces ◽  
Embedding Theorems ◽  
Approximation Numbers ◽  
Selection Of

This chapter presents a selection of some of the most important results in the theory of Sobolev spacesn. Special emphasis is placed on embedding theorems and the question as to whether an embedding map is compact or not. Some results concerning the k-set contraction nature of certain embedding maps are given, for both bounded and unbounded space domains: also the approximation numbers of embedding maps are estimated and these estimates used to classify the embeddings.

scholarly journals Viscous Flow Around a Rigid Body Performing a Time-periodic Motion

2021 ◽  
Vol 23 (1) ◽  
Author(s):  
Thomas Eiter ◽  
Mads Kyed
Keyword(s):  
Rigid Body ◽  
Sobolev Spaces ◽  
Periodic Motion ◽  
Viscous Incompressible Fluid ◽  
Body Motion ◽  
Translational Velocity ◽  
Ill Posed ◽  
Homogeneous Sobolev Spaces ◽  
The Mean ◽  
Time Periodic

AbstractThe equations governing the flow of a viscous incompressible fluid around a rigid body that performs a prescribed time-periodic motion with constant axes of translation and rotation are investigated. Under the assumption that the period and the angular velocity of the prescribed rigid-body motion are compatible, and that the mean translational velocity is non-zero, existence of a time-periodic solution is established. The proof is based on an appropriate linearization, which is examined within a setting of absolutely convergent Fourier series. Since the corresponding resolvent problem is ill-posed in classical Sobolev spaces, a linear theory is developed in a framework of homogeneous Sobolev spaces.

scholarly journals The Dirichlet problem for the Jacobian equation in critical and supercritical Sobolev spaces

2021 ◽  
Vol 60 (1) ◽  
Author(s):  
André Guerra ◽  
Lukas Koch ◽  
Sauli Lindberg
Keyword(s):  
Dirichlet Problem ◽  
Sobolev Spaces ◽  
Regularity Of Solutions ◽  
Jacobian Equation ◽  
L Log L

AbstractWe study existence and regularity of solutions to the Dirichlet problem for the prescribed Jacobian equation, $$\det D u =f$$ det D u = f , where f is integrable and bounded away from zero. In particular, we take $$f\in L^p$$ f ∈ L p , where $$p>1$$ p > 1 , or in $$L\log L$$ L log L . We prove that for a Baire-generic f in either space there are no solutions with the expected regularity.

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