On the images of Dunkl–Sobolev spaces under the Schrödinger semigroup associated to Dunkl operators

2017 ◽  
Vol 10 (1) ◽  
pp. 93-120
Author(s):  
C. Sivaramakrishnan ◽  
D. Sukumar ◽  
D. Venku Naidu
Keyword(s):  
Sobolev Spaces ◽  
Dunkl Operators ◽  
Schrödinger Semigroup

On the images of Sobolev spaces under the Schrödinger semigroup

2019 ◽  
Vol 10 (1) ◽  
pp. 65-79 ◽  
Author(s):  
Sivaramakrishnan C ◽  
Sukumar D ◽  
Venku Naidu Dogga
Keyword(s):  
Sobolev Space ◽  
Bergman Space ◽  
Sobolev Spaces ◽  
Weighted Bergman Space ◽  
Hermite Operator ◽  
Schrödinger Semigroup ◽  
Similar Characterization

Abstract In this article, we consider the Schrödinger semigroup for the Laplacian Δ on {\mathbb{R}^{n}} , and characterize the image of a Sobolev space in {L^{2}(\mathbb{R}^{n},e^{u^{2}}du)} under this semigroup as weighted Bergman space (up to equivalence of norms). Also we have a similar characterization for Hermite Sobolev spaces under the Schrödinger semigroup associated to the Hermite operator H on {\mathbb{R}^{n}} .

scholarly journals Dunkl-Sobolev spaces of exponential type and applications

2011 ◽  
Vol 9 (1) ◽  
pp. 41-66
Author(s):  
Hatem Mejjaoli
Keyword(s):  
Sobolev Spaces ◽  
Exponential Type ◽  
Reproducing Kernels ◽  
Difference Operators ◽  
Dunkl Operators

We study the Sobolev spaces of exponential type associated with the Dunkl operators. Some properties including completeness and imbedding theorem are proved. Next we introduce a classes of symbols of exponential type and the associated pseudo-differential-difference operators, which naturally act on the Dunkl-Sobolev spaces of exponential type. Finally using the theory of reproducing kernels some applications are given for these spaces.

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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