Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 
 

 

Beltrami equations with coefficient in the fractional Sobolev space $ W^{\theta, \frac2{\theta}}$


Authors: Antonio L. Baisón, Albert Clop and Joan Orobitg
Journal: Proc. Amer. Math. Soc. 145 (2017), 139-149
MSC (2010): Primary 30C62, 35J46, 42B20, 42B37
DOI: https://doi.org/10.1090/proc/13204
Published electronically: June 30, 2016
MathSciNet review: 3565367
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we look at quasiconformal solutions $ \phi :\mathbb{C}\to \mathbb{C}$ of Beltrami equations

$\displaystyle \partial _{\overline {z}} \phi (z)=\mu (z)\,\partial _z \phi (z), $

where $ \mu \in L^\infty (\mathbb{C})$ is compactly supported on $ \mathbb{D}$, and $ \Vert\mu \Vert _\infty <1$ and belongs to the fractional Sobolev space $ W^{\alpha , \frac 2\alpha }(\mathbb{C})$. Our main result states that

$\displaystyle \log \partial _z\phi \in W^{\alpha , \frac 2\alpha }(\mathbb{C})$

whenever $ \alpha \ge \frac 12$. Our method relies on an $ n$-dimensional result, which asserts the compactness of the commutator

$\displaystyle [b,(-\Delta )^\frac {\beta }{2}]:L^\frac {np}{n-\beta p}(\mathbb{R}^n)\to L^p(\mathbb{R}^n)$

between the fractional laplacian $ (-\Delta )^\frac \beta 2$ and any symbol $ b\in W^{\beta ,\frac {n}\beta }(\mathbb{R}^n)$, provided that $ 1<p<\frac {n}{\beta }$.

References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 30C62, 35J46, 42B20, 42B37

Retrieve articles in all journals with MSC (2010): 30C62, 35J46, 42B20, 42B37


Additional Information

Antonio L. Baisón
Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193- Bellaterra (Catalonia)
Email: baison@mat.uab.cat

Albert Clop
Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193- Bellaterra (Catalonia)
Email: albertcp@mat.uab.cat

Joan Orobitg
Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193- Bellaterra (Catalonia)
Email: orobitg@mat.uab.cat

DOI: https://doi.org/10.1090/proc/13204
Keywords: Quasiconformal mapping, Beltrami equation, fractional Sobolev spaces, Beltrami operators
Received by editor(s): July 21, 2015
Received by editor(s) in revised form: February 29, 2016
Published electronically: June 30, 2016
Communicated by: Jeremy Tyson
Article copyright: © Copyright 2016 American Mathematical Society