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)

 
 

 

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?)

  • [1] Lars V. Ahlfors, Lectures on quasiconformal mappings, 2nd ed., University Lecture Series, vol. 38, American Mathematical Society, Providence, RI, 2006. With supplemental chapters by C. J. Earle, I. Kra, M. Shishikura and J. H. Hubbard. MR 2241787
  • [2] Kari Astala, Tadeusz Iwaniec, and Gaven Martin, Elliptic partial differential equations and quasiconformal mappings in the plane, Princeton Mathematical Series, vol. 48, Princeton University Press, Princeton, NJ, 2009. MR 2472875
  • [3] Kari Astala, Tadeusz Iwaniec, István Prause, and Eero Saksman, Bilipschitz and quasiconformal rotation, stretching and multifractal spectra, Publ. Math. Inst. Hautes Études Sci. 121 (2015), 113-154. MR 3349832, https://doi.org/10.1007/s10240-014-0065-6
  • [4] Albert Clop and Victor Cruz, Weighted estimates for Beltrami equations, Ann. Acad. Sci. Fenn. Math. 38 (2013), no. 1, 91-113. MR 3076800, https://doi.org/10.5186/aasfm.2013.3818
  • [5] A. Clop, D. Faraco, J. Mateu, J. Orobitg, and X. Zhong, Beltrami equations with coefficient in the Sobolev space $ W^{1,p}$, Publ. Mat. 53 (2009), no. 1, 197-230. MR 2474121, https://doi.org/10.5565/PUBLMAT_53109_09
  • [6] D. H. Hamilton, BMO and Teichmüller space, Ann. Acad. Sci. Fenn. Ser. A I Math. 14 (1989), no. 2, 213-224. MR 1024426, https://doi.org/10.5186/aasfm.1989.1409
  • [7] Tadeusz Iwaniec, $ L^p$-theory of quasiregular mappings, Quasiconformal space mappings, Lecture Notes in Math., vol. 1508, Springer, Berlin, 1992, pp. 39-64. MR 1187088, https://doi.org/10.1007/BFb0094237
  • [8] Carlos E. Kenig, Gustavo Ponce, and Luis Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math. 46 (1993), no. 4, 527-620. MR 1211741, https://doi.org/10.1002/cpa.3160460405
  • [9] A. Koski, Singular integrals and Beltrami type operators in the plane and beyond. Master Thesis, Department of Mathematics, University of Helsinki, 2011.
  • [10] Steven G. Krantz and Song-Ying Li, Boundedness and compactness of integral operators on spaces of homogeneous type and applications. II, J. Math. Anal. Appl. 258 (2001), no. 2, 642-657. MR 1835564, https://doi.org/10.1006/jmaa.2000.7403
  • [11] H. M. Reimann, Functions of bounded mean oscillation and quasiconformal mappings, Comment. Math. Helv. 49 (1974), 260-276. MR 0361067
  • [12] Hans Martin Reimann and Thomas Rychener, Funktionen beschränkter mittlerer Oszillation, Lecture Notes in Mathematics, Vol. 487, Springer-Verlag, Berlin-New York, 1975 (German). MR 0511997
  • [13] Thomas Runst and Winfried Sickel, Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations, de Gruyter Series in Nonlinear Analysis and Applications, vol. 3, Walter de Gruyter & Co., Berlin, 1996. MR 1419319
  • [14] Armin Schikorra, Tien-Tsan Shieh, and Daniel Spector, $ L^p$ theory for fractional gradient PDE with $ VMO$ coefficients, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 26 (2015), no. 4, 433-443. MR 3420498

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

American Mathematical Society