BELTRAMI EQUATIONS WITH COEFFICIENT IN THE FRACTIONAL SOBOLEV SPACE

In this paper, we look at quasiconformal solutions φ: C → C of Beltrami equations ∂ z ̄ φ ( z ) = μ ( z ) ∂ z φ ( z ) , where μ ∈ L∞(C) is compactly supported on D, and ‖μ‖∞ < 1 and belongs to the fractional Sobolev space W α , 2 α ( C ) . Our main result states that log ∂ z φ ∈ W α , 2 α ( C )...

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Bibliographic Details
Published in:Proceedings of the American Mathematical Society 2017-01, Vol.145 (1), p.139-149
Main Authors: BAISÓN, ANTONIO L., CLOP, ALBERT, OROBITG, JOAN
Format: Article
Language:English
Subjects:
B. ANALYSIS
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Summary:In this paper, we look at quasiconformal solutions φ: C → C of Beltrami equations ∂ z ̄ φ ( z ) = μ ( z ) ∂ z φ ( z ) , where μ ∈ L∞(C) is compactly supported on D, and ‖μ‖∞ < 1 and belongs to the fractional Sobolev space W α , 2 α ( C ) . Our main result states that log ∂ z φ ∈ W α , 2 α ( C ) whenever α ≥ 1 2 . Our method relies on an n-dimensional result, which asserts the compactness of the commutator [ b , ( − Δ ) β 2 ] : L n p n − β p ( R n ) → L p ( R n ) between the fractional laplacian ( − Δ ) β 2 and any symbol b ∈ W β , n β ( R n ) , provided that 1 < p < n β .
ISSN:0002-9939
1088-6826