Abstract
We establish a class of sharp logarithmic estimates for the Beurling-Ahlfors transform B on the complex plane. For any K > 0 we determine the optimal constant \({L = L(K) \in (0, \infty]}\) such that the following holds. If \({F : \mathbb{C} \rightarrow \mathbb{C}}\) is a radial function, then for any R > 0,
where Ψ(t) = (t + 1) log(t + 1) – t and \({\mathcal{B}(0, R) \subset \mathbb{C}}\) denotes the ball of center 0 and radius R. A related result in higher dimensions is also established. The proof rests on probabilistic methods and exploits a certain sharp inequality for martingales.
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References
K. Astala, Area distortion of quasiconformal mappings, Acta Math. 173 (no. 1) (1994), 37–60.
K. Astala, T. Iwaniec and G. Martin, Elliptic partial differential equations and quasiconformal mappings in the plane, Princeton University Press, Princeton, NJ, 2009.
R. Bañuelos, The foundational inequalities of D. L. Burkholder and some of their ramifications, Illinois J. Math. (no. 3) 54 (2010), 789–868.
Bañuelos R., Janakiraman P.: L p -bounds for the Beurling-Ahlfors transform. Trans. Amer. Math. Soc. 360, 3603–3612 (2008)
Bañuelos R., Janakiraman P.: On the weak-type constant of the Beurling-Ahlfors Transform. Michigan Math. J. 58, 239–257 (2009)
R. Bañuelos and A. Ose̦kowski, Sharp inequalities for the Beurling-Ahlfors transform on radial functions, Duke Math. J., to appear.
A. Baernstein II and S. J. Montgomery-Smith, Some conjectures about integral means of \({{\partial}f}\) and \({\bar{\partial}f}\) , in: Complex Analysis and Differential Equations (Uppsala, Sweden, 1997), Ch. Kiselman ed., Acta. Univ. Upsaliensis Univ. C Organ. Hist. 64 (1999), Uppsala Univ. Press, Uppsala, Sweden, 92–109.
A. Borichev, P. Janakiraman and A. Volberg, Subordination by conformal martingales in L p and zeros of Laguerre polynomials, preprint.
Burkholder D.L.: Boundary value problems and sharp inequalities for martingale transforms. Ann. Probab. 12, 647–702 (1984)
D. L. Burkholder, Explorations in martingale theory and its applications, École d’Ete de Probabilités de Saint-Flour XIX—1989, Lecture Notes in Math. 1464, Springer, Berlin, 1991, 1–66.
Dellacherie C., Meyer P.-A.: Probabilities and potential B: Theory of martingales. North Holland, Amsterdam (1982)
F. W. Gehring, E. Reich, Area distortion under quasiconformal mappings, Ann. Acad. Sci. Fenn. Ser. A I 388 (1966) pp. 1– 15.
A. Hinkkanen, On the norm of the Beurling-Ahlfors transformation, preprint.
Iwaniec T.: Extremal inequalities in Sobolev spaces and quasiconformal mappings. Z. Anal. Anwendungen 1, 1–16 (1982)
T. Iwaniec, Hilbert transform in the complex plane and area inequalities for certain quadratic differentials, Michigan Math. J. 34 (no. 3) (1987), 407–434.
Lehto O.: Remarks on the integrability of the derivatives of quasiconformal mappings. Ann. Acad. Sci. Fenn. Series A I Math. 371, 3–8 (1965)
Lehto O., Virtanen K.I.: Quasiconformal mappings in the plane, Second edition. Springer-Verlag, New York-Heidelberg (1973)
A. Ose̦kowski, Logarithmic inequalities for Fourier multipliers, Math. Z., to appear.
E. M. Stein, Singular integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, 1970.
G. Wang, Differential subordination and strong differential subordination for continuous-time martingales and related sharp inequalities, Ann. Probab. 23 (no. 2) (1995), 522–551.
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Partially supported by Polish Ministry of Science and Higher Education (MNiSW) grant IP2011 039571 ‘Iuventus Plus’.
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Ose̦kowski, A. Sharp Logarithmic Bounds for Beurling-Ahlfors Operator Restricted to the Class of Radial Functions. Mediterr. J. Math. 10, 1883–1894 (2013). https://doi.org/10.1007/s00009-013-0270-4
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DOI: https://doi.org/10.1007/s00009-013-0270-4