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A remark on singular integrals with complex homogeneity

Author: Roger L. Jones
Journal: Proc. Amer. Math. Soc. 114 (1992), 763-768
MSC: Primary 42A50
MathSciNet review: 1100655
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Abstract: The Hilbert transform can be approximated by operators with Fourier multiplier given by $ {c_\gamma }\operatorname{sgn} (\xi )\vert\xi {\vert^{i\gamma }}$. If we let $ \gamma \to 0$ it is known that these operators converge to the Hilbert transform in $ {L^2}$ norm. It has also been shown that as $ \gamma \to 0$ these operators may diverge even for functions in $ {L^2}$. In this paper it is shown that we also will have divergence if we select any sequence $ \{ {\gamma _n}\} _{n = 1}^\infty $ such that $ {\gamma _n} \to 0$. The proof makes use of Bourgain's entropy method.

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Keywords: Singular integral, Hilbert transform, multipliers, conjugate function, complex homogeneity
Article copyright: © Copyright 1992 American Mathematical Society

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