A remark on singular integrals with complex homogeneity
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- by Roger L. Jones PDF
- Proc. Amer. Math. Soc. 114 (1992), 763-768 Request permission
Abstract:
The Hilbert transform can be approximated by operators with Fourier multiplier given by ${c_\gamma }\operatorname {sgn} (\xi )|\xi {|^{i\gamma }}$. 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.References
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Additional Information
- © Copyright 1992 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 114 (1992), 763-768
- MSC: Primary 42A50
- DOI: https://doi.org/10.1090/S0002-9939-1992-1100655-8
- MathSciNet review: 1100655