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 . If we let it is known that these operators converge to the Hilbert transform in norm. It has also been shown that as these operators may diverge even for functions in . In this paper it is shown that we also will have divergence if we select any sequence such that . The proof makes use of Bourgain's entropy method.

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9939-1992-1100655-8

Keywords:
Singular integral,
Hilbert transform,
multipliers,
conjugate function,
complex homogeneity

Article copyright:
© Copyright 1992
American Mathematical Society