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A family of strongly singular operators

Part of: Integral transforms, operational calculus Abstract harmonic analysis Harmonic analysis in one variable

Published online by Cambridge University Press:  09 April 2009

Magali Folch-Gabayet
Magali Folch-Gabayet
Affiliation:
Instituto de Matemáticas Universidad Nacional Autónoma de México Circuito Exterior, Cd. Universitaria México D.F. 04510 México e-mail: [email protected]
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Abstract

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Let ψ be a positive function defined near the origin such lim1 →0+ ψ(t) = 0. We consider the operator Tzƒ, defined as the pricipal value of the convolution of function ƒ and a kernel K(t) = eiy(t)t−z /ψ(t)1−z, where z is a complex number, 0 ≤ Re(z) ≤ 1, 0 < t ≤ 1 and γ is a real function. Assuming certain regularity conditions on ψ and γ and certain relations between ψ and γ we show that Tθ is a bounded operator on Lp (R) for 1/p = (1+ θ) /2 and 0 ≤ θ < 1, and T1 is bounded from H1 (R) to L1 (R).

MSC classification

Secondary: 42A45: Multipliers 42A85: Convolution, factorization 43A32: Other transforms and operators of Fourier type 44A05: General transforms 44A35: Convolution
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 67 , Issue 1 , August 1999 , pp. 58 - 84
Copyright
Copyright © Australian Mathematical Society 1999

References

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