Operators of cotlar type with unilateral vanishing moments

Research Report MRR96-004

Operators of cotlar type with unilateral vanishing moments

John E. Gilbert, Jeffrey A. Hogan and Joseph D. Lakey

Abstract One method of construction of wavelet frames is to write Riemann sum approximations of the Calderón-Zygmund singular integral operators which arise as inversion formulae (Calderón reproducing formulae) for the continuous wavelet transform. When the analysing and synthesising functions are smooth and have a vanishing moment, boundedness of the approximations (on $L^p (\Bbb R^n) (1 < p < \infty), H^1 (\Bbb R^n)$ and BMO $(\Bbb R^n)$ ) can be demonstrated through the use of classical techniques. In this paper, the situation which arises when only one of the analysing/synthesising pair possesses a vanishing moment is investigated. It is shown that the dyadic discretisation of the continuous transform is no longer automatically bounded. The T( 1)-theorem is used to find conditions on the analysing/synthesising pair which ensure boundedness and invertibility.

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