Singular integral operators generated by wavelet transforms

Let g ∈ L¹(ℝⁿ), gt (cursive Greek chi) = t^-ng(cursive Greek chi/t). If ∫ g = 0, then g is called the wavelet function, and the convolution operator f → f * gt is called the wavelet transform of f generated by the wavelet g. For a large class of functions g and f ∈ L^p(ℝⁿ), 1 < p < ∞, it is shown that ∫^ρ_ε(f*gt)(cursive Greek chi)dt/t converges as ε → 0 and ρ → ∞ in the L^p-norm and in the a.e. sense to a limit I(f,g) = cf+Tf, where c ≡ c(g) = const, and T ≡ T(g) is the Calderón-Zygmund singular integral operator. The particular case T ≡ 0 corresponds to Calderón's reproducing formula. Each "rough" singular integral operator T_θf = p.v. |cursive Greek chi|^-nθ(cursive Greek chi/|cursive Greek chi|) * f, with θ ∈ H¹(Σ_n-1) (the Hardy space on the unit sphere) can be represented as I(f,g) with a suitable wavelet g. A new proof of the L^p-boundedness of T_θ, θ ∈ H¹(Σ_n-1), is given.