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An improvement of a lemma of Calderón and Zygmund involving singular spherical harmonic kernels is obtained and a counter-example is given to show that this result is best possible. In a particular case when the singularity is O(|log r|), let and suppose f vanishes outside of a compact subset of , N ≥ 2. Also, let k(x) be a Calderón-Zygmund kernel of spherical harmonic type. Suppose f(x) = O(|log r|) as r → 0 in the -sense. Set
.
Then F(x) = O(log²r) as r → 0 in the -sense, 1 < p < ∞....
In this paper we obtain the -boundedness of Riesz transforms for the Dunkl transform for all .
We prove the boundedness of the oscillatory singular integrals for arbitrary real-valued functions and for rather general domains whose dependence upon x satisfies no regularity assumptions.
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