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### A counter-example in singular integral theory

Studia Mathematica

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 $f\in C¹\left({ℝ}^{N}\setminus 0\right)$ and suppose f vanishes outside of a compact subset of ${ℝ}^{N}$, 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 ${L}^{p}$-sense. Set $F\left(x\right)={\int }_{{ℝ}^{N}}k\left(x-y\right)f\left(y\right)dy\forall x\in {ℝ}^{N}\setminus 0$. Then F(x) = O(log²r) as r → 0 in the ${L}^{p}$-sense, 1 < p < ∞....

### Algebraic estimates, stability of local zeta functions, and uniform estimates for distribution functions.

Annals of Mathematics. Second Series

### Riesz transforms for Dunkl transform

Annales mathématiques Blaise Pascal

In this paper we obtain the ${L}^{p}$-boundedness of Riesz transforms for the Dunkl transform for all $1<p<\infty$.

### Singular integrals with highly oscillating kernels on product spaces

Colloquium Mathematicae

We prove the ${L}^{2}{\left(}^{2}\right)$ boundedness of the oscillatory singular integrals ${P}_{0}f\left(x,y\right)={\int }_{{D}_{x}}{e}^{i\left({M}_{2}\left(x\right){y}^{\text{'}}+{M}_{1}\left(x\right){x}^{\text{'}}\right)}οver{x}^{\text{'}}{y}^{\text{'}}f\left(x-{x}^{\text{'}},y-{y}^{\text{'}}\right)d{x}^{\text{'}}d{y}^{\text{'}}$ for arbitrary real-valued ${L}^{\infty }$ functions ${M}_{1}\left(x\right),{M}_{2}\left(x\right)$ and for rather general domains ${D}_{x}{\subseteq }^{2}$ whose dependence upon x satisfies no regularity assumptions.

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