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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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