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A counter-example in singular integral theory

Lawrence B. Difiore, Victor L. Shapiro (2012)

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 C ¹ ( N 0 ) and suppose f vanishes outside of a compact subset of 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 ( x ) = N k ( x - y ) f ( y ) d y x N 0 . Then F(x) = O(log²r) as r → 0 in the L p -sense, 1 < p < ∞....

Riesz transforms for Dunkl transform

Bechir Amri, Mohamed Sifi (2012)

Annales mathématiques Blaise Pascal

In this paper we obtain the L p -boundedness of Riesz transforms for the Dunkl transform for all 1 &lt; p &lt; .

Singular integrals with highly oscillating kernels on product spaces

Elena Prestini (2000)

Colloquium Mathematicae

We prove the L 2 ( 2 ) boundedness of the oscillatory singular integrals P 0 f ( x , y ) = D x e i ( M 2 ( x ) y ' + M 1 ( x ) x ' ) ο v e r x ' y ' f ( x - x ' , y - y ' ) d x ' d y ' for arbitrary real-valued L functions M 1 ( x ) , M 2 ( x ) and for rather general domains D x 2 whose dependence upon x satisfies no regularity assumptions.

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