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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 < ∞....

A Hörmander-type spectral multiplier theorem for operators without heat kernel

Sönke Blunck (2003)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Hörmander’s famous Fourier multiplier theorem ensures the L p -boundedness of F ( - Δ D ) whenever F ( s ) for some s &gt; D 2 , where we denote by ( s ) the set of functions satisfying the Hörmander condition for s derivatives. Spectral multiplier theorems are extensions of this result to more general operators A 0 and yield the L p -boundedness of F ( A ) provided F ( s ) for some s sufficiently large. The harmonic oscillator A = - Δ + x 2 shows that in general s &gt; D 2 is not sufficient even if A has a heat kernel satisfying gaussian estimates. In this paper,...

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