Oliver Dragicevic -- Sharp L^p asymptotics for powers of the complex Riesz transform

Author information

• Oliver Dragicevic
• University of Ljubljana, Slovenia

Abstract:

Let R_1,R_2 be the Riesz transforms on the complex plane. The singular integral R=R_2+iR_1 is called the complex Riesz transform. We establish the estimates on $L^p$, $1<p<\mathrm{\infty }$ for the integer powers $R^k$ of $R$ that are sharp simultaneously in $k$ and $p$. This answers a question suggested in a 1996 work by Iwaniec and Martin. We present three different proofs of this result. We also conjecture the exact values of the $L^p$ norms of $R^k$. This is a joint work with Andrea Carbonaro and Vjekoslav Kovac.

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