Martingales and Sharp Bounds for Fourier multipliers

Using the argument of Geiss, Montgomery-Smith and Saksman \cite{GMSS}, and a new martingale inequality, the $L^p$--norms of certain Fourier multipliers in $\R^d$, $d\geq 2$, are identified. These include, among others, the second order Riesz transforms $R_j^2$, $j=1, 2,..., d$, and some of the Lévy multipliers studied in \cite{BBB}, \cite{BB}

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