Resumen de Sharp [L(p)] estimates for the segment multiplier

Laura De Carli, Enrico Laeng

• Let $S$ be the segment multiplier on the real line, i.e., the linear operator obtained by taking the inverse Fourier transform of $\hat f\chi_{[a,b]}$ where we denote by $\hat f$ the Fourier transform of a function $f$ and by $\chi_{[a,b]}$ the characteristic function of the segment [$a,b$] (finite with positive measure). Our main result consists in computing, for all $1 < p <\infty$, the best constant $c_p$ in the inequality $\parallel Sf\parallel_p\leq c_p\parallel f\parallel_p$. We obtain along the way some results on the Hilbert Transform and on the "gap Hilbert transform"which might have some independent interest. Also we compute the best constant in the $L^p(\mathbb{R})$ estimate for the "box multiplier", which is a higher dimensional version of the segment multiplier.