Best constants for the Riesz projection
Recall that for \(f\in L^{p}(T)\) with Fourier expansion \(f(\zeta)=\sum_{k} \widehat f(k)\zeta^{k}\) the Riesz projection onto the Hardy space \(H^{p}(T)\) is defined by \(P_{+}f(\zeta)=\sum_{k\geq 0} \widehat f(k)\zeta^{k}\), and the co-analytic projection by \(P_{-}=I-P_{+}\). The estimate \(\|\max(|P_{+}f|,|P_{-}f|)\|_{p}\leq c_{p}\|f\|_{p}\) \((1<p\leq 2)\) is proved where \(c_{p}=(\sin(\pi/p))^{-1}\). This implies \(\|P_{+}\|= c_{p}\) \((1<p<\infty)\). Analogous results for the analytic projection on the real line (half-line Fourier multiplier) and in Euclidean space and for weighted \(L^{p}\) spaces with weight function \(|\zeta-\zeta_0|^\alpha\) are obtained. For the proof, the plurisubharmonic estimate \(\max(|w|^{p}, |z|^{p})\leq c_{p}^{p} |w+\overline z|^{p}-b_{p} \text{Re}[(wz)^{p/2}]\) is shown.
scientific article; zbMATH DE number 1525188
scientific article