An inequality of Bohr and Favard for Orlicz spaces (Q2777733)

If \(f, f',\dots ,f^{(n)}\) are continuous and bounded on \(\mathbb R\) and supp\(\hat f\cap (-\sigma ,\sigma)=\emptyset\), where \(\hat f\) is the Fourier transform of \(f\), then the Bohr-Favard inequality NEWLINE\[NEWLINE \|f\|_{\infty }\leq \sigma^{-n}K_n\|f^{(n)}\|_{\infty } NEWLINE\]NEWLINE holds. The Favard constant \(K_n\) does not depend on \(f\) and is best possible. NEWLINENEWLINENEWLINEBy mean of Stein's methods [\textit{E. M. Stein}, ``Functions of exponential type'', Ann. Math. (2) 65, 582-592 (1957; Zbl 0079.13103)], this result can be extended to \(L^p\)-norms \((1\leq p<\infty)\). NEWLINENEWLINENEWLINEIn this paper, the author proves an Orlicz space version of the above inequality. More precisely, it is shown that if \(\Phi\) is a Young function, \(f\) and its generalized derivative \(f^{(n)}\) are in the Orlicz space \(L_{\Phi }(\mathbb R)\) and supp\(\hat f\cap (-\sigma ,\sigma)=\emptyset\), then \(f^{(k)}\in L_{\Phi }(\mathbb R)\) for all \(0<k<n\) and NEWLINE\[NEWLINE \|f\|_{\Phi }\leq \sigma^{-n}K_n\|f^{(n)}\|_{\Phi }, NEWLINE\]NEWLINE where \(\|\cdot \|_{\Phi }\) is the Orlicz norm in \(L_{\Phi }(\mathbb R)\). NEWLINENEWLINENEWLINEThe proof works essentially developing Stein's method.