Some spectral formulas for functions generated by differential and integral operators in Orlicz spaces - Bang, H.H.; Huy, V.N. (Carpathian Mathematical Publications, 2021)
In this paper, we investigate the behavior of the sequence of $L^Φ$-norm of functions, which are generated by differential and integral operators through their spectra (the support of the Fourier transform of a function $f$ is called its spectrum and denoted by sp$(f)$). With $Q$ being a polynomial, we introduce the notion of $Q$-primitives, which will return to the notion of primitives if ${Q}(x)= x$, and study the behavior of the sequence of norm of $Q$-primitives of functions in Orlicz space $L^Φ(\mathbb R^n)$. We have the following main result: let $Φ $ be an arbitrary Young function, ${Q}({\bf x} )$ be a polynomial and $(\mathcal{Q}^mf)_{m=0}^∞ \subset L^Φ(\mathbb R^n)$ satisfies $\mathcal{Q}^0f=f, Q(D)\mathcal{Q}^{m+1}f=\mathcal{Q}^mf$ for $m\in\mathbb{Z}_+$. Assume that sp$(f)$ is compact and $sp(\mathcal{Q}^{m}f)= sp(f)$ for all $m\in \mathbb{Z}_+.$ Then $$ \lim\limits_{m→ ∞ } \|\mathcal{Q}^m f\|_{Φ}^{1/m}= \sup\limits_{{\bf x} \in sp(f)} \bigl|1/ Q({\bf x}) \bigl|. $$ The corresponding results for functions generated by differential operators and integral operators are also given.