# Some spectral formulas for functions generated by differential and integral operators in Orlicz spaces

• H.H. Bang Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet str., Hanoi, Vietnam
• V.N. Huy Hanoi University of Science, Vietnam National University, 334 Nguyen Trai str., Hanoi, Vietnam; Thang Long Institute of Mathematics and Applied Sciences, Thang Long University, Nghiem Xuan Yem str., Hanoi, Vietnam
Keywords: Orlicz space, inequality in approximation, Fourier transform, generalized function
Published online: 2021-08-03

### Abstract

In this paper, we investigate the behavior of the sequence of $L^\Phi$-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 $L^\Phi(\mathbb R^n)$. We have the following main result: let $\Phi$ be an arbitrary Young function, ${Q}({\bf x} )$ be a polynomial and $(\mathcal{Q}^mf)_{m=0}^\infty \subset L^\Phi(\mathbb R^n)$ 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 $$\lim\limits_{m\to \infty } \|\mathcal{Q}^m f\|_{\Phi}^{1/m}= \sup\limits_{{\bf x} \in sp(f)} \bigl|1/ {Q}({\bf x}) \bigl|.$$ The corresponding results for functions generated by differential operators and integral operators are also given.

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