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A study of behavior of the sequence of norm of primitives of functions in Orlicz spaces depending on their spectrum. (English) Zbl 1345.46021

H. H. Bang [Trans. Am. Math. Soc. 347, No. 3, 1067–1080 (1995; Zbl 0828.42009)] proved that for \(1\leq p\leq\infty\) any function \(f\in L_p(\mathbb R^n)\), \(f\not\equiv0\), with \(\text{supp}\hat f\) bounded, satisfies \(\lim_{|\alpha|\to\infty}(\| D^\alpha f\|_p / \sup_{\xi\in\text{supp}\hat f}|\xi^\alpha|)^{1/|\alpha|}=1\). This allows to approximate the behaviour of the sequence \(\{\| D^\alpha f\|_p^{1/|\alpha|}\}\) (which is difficult to calculate directly) by the sequence \(\{\sup_{\xi\in\text{supp}\hat f}|\xi^\alpha|^{1/|\alpha|}\}\).
The natural question is about the situation when derivatives are replaced by integrals. V. K. Tuan answered the question in [J. Fourier Anal. Appl. 7, No. 3, 319–323 (2001; Zbl 0988.94011)] for \(p=2\) and \(n=1\), and the authors did it for \(1\leq p\leq\infty\), \(n=1\), in [J. Approx. Theory 162, No. 6, 1178–1186 (2010; Zbl 1204.46017)], and for \(1\leq p\leq\infty\), \(n\geq1\), in [Dokl. Math. 84, No. 2, 672–674 (2011; Zbl 1243.46015); translation from Dokl. Akad. Nauk 440, No. 4, 456–458 (2011)].
In the present paper, they deal with the \(n\)-dimensional case in Orlicz spaces. For instance, they prove that, for a function \(f\not\equiv0\) from an Orlicz space \(L_\Phi(\mathbb R^n)\) whose spectrum \(\text{supp}\hat f\) is contained in \(\{\xi\in\mathbb R^n: \min\{|\xi_1|,\dots,|\xi_n|\}>\Delta\}\) for some \(\Delta>0\), there exists exactly one sequence of primitives \((I^\alpha f)_{\alpha\in\mathbb Z^n_+}\subset L_\Phi(\mathbb R^n)\) such that the spectrum of \(I^\alpha f\) coincides with the spectrum of \(f\) for all \(\alpha\in\mathbb Z^n_+\), and \(\lim_{|\alpha|\to\infty}(\inf_{\xi\in\text{supp}\hat f}|\xi^\alpha|\| I^\alpha f\|_\Phi)^{1/|\alpha|}=1\).

MSC:

46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
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