Gol’dman, M. L.; Zabreĭko, P. P. Optimal Banach function space generated with the cone of nonnegative increasing functions. (Russian. English summary) Zbl 1464.46030 Tr. Inst. Mat., Minsk 22, No. 1, 24-34 (2014). Summary: The article deals with the effective constructions for the optimal Banach ideal and symmetric spaces (of functions \(f: [0,T]\to\mathbb{R}\)) containing a cone of nonnegative and monotone inceasing functions with respect to the natural functionally generated \(L_p\)-norm (\(1\leq p<\infty\)). The first of these spaces turns out to be the space of measurable functions \(f\) such that \(\|f\|_{L_\infty(\cdot,T)}\in L_p(0,T)\); this space can be endowed with the norm \(\|\,\|f\|_{L_\infty(\cdot,T)}\|f\|_{L_p(0,T)}\). \( \bigl\|\,\|f\|_{L_\infty(\cdot,T)} \,\bigr\|_{L_p(0,T)} \). The second coincides with the usual space \(L_p\). Cited in 3 Documents MSC: 46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) Keywords:optimal Banach function space; cone of nonnegative increasing functions PDFBibTeX XMLCite \textit{M. L. Gol'dman} and \textit{P. P. Zabreĭko}, Tr. Inst. Mat., Minsk 22, No. 1, 24--34 (2014; Zbl 1464.46030) Full Text: DOI MNR References: [1] [1] Bennett C., Sharpley R., Interpolation of Operators, Academic, New York, 1988 · Zbl 0647.46057 [2] [2] Krein S. G., Petunin Yu. I., Semenov E. M., Interpolyatsiya lineinykh operatorov, Nauka, M., 1978 [3] [3] Zabreiko P. P., “Nelineinye integralnye operatory”, Trudy seminara po funktsionalnomu analizu, 8, Voronezh, 1966, 3–148 [4] [4] Zabreiko P. P., Issledovaniya po teorii integralnykh operatorov v idealnykh prostranstvakh funktsii, Dis. … d-ra fiz.-mat. nauk, Voronezh. gos. un-t, Voronezh, 1968 [5] [5] Zabreiko P. P., “Idealnye prostranstva funktsii, I”, Vestn. Yarosl. un-ta, 1974, no. 8, 12–52 [6] [6] Goldman M. L., Zabreiko P. P., “Optimalnoe vosstanovlenie banakhova funktsionalnogo prostranstva po konusu neotritsatelnykh funktsii”, Trudy Matematich. in-ta im. V. A. Steklova (to appear) [7] [7] Burenkov V. I., Goldman M. L., “Vychislenie normy polozhitelnogo operatora na konuse monotonnykh funktsii”, Trudy Matematich. in-ta im. V. A. Steklova, 210, 1995, 65–89 [8] [8] Goldman M. L., Haroske D., “Estimates for continuity envelopes and approximation numbers of Bessel potentials”, Journal of Approximation Theory, 172 (2013), 58–85 · Zbl 1293.46020 [9] [9] Sawyer E., “Boundedness of classical operators on classical Lorentz spaces”, Studia Math., 96 (1990), 145–158 · Zbl 0705.42014 [10] [10] Goldman M. L., Heinig H. P., Stepanov V. D., “On the principle of duality in Lorentz spaces”, Canadian J. Math., 48 (1996), 959–979 · Zbl 0874.47011 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.