Geodesic
Optimal Banach function space generated with the cone of nonnegative increasing functions
Trudy Instituta matematiki, Tome 22 (2014) no. 1, pp. 24-34.

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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 increasingly monotone functions with respect to the natural functional generated $L_p$-norm ($1\le p\infty$). 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 $\|\,\|f\|_{L_\infty(\cdot,T)}\|f\|_{L_p(0,T)}$. The second coincides with the usual space $L_p$. 
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M. L. Goldman; P. P. Zabreiko. Optimal Banach function space generated with the cone of nonnegative increasing functions. Trudy Instituta matematiki, Tome 22 (2014) no. 1, pp. 24-34. http://geodesic.mathdoc.fr/item/TIMB_2014_22_1_a2/

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