Strong and weak sharp bounds for Neural Network Operators in Sobolev-Orlicz spaces and their quantitative extensions to Orlicz spaces

In this paper, we establish sharp bounds for a family of Kantorovich-type neural network operators within the general frameworks of Sobolev-Orlicz and Orlicz spaces. We establish both strong (in terms of the Luxemburg norm) and weak (in terms of the modular functional) estimates, using different approaches. The strong estimates are derived for spaces generated by $φ$-functions that are $N$-functions or satisfy the $Δ^\prime$-condition. Such estimates also lead to convergence results with respect to the Luxemburg norm in several instances of Orlicz spaces, including the exponential case. Meanwhile, the weak estimates are achieved under less restrictive assumptions on the involved $φ$-function. To obtain these results, we introduce some new tools and techniques in Orlicz spaces. Central to our approach is the Orlicz Minkowski inequality, which allows us to obtain unified strong estimates for the operators. We also present a weak (modular) version of this inequality holding under weaker conditions. Additionally, we introduce a novel notion of discrete absolute $φ$-moments of the hybrid type, and we employ the Hardy-Littlewood maximal operator within Orlicz spaces for the asymptotic analysis. Furthermore, we introduce the new space $\mathcal{W}^{1,φ}(I)$, which is embedded in the Sobolev-Orlicz space $W^{1,φ}(I)$ and modularly dense in $L^φ(I)$. This allows to achieve asymptotic estimates for a wider class of $φ$-functions, including those that do not meet the $Δ_2$-condition. For the extension to the whole Orlicz-setting, we generalize a Sobolev-Orlicz density result given by H. Musielak using Steklov functions, providing a modular counterpart.

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