### A Liouville-type theorem for very weak solutions of nonlinear partial differential equations.

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Variable (exponent) Lebesgue spaces represent a relevant research area within the theory of Banach function spaces. Much attention is devoted to look for sufﬁcient conditions on the variable exponent to establish the assertions within the theory. In this Note we try to show the beauty of the research in this ﬁeld, mainly quoting some known results organized into “categories", each of them characterized by a common typology of conditions on the variable exponent. New results involve the failure of...

We study the sequence ${u}_{n}$, which is solution of $-div\left(a(x,\mathbb{D}{u}_{n})\right)+{\Phi}^{\text{'}\text{'}}\left(\right|{u}_{n}\left|\right)\phantom{\rule{0.166667em}{0ex}}{u}_{n}={f}_{n}+{g}_{n}$ in $\Omega $ an open bounded set of ${\mathbf{R}}^{N}$ and ${u}_{n}=0$ on $\partial \Omega $, when ${f}_{n}$ tends to a measure concentrated on a set of null Orlicz-capacity. We consider the relation between this capacity and the $N$-function $\Phi $, and prove a non-existence result.

We study the sequence , which is solution of $-\mathrm{div}\left(a(x,\nabla {u}_{n})\right)+{\Phi}^{\text{'}\text{'}}\left(\right|{u}_{n}\left|\right)\phantom{\rule{0.166667em}{0ex}}{u}_{n}={f}_{n}+{g}_{n}$ in an open bounded set of and = 0 on ∂Ω, when tends to a measure concentrated on a set of null Orlicz-capacity. We consider the relation between this capacity and the -function , and prove a non-existence result.

We study connections between the Boyd indices in Orlicz spaces and the growth conditions frequently met in various applications, for instance, in the regularity theory of variational integrals with non-standard growth. We develop a truncation method for computation of the indices and we also give characterizations of them in terms of the growth exponents and of the Jensen means. Applications concern variational integrals and extrapolation of integral operators.

We study the Hardy inequality and derive the maximal theorem of Hardy and Littlewood in the context of grand Lebesgue spaces, considered when the underlying measure space is the interval (0,1) ⊂ ℝ, and the maximal function is localized in (0,1). Moreover, we prove that the inequality ${\left|\right|Mf\left|\right|}_{p),w}{\le c\left|\right|f\left|\right|}_{p),w}$ holds with some c independent of f iff w belongs to the well known Muckenhoupt class ${A}_{p}$, and therefore iff ${\left|\right|Mf\left|\right|}_{p,w}{\le c\left|\right|f\left|\right|}_{p,w}$ for some c independent of f. Some results of similar type are discussed for the case of small Lebesgue spaces....

Given $\alpha $, $0<\alpha <n$, and $b\in \mathrm{B}MO$, we give sufficient conditions on weights for the commutator of the fractional integral operator, $[b,{I}_{\alpha}]$, to satisfy weighted endpoint inequalities on ${\mathbb{R}}^{n}$ and on bounded domains. These results extend our earlier work [3], where we considered unweighted inequalities on ${\mathbb{R}}^{n}$.

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