Fully measurable small Lebesgue spaces (Q335443)

The interval \([0,1]\) of the real line \(\mathbb R=[-\infty,\infty]\) is denoted by \(I\), the class of Lebesgue measurable functions on \(I\) is denoted by \({\mathcal M}\) and the class of essentially bounded functions on \(I\) is denoted by \(L^\infty(I)\), so that \(L^\infty(I)= \{f\in{\mathcal M}(I):\| f\|_\infty<\infty\}\), where \(\| f\|_\infty= \inf\{a: \text{meas }(x\in I:|f(x)|> a)= 0\}\). If \(p(.)\in\mathcal{M}\), \(p(.)>1\) a.e., \(\delta>0\) a.e., \(\delta\in L^\infty(I)\), \(\|\delta\|_\infty\leq 1\), then the Banach space \(L^{p[.])\delta(.)}(I)\) is defined to be NEWLINENEWLINE\[NEWLINEL^{p[.])\delta(.)}(I)=\{f\in{\mathcal M},\;f\;\text{finite a.e.}:\| f\|_{p[.]),\delta(.)}= \rho_{p[.]),\delta(.)}(|f|)<\infty\},NEWLINE\]NEWLINE NEWLINEwhere NEWLINENEWLINE\[NEWLINE\begin{multlined}\rho_{p[.]),\delta(.)}(f)= \text{ess\,sup}\{\rho_{p(x)}(\delta(x) f(.)): x\in I\}\\NEWLINE= \text{ess\,sup}\Biggl\{(\int_I(\delta(x) f(t))^{p(x)}\,dt)^{1/p(x)}:\;1\leq p(x)<\infty,\, x\in I\Biggr\},\end{multlined}NEWLINE\]NEWLINE NEWLINEand is called a fully measurable grand Lebesgue space.NEWLINENEWLINENEWLINE NEWLINEIn addition, the fully measurable small Lebesgue space \(L^{(p[.],\delta(.)}(I)\) is defined by NEWLINENEWLINE\[NEWLINEL^{(p[.],\delta(.)}(I)= \{f\in{\mathcal M}: \;f\text{ finite a.e.}, \rho_{(p[.],\delta(.)}(|f|)<\infty\},NEWLINE\]NEWLINE where NEWLINE\[NEWLINE\rho_{(p[.],\delta(.)}(f)= \inf\Biggl\{\sum^\infty_{k=1} \{\text{ess\,inf\,}\rho_{p(x)} (\delta(x)^{-1} f_k(.)):\, x\in I\}:f= \sum^\infty_{k=1} f_k\Biggr\}.NEWLINE\]NEWLINE NEWLINEThe main results of this paper include statements thatNEWLINENEWLINENEWLINE (1) \(L^{(p[.]\delta(.)}(I)\) is a Banach space;NEWLINENEWLINENEWLINE (2) the monotone convergence theorem is valid in \(L^{(p[.],\delta(.)}(I)\);NEWLINENEWLINENEWLINE (3) a Hölder-type inequality is valid in the form NEWLINE\[NEWLINE\int_I f(t)\,g(t)\,dt\leq \rho_{(p[.],\delta(.)}(f)\rho_{(p'[.],\delta(.)}(g),\;1/p(x)+ 1/p'(x)= 1,\;x\in I;\;f\geq 0,\;g\geq 0.NEWLINE\]