New properties of small Lebesgue spaces and their applications
Let \(n\) be a positive integer and let \(\Omega\) be an open subset of Euclidean space \(\mathbb{R}^n\) with Lebesgue measure \(m\). The space of measurable functions on \(\Omega\) is denoted by \(L^0(\Omega)\), and, for \(1< p<\infty\), \(L^p(\Omega)\) denotes the space consisting of \(p\)th power integrable functions with \(\| f\|_p= (\int_W| f(t)|^p\,dt)^{1/p}\), \(f\in L^p(\Omega)\). If \(1< p<\infty\) and the conjugate pair \((p,p')\) is related by \((1/p)+ (1/p')= 1\), let the expression \(\kappa_{p'}(h)\) be defined for \(h\in L^0(\Omega)\) by \[ \kappa_{p'}(h)= \text{inf}\Biggl\{\varepsilon^{-(l/(p-\varepsilon)} \Biggl(m(\Omega)^{-1} \inf_\Omega\;h(x)^{(p-\varepsilon)'} dx\Biggr)^{1/(p-\varepsilon)'}\;\Biggl|\;0< \varepsilon< p-1\Biggr\}. \] Statements in the introduction of this paper indicate that an earlier paper by one of the authors [\textit{A. Fiorenza}, Collect. Math. 51, 131--148 (2000; Zbl 0960.46022)], includes an introduction of the expression \[ \| g\|_{(p')}= \text{inf} \Biggl\{\sum^\infty_{k=1} \kappa_p(g_k)\;\Biggl|\;g= \sum^\infty_{k=1} g_k,\,g_j\geq 0\Biggr\}, \] which provides the definition of a small Lebesgue space \[ L^{(p')}(\Omega)= \{g\in L^1(\Omega):\| g\|_{(g')}<\infty\}. \] If \(g^*\) is the non-increasing rearrangement of \(g\) so that \(g^*(s)= \inf\{t> 0:\lambda_f(t)< s\}\), where \(\lambda_f(t)= m\{x\in\Omega:| g(x)|> t\}\) and \(g^{**}(s)= s^{-1} \int_{[0,s]} g^*(u)\,du\), then it is indicated in the introduction of this paper that \(\| g\|_{(p)}\leq\| g^{**}\|_{(p)}< p'\| g\|_{(p)}\). The main results of the paper include statements of the Levi monotone convergence theorem in small Lebesgue spaces. In particular, if the monotone increasing sequence \(\{f_j,\,j= 1,2,3,\dots\}\) in \(L^0(\Omega)\) is such that \(\sup\{\| f_j\|_{(p)},\,j= 1,2,\dots\}= M<\infty\), and if \(f(x)= \sup\{f_j(x),\,j= 1,2,3,\dots\}\), then \(\| f_j- f\|_{(p)}\to 0\) as \(j\to\infty\). Estimates are also derived for functions in small Lebesgue-Sobolev spaces or for quasilinear equations with coefficients in small Lebesgue spaces.
scientific article; zbMATH DE number 1982583