Weak compactness and representation in variable exponent Lebesgue spaces on infinite measure spaces
The paper is devoted to the study of weak compactness in variable Lebesgue spaces \(L^{p(\cdot)}(\Omega)\) over infinite measure spaces \((\Omega,\mu)\). A subset \(S\subset L^{p(\cdot)}(\Omega)\) is said to be equi-integrable if for every decreasing sequence of measurable sets \((A_n)\) in \(\Omega\) with zero intersection (denoted by \(A_n\downarrow\emptyset\)), one has \(\lim\limits_{A_n\downarrow\emptyset}\sup\limits_{f\in S} \|f\chi_{A_n}\|=0\). Let \(L^{p(\cdot)}(\Omega)\) and \(L^{q(\cdot)}(\Omega)\) be variable Lebesgue spaces. Recall that an inclusion operator \(L^{q(\cdot)}(\Omega)\subset L^{p(\cdot)}(\Omega)\) is said to be \(L\)-weakly compact if the unit ball in \(L^{q(\cdot)}(\Omega)\) is an equi-integrable set in \(L^{p(\cdot)}(\Omega)\). It is shown that if \(\mu(\Omega)=\infty\), then the inclusion \(L^{q(\cdot)}(\Omega)\subset L^{p(\cdot)}(\Omega)\) is not \(L\)-weakly compact. Further, it is proved that a subset \(S\subset L^{p(\cdot)}(0,\infty)\), where \(p_+=\mathop{\operatorname{ess\,sup}}\limits_{x\in(0,\infty)}p(x)<\infty\), is relatively weakly compact if and only if it is norm-bounded, and for each \(g\in L^{p^*(\cdot)}(0,\infty)\), where \(1/p(x)+1/p^*(x)=1\), and each sequence of measurable sets \(A_n\downarrow \emptyset\), one has \(\lim\limits_{A_n\downarrow \emptyset}\sup\limits_{f\in S} \int_{A_n}|fg| \, dt=0\). Finally, it is shown that every space \(L^{p(\cdot)}(\Omega)\) with \(p_+<\infty\) on a non-atomic separable \(\sigma\)-finite measure space \((\Omega,\mu)\) is lattice isomorphic to \(L^{q(\cdot)}(0,1)\) for a suitable exponent \(q(\cdot)\) with the same essential range as \(p(\cdot)\). Some applications of this representation theorem are given.
scientific article; zbMATH DE number 7600350
scientific article