A Landau-type construction concerning \((L^{p})^{\prime} \subset L^{q}\) (Q2503001)

Fix \(1\leq p <\infty\), take \(q\) such that \(1/p+1/q=1\), and consider a locally finite positive measure space \((X,\Sigma,\mu)\). In this paper, for every function \(f\notin L_q\), a function \(g\in L_q\) is constructed explicitly such that \[ \int_X | f(x)\,g(x)| \,d\mu(x)=\infty. \] The same author gave the above result for sequence spaces in [Am.\ Math.\ Mon.\ 111, No.\,6, 518--520 (2004; Zbl 1076.46013)].