Weighted spaces related to Bochner integrable functions (Q2260272)

The paper deals with the duality of weighted \(p\)-Bochner integrable functions. Though the results are as one expects, there are some flaws. 1. In Example 2.2: The function is not the zero function a.e., \(w(x) = x\), so \(w(0) = 0\). Hence \(\|(wf )(0)\|_X = 0\). But the authors calculate \(V_w (f ) = 0\). So \(\| f \|_V = 0\). 2. Lemma 3.1 is not precise : Let \(p = 2\), \(f (t) = \frac{1}{\sqrt{t}}\) and \(w(t) = t\). Then \(w^{\frac{-1}{2}}(t) = \frac{1}{\sqrt{t}}\). Now \(w^{\frac{-1}{2}}(t) f (t) =\frac1t \notin L^2_w (I)\), where \(X = \mathbb R\).