On sharpness of inequalities for independent random variables in Lorentz spaces (Q2366411)

Let \(L_{p,q}(I)\) be a Lorentz functional space, i.e., the space of all measurable functions \(f\) on \(I\) such that \(\| f \|_{p,q}< \infty\), where \[ \| f \|_{p,q} \equiv \left(\int_ I \bigl(f^*(t)\bigr)^ qd(t^{q/p})\right)^{1/q} <\infty, \] where \(0<p < \infty\), \(0<q<\infty\) and \(I=[0,1]\) or \([0,\infty)\). Here \(f^*\) is a nondecreasing rearrangement of the function \(| f |\). Earlier \textit{N. L. Carothers} and \textit{S. J. Dilworth} [J. Funct. Anal. 84, No. 1, 146-159 (1989; Zbl 0691.46015)] have proved inequalities: For \(1<p<2\), \(1 \leq q< \infty\) there exists a constant \(C=C(p,q)>0\) such that \[ C^{- 1}K^{-1}\|(a_ i)\|_ 2 \leq \Bigl\| \sum a_ if_ i \Bigr\|_{p,q} \leq CK\| (a_ i) \|_{p,q} \qquad (p<q), \] \[ C^{-1}K^{-1} \|(a_ i) \|_ 2 \geq \Bigl\| \sum a_ if_ i \Bigl\|_{p,q} \geq CK \|(a_ i) \|_{p,q} \qquad (q \leq p). \] The author investigates the following question: whether the right estimates in this inequalities can be improved for i.i.d. centered random variables? The answer to this question is negative.