Sums of independent functions in symmetric spaces with the Kruglov property (Q881149)

If \(f\) is a Lebesgue measurable function on \([0,1]\), then the distribution function \(\lambda_f\) is defined on \([0,\infty)\) by \(\lambda_f(t)=\text{meas}(\{x\in(0,1):|f(x)|>t\})\), and the non-increasing rearrangement \(f^*\) of \(f\) is defined by \(f^*(s)=\text{inf} \{t>0:\lambda_f(t)\leq s\}\). The functions \(f,g\) are said to be identically distributed if \(\lambda_g=\lambda_f\). If \(\{f_k,k=1,2,3,\dots\}\) is a sequence of disjunctive identically distributed functions on \([0,1)\), and \(\{F_k,k=1,2,3,\dots\}\) is also a sequence of disjunctively identically distributed functions on \([0,\infty)\), then the statements of this paper relate to the application of Kruglov operators in deriving estimates of the form \[ \biggl\|\sum^n_{k=1}f_k\biggr\|_Y\leq C\biggl\|\sum^n_{k=1}F_k \biggr\|_Z, \] where either \(Z=X\text{ or }Z=Z^2_X,\) \(Z^2_X=\{f\in (L^1+ L^\infty)(0,\infty): \|f^*\chi_{(0,1)} \|_X+\|f^*\chi_{(1,\infty)} \|_2<\infty\}\), \(\|\cdot\|_2\) is the norm on \(L^2(1,\infty)\), and \(\chi_E\) is the characteristic function of the set \(E\). Spaces with the Kruglov property were introduced by \textit{V.\,M.\thinspace Kruglov} [Teor.\ Veroyatn.\ Primen.\ 15, No.\,2, 330--336 (1970; Zbl 0301.60014)] as the class \(K=\{X:X\) is a symmetric space and \(f\in X\) implies that \(\pi(f) \in X\}\), where \(\pi(f)\) is expressed as a sum of independent copies of \(f\) in the form \(\sum^N_{j=1}f_j\), and a Kruglov operator, introduced by the authors in [Isr.\ J.\ Math.\ 145, 125--156 (2005; Zbl 1084.46020)], is a bounded linear operator in \(X\), where \(X\in K\). Statements in the introduction of the paper indicate that earlier results of \textit{W.\,B.\thinspace Johnson} and \textit{G.\,Schechtman} [Ann.\ Probab.\ 17, 789--808 (1989; Zbl 0674.60051)] include special cases in which \(F_k(t)=f_k(t-k+1)\), \(k=1,2,\dots\;\).