On dimension-free variational inequalities for averaging operators in \({\mathbb{R}^d}\) (Q1746609)

The authors initiate the study of dimension-free \(r\)-variational estimates for averaging operators defined over symmetric convex bodies \(G\subset\mathbb R^d\). In particular, if \[ M_t^Gf(x)=\frac1{|G_t|}\int_{G_t}f(x-y)\,dy \] is the Hardy-Littlewood averaging operator, \(G_t\) is the \(t\)-dilation of \(G\) and \[ V_r(a_t(x):t\in Z)=\sup_{\{t_0<\dots < t_J:t_j\in Z\}}\bigg(\sum_{j=0}^J|a_{t_{j+1}}(x)-a_{t_{j}}(x)|^r\bigg)^{1/r} \] is the \(r\)-variational seminorm, for a fixed \(x\in\mathbb R^d\), of the function \(a_t(x)\), with \(1\leq r<\infty\) and \(Z\subset(0,\infty)\), then one of the main results of this paper afirms that for \(p\in(3/2,4)\) and \(r>2\), there exists \(C_{p,r}>0\), independent of the dimension \(d\in\mathbb N\) and \(G\) as above, such that \[ \|V_r(M_t^Gf:t>0)\|_{L^p}\leq C_{p,r}\|f\|_{L^p}. \] The question, for other values of \(p\in(1/3,2]\cup[4,\infty)\), remains still open. If \(Z=\{2^n:n\in\mathbb N\}\), the \(r\)- variation \(V_r\) is called the long \(r\)-variation. In this case, the previous result can be improved to consider \(p\in(1,\infty)\) and \(r\in(2,\infty)\). Again, if \(G=B_q\) is the unit ball for the \(q\)-norm in \(\mathbb R^d\), \(1\leq q\leq\infty\), the result also holds, with a constant \(C_{p,q,r}>0\), whenever \(p\in(1,\infty)\) and \(r\in(2,\infty)\). Dimension dependent versions of these results, with sharp ranges of parameters \(p\in(1,\infty)\) and \(r\in(2,\infty)\), were already proved in [\textit{M. Mirek}, \textit{E. M. Stein}, and \textit{B. Trojan}, Invent. Math. 209, No. 3, 665--748 (2017; Zbl 1397.42006)].