Perturbation of plane curves and sequences of integers (Q1129408)

A continuous curve \(\Gamma:[0,\infty)\to\mathbb R\) is said to differentiate \(L_{\text{loc}}^{p}({\mathbb R}^2)\) if, for any \(f\in L_{\text{loc}}^{p}({\mathbb R}^2)\), \( \lim_{s\to 0}{1\over s}\int_{0}^{s} f(x+\Gamma(t))dt=f(x) \) for a.e. \(x\in{\mathbb R}^2\), and is \(\infty\)-sweeping out for \(L^p({\mathbb R}^2)\) if there exists an \(f\in L^p({\mathbb R}^2)\) for which \( \limsup_{s\to 0^{+}}{1\over s}\int_0^sf(x+\Gamma(t))dt=\infty \) for a.e. \(x\in{\mathbb R}^2\). \textit{M. Christ} [Publ. Mat., Barc. 35, No. 1, 269-279 (1991; Zbl 0722.42009)] exhibited a large class of convex curves that differentiate \(L_{\text{loc}}^p\) for \(p>1\) and asked if there is a curve that differentiates \(L_{\text{loc}}^2\) but not \(L_{\text{loc}}^p\) for \(p<2\). In this paper a rather general construction is given of the following sort. If \(\Gamma\) differentiates \(L_{\text{loc}}^q\) for some \(q\in(1,\infty]\), then there is a `nearby' curve \(\Gamma'\) that still differentiates \(L_{\text{loc}}^q\) but is \(\infty\)-sweeping out for \(L_{\text{loc}}^p\), \(p\in[1,q)\). Fix \(q\in[1,\infty)\); then if \(\Gamma\) differentiates \(L_{\text{loc}}^p\) for each \(p>q\) there is a `nearby' curve \(\Gamma'\) that differentiates \(L_{\text{loc}}^p\) for \(p>q\) but is \(\infty\)-sweeping out for \(L^q\). The techniques used come from the ergodic theory of pointwise (non-)convergence results for non-conventional averages.