An inequality for first-order differences (Q558506)

Let \(1\leq p<\infty\) and suppose \(f:[0,\infty)\to\mathbb R\). For \(h>0\), set \[ F(h)\equiv\| \Delta_hf\| _{L^ p(0,h)}=\left(\int_0^ h| f(x+h)-f(x)| ^ p\,dx\right)^{1/p} \] and \[ G(h)\equiv\| \Delta_hf\| _{L^ p(h,3h)}=\left(\int_h^{3h}| f(x+h)-f(x)| ^ p\,dx\right)^{1/p}. \] The author studies the question of comparison of \(F(h)\) and \(G(h)\). It is obvious that pointwise comparisons are impossible, but norm comparisons are possible. In particular, \textit{V. I. Burenkov} and \textit{A. Senouci} [J. Comput. Appl. Math. 171, No.~1/2, 141--149 (2004; Zbl 1069.26014)] proved such~comparison in weighted Lebesgue norms with certain power weights. More precisely, they showed that, for \(\delta\in(0,\infty]\), \(l>0\), and \(\theta\in(1,\infty]\), \[ \left(\int_0^{\delta}F(h)^{\theta}h^{-l\theta}\frac{dh}h\right)^{1/{\theta}} \leq A \left(\int_0^{\delta}G(h)^{\theta}h^{-l\theta}\frac{dh}h\right)^{1/{\theta}} \] for some \(A>0\) and all measurable \(f\). In the same paper, they raised a~question whether their inequality still holds when the constant 3 in the definition of \(G\) is replaced by a smaller number. The author of the paper under review answers this question in the positive. He shows an ingenious key lemma comparing terms of the form \(\| \Delta_{th}\| _{L^ p(b,c)}\), which is then used to establish several interesting results, including a~fairly general theorem giving a~`norm-in-norm' estimate from which it follows that norms of \(F\) and \(G\) and other similar expressions are all comparable in great generality. In particular, it follows that the above inequality holds when 3 is replaced by any \(\beta>1\). The author states an interesting question of extending his results to higher order differences (his techniques do not apply to that case).