论文标题
振荡不平等现象，$ h^1 $空间
Oscillation inequalities on real and ergodic $H^1$ spaces
论文作者
论文摘要
令$（x_n）$为序列，$ρ\ geq 1 $。对于固定序列$ n_1 <n_2 <n_3 <\点$，$ m $定义振荡操作员$$ \ MATHCAL {o}_ρ（x_n）= \ left（\ sum_ {k = 1}^\ sup iftty \ sup iftty \ sup _ sup _ sup _ sup _ { M}}\left|x_m-x_{n_k}\right|^ρ\right)^{1/ρ}.$$ Let $(X,\mathscr{B} ,μ, τ)$ be a dynamical system with $(X,\mathscr{B} ,μ)$ a probability space and $τ$ a measurable, invertible, measure preserving point transformation从$ x $到自身。然后，我们证明了$ρ\ geq 2 $的以下结果： （i）定义$ drac {1} {n}χ_ {[0，n]}（x）$上的$ drac {1} {1} {1} {n}χ_ = \ frac {1} {n} {n}χ_ {n} {n} = \ frac {n} {n} {n} {n} {n} {n} {n} {n} {n} {n} {n} {n}（x）$ on $ \ m athbb {r} $。然后存在一个常数$ c> 0 $，以至于$ \ | \ mathcal {o}_ρ（ϕ_n \ ast f） h^1（\ mathbb {r}）$。 （ii）让$ a_nf（x）= \ frac {1} {n} \ sum_ {k = 1}^nf（τ^kx）$是ergodic理论中通常的ergodic平均值。然后，$ \ | \ | \ MATHCAL {O}_ρ（a_nf）\ | _ {l^1（x）} \ leq c \ | f \ | _ {h^1（x）} $ for H^1（x）$ in H^1（x）$。 （iii）如果$ [f（x）\ log（x）]^+$是可集成的，则$ \ mathcal {o}_ρ（a_nf）$是可集成的。
Let $(x_n)$ be a sequence and $ρ\geq 1$. For a fixed sequences $n_1<n_2<n_3<\dots$, and $M$ define the oscillation operators $$\mathcal{O}_ρ(x_n)=\left(\sum_{k=1}^\infty\sup_{\substack{n_k\leq m< n_{k+1}\\m\in M}}\left|x_m-x_{n_k}\right|^ρ\right)^{1/ρ}.$$ Let $(X,\mathscr{B} ,μ, τ)$ be a dynamical system with $(X,\mathscr{B} ,μ)$ a probability space and $τ$ a measurable, invertible, measure preserving point transformation from $X$ to itself.\\ Suppose that the sequences $(n_k)$ and $M$ are lacunary. Then we prove the following results for $ρ\geq 2$: (i) Define $ϕ_n(x)=\frac{1}{n}χ_{[0,n]}(x)$ on $\mathbb{R}$. Then there exists a constant $C>0$ such that $\|\mathcal{O}_ρ(ϕ_n\ast f)\|_{L^1(\mathbb{R})}\leq C\|f\|_{H^1(\mathbb{R})}$ for all $f\in H^1(\mathbb{R})$. (ii) Let $A_nf(x)=\frac{1}{n}\sum_{k=1}^nf(τ^kx)$ be the usual ergodic averages in ergodic theory. Then $\|\mathcal{O}_ρ(A_nf)\|_{L^1(X)}\leq C\|f\|_{H^1(X)}$ for all $f\in H^1(X)$. (iii) If $[f(x)\log (x)]^+$ is integrable, then $\mathcal{O}_ρ(A_nf)$ is integrable.
