论文标题
$ h^p $空间上的variaiton和$λ$ - 跳跃不平等
Variaiton and $λ$-jump inequalities on $H^p$ spaces
论文作者
论文摘要
让$ \ intscr {s} $与$ \ intcr（x）\，dx = 1 $，并定义$$ ϕ_t（x）= \ frac {1} {t^n} ϕ（\ frac {x} {t} {t}），$ thefiald function $ \ \ ars_t f（x）\} _ {t> 0} $ by $φ\ ast f（x）$。假设存在一个常数$ C_1 $，以便$ \ sum_ {t> 0} | \ hat ϕ_t（x）|^2 <c_1 $$ for ash \ mathbb {r}^n $中的所有$ x \。然后 （i）存在常数$ c_2> 0 $，以便$ \ | \ mathscr {v} _2（φ\ ast f）\ | _ {l^p} \ leq c_2 \ | f \ | f \ | __________________ {h^p} h^p（\ mathbb {r}^n）$，$ \ frac {n} {n+1} <p \ leq 1 $。 （ii）$λ$ -JUMP操作员$n_λ（φ\ ast f）$满足$$ \ |λ[n_λ（φ\ ast f）]^{1/2} \ | _ {l^p} \ leq c_3 \ | f \ | _ {h^p}，\; \; \ frac {n} {n+1} <p \ leq 1，$λ> 0 $ in $ c_3> 0 $均匀地在$λ> 0 $中。
Let $ϕ\in \mathscr{S}$ with $\intϕ(x)\, dx=1$, and define $$ϕ_t(x)=\frac{1}{t^n}ϕ(\frac{x}{t}),$$ and denote the function family $\{ϕ_t\ast f(x)\}_{t>0}$ by $Φ\ast f(x)$. Suppose that there exists a constant $C_1$ such that $$\sum_{t>0} |\hatϕ_t(x)|^2<C_1$$ for all $x\in \mathbb{R}^n$. Then (i) There exists a constant $C_2>0$ such that $$\|\mathscr{V}_2(Φ\ast f)\|_{L^p}\leq C_2\|f\|_{H^p},\;\;\frac{n}{n+1}<p\leq 1$$ for all $f\in H^p(\mathbb{R}^n)$, $\frac{n}{n+1}<p\leq 1$. (ii) The $λ$-jump operator $N_λ(Φ\ast f)$ satisfies $$\|λ[N_λ(Φ\ast f)]^{1/2}\|_{L^p}\leq C_3\|f\|_{H^p},\;\;\frac{n}{n+1}<p\leq 1,$$ uniformly in $λ>0$ for some constant $C_3>0$.
