论文标题
变异操作员的不平等
Inequalities For Variation Operator
论文作者
论文摘要
令$ f $为在$ \ mathbb {r} $上定义的可测量函数。对于每个$ n \ in \ mathbb {z} $定义运算符$ a_n $ by $$ a_nf（x）= \ frac {1} {2^n} {2^n} \ int_x^{x+2^n} f（y） $ \ MATHCAL {V} f（x）= \ left（\ sum_ {n = - \ infty}^\ infty | a_nf（x）-a_ {n-1} f（x）|^s \ right）^{1/s} $$ for $ 2 \ \ leq s <\ leq s <\ infty $。已在\ cite {jkw1}中证明了$ \ mathcal {v} $是强型$（p，p）$的$ 1 <p <\ infty $，并且类型$（1,1）$，它将$ l^\ l^\ iffty映射到bmo。我们首先为这些已知结果提供了完全不同的证据，此外，我们还证明了$ \ Mathcal {v} $ maps $ h^1 $ to $ l^1 $。此外，我们证明它满足了矢量值加权强的类型和弱类型不平等。作为一种特殊情况，$ \ MATHCAL {V} $满足加权强型和弱类型不等式。
Let $f$ be a measurable function defined on $\mathbb{R}$. For each $n\in\mathbb{Z}$ define the operator $A_n$ by $$A_nf(x)=\frac{1}{2^n}\int_x^{x+2^n}f(y)\, dy.$$ Consider the variation operator $$\mathcal{V}f(x)=\left(\sum_{n=-\infty}^\infty|A_nf(x)-A_{n-1}f(x)|^s\right)^{1/s}$$ for $2\leq s<\infty$. It has been proved in \cite{jkw1} that $\mathcal{V}$ is of strong type $(p,p)$ for $1<p<\infty$ and is of weak type $(1,1)$, it maps $L^\infty$ to BMO. We first provide a completely different proofs for these known results and in addition we prove that $\mathcal{V}$ maps $H^1$ to $L^1$. Furthermore, we prove that it satisfies vector-valued weighted strong type and weak type inequalities. As a special case it follows that $\mathcal{V}$ satisfies weighted strong type and weak type inequalities.
