论文标题
二元性最大功能的变化
Variation of the dyadic maximal function
论文作者
论文摘要
我们证明，对于二元最大运算符$ \ mathrm m $和每个本地集成的函数$ f \ in l^1 _ {\ mathrm {loc}}}（\ Mathbb r^d）$，具有有界变化，$ \ \\ Mathrm m f $也是本地集成的，也是$ \ mathop and Mathop and Mathop and Mathop {var mathrm} C_D \ Mathop {\ Mathrm {var}} f $对于任何维度$ d \ geq1 $。 It means that if $f\in L^1_{\mathrm{loc}}(\mathbb R^d)$ is a function whose gradient is a finite measure then so is $\nabla \mathrm M f$ and $\|\nabla \mathrm M f\|_{L^1(\mathbb R^d)}\leq c_d \ | \ nabla f \ | _ {l^1（\ mathbb r^d）} $。 我们还为局部二元最大运算符证明了这一点。
We prove that for the dyadic maximal operator $\mathrm M$ and every locally integrable function $f\in L^1_{\mathrm{loc}}(\mathbb R^d)$ with bounded variation, also $\mathrm M f$ is locally integrable and $\mathop{\mathrm{var}}\mathrm M f\leq C_d\mathop{\mathrm{var}} f$ for any dimension $d\geq1$. It means that if $f\in L^1_{\mathrm{loc}}(\mathbb R^d)$ is a function whose gradient is a finite measure then so is $\nabla \mathrm M f$ and $\|\nabla \mathrm M f\|_{L^1(\mathbb R^d)}\leq C_d\|\nabla f\|_{L^1(\mathbb R^d)}$. We also prove this for the local dyadic maximal operator.
