论文标题
端点Sobolev的界点范围用于分数硬木极限运算符
Endpoint Sobolev Bounds for Fractional Hardy-Littlewood Maximal Operators
论文作者
论文摘要
令$ 0 <α<d $和$ 1 \ leq p <d/α$。我们提供了一个证据，证明了所有$ f \ in w^{1，p}（\ Mathbb {r}^d）$均与中心和无hardy-littlewood的最大算子$ \ mathcalm_αf$是微弱的，$ \ | c_ {d，α，p} \ | \ nabla f \ | _p，$ whene $ p^* =（p^{ - 1}-α/d）^{ - 1}。 $特别涵盖了端点案例$ p = 1 $ for $ 0 <α<1 $以前未知。对于$ p = 1 $，我们可以替换$ w^{1,1}（\ mathbb {r}^d）$ by $ \ mathrm {bv}（\ mathbb {r}^d）$。所使用的成分是分数最大函数梯度，层蛋糕公式，维塔利类型的参数，从球到二元立方体的减少，coarea公式，相对等级不平等的相对等级不平等和早期确定的结果$ $α= 0 $α= 0 $。我们将其用于$α> 0 $，分数最大功能不使用某些小球。对于$α= 0 $，证明倒塌了。
Let $0<α<d$ and $1\leq p<d/α$. We present a proof that for all $f\in W^{1,p}(\mathbb{R}^d)$ both the centered and the uncentered Hardy-Littlewood fractional maximal operator $\mathcal M_αf$ are weakly differentiable and $ \|\nabla\mathcal M_αf\|_{p^*} \leq C_{d,α,p} \|\nabla f\|_p , $ where $ p^* = (p^{-1}-α/d)^{-1} . $ In particular it covers the endpoint case $p=1$ for $0<α<1$ where the bound was previously unknown. For $p=1$ we can replace $W^{1,1}(\mathbb{R}^d)$ by $\mathrm{BV}(\mathbb{R}^d)$. The ingredients used are a pointwise estimate for the gradient of the fractional maximal function, the layer cake formula, a Vitali type argument, a reduction from balls to dyadic cubes, the coarea formula, a relative isoperimetric inequality and an earlier established result for $α=0$ in the dyadic setting. We use that for $α>0$ the fractional maximal function does not use certain small balls. For $α=0$ the proof collapses.
