论文标题
急剧加权的分数硬度不平等
Sharp weighted fractional Hardy inequalities
论文作者
论文摘要
我们调查加权分数订单Hardy不平等$$ \int_Ω\int_Ω\ frac {| f（x）-f（y）|^{p}} {| x-y |^{d+sp}} \ text {dist}（x，x，\partialΩ）^{ - α} { - α} c \int_Ω\ frac {| f（x）|^{p}}} {\ text {dist}（x，x，x，\partialΩ）^{sp+α+β}} \，dx，$ $ $ω= \ Mathbb {r}^d \ setMinus \ {0 \} $。在所有情况下，我们的工作着重于找到最佳（即尖锐）常数$ c = c（d，s，p，α，β）$。我们还获得了分数Hardy-Sobolev-Maz'ya的加权版本。这些证明是基于弗兰克和塞金林格建立的一般性不平等和非线性基态表示。
We investigate the weighted fractional order Hardy inequality $$ \int_Ω\int_Ω\frac{|f(x)-f(y)|^{p}}{|x-y|^{d+sp}}\text{dist}(x,\partialΩ)^{-α}\text{dist}(y,\partialΩ)^{-β}\,dy\,dx\geq C\int_Ω\frac{|f(x)|^{p}}{\text{dist}(x,\partialΩ)^{sp+α+β}}\,dx, $$ for $Ω=\mathbb{R}^{d-1}\times(0,\infty)$, $Ω$ being a convex domain or $Ω=\mathbb{R}^d\setminus\{0\}$. Our work focuses on finding the best (i.e. sharp) constant $C=C(d,s,p,α,β)$ in all cases. We also obtain weighted version of the fractional Hardy-Sobolev-Maz'ya inequality. The proofs are based on general Hardy inequalities and the non-linear ground state representation, established by Frank and Seiringer.
