论文标题
加权Sobolev不平等的可接受功能空间
Admissible function spaces for weighted Sobolev inequalities
论文作者
论文摘要
令$ k，n \ in \ mathbb {n} $带$ 1 \ le k \ le n $，让$ω=ω_1\ timesω_2$是$ \ mathbb {r}^k \ times \ times \ mathbb {r}^r}^{r}^{n-k} $中的打开集。对于$ p \ in（1，\ infty）$和$ q \ in（0，\ infty），我们考虑以下Hardy-Sobolev类型不等式： \ begin {align} \int_Ω| g_1（y）g_2（z）| | u（y，z）|^q \，dy \，dz \ leq c \ left（\int_Ω| \ nabla u（y，z）|^p \，dy \，dz \ right）^{\ frac {q} {q} {q} {p}} {p}} {p}} {p}} { \ end {align} 对于某些$ C> 0 $。根据$ n，k，p，q，$的值，我们已经确定了各对lorentz空间，lorentz-zygmund空间和加权的Lebesgue Space，以$（g_1，g_2）$，以便上述不平等。此外，我们在$ g_1，g_2 $上给出了足够的条件u \ | _ {l^p（ω）} $。
Let $k,N \in \mathbb{N}$ with $1\le k\le N$ and let $Ω=Ω_1 \times Ω_2$ be an open set in $\mathbb{R}^k \times \mathbb{R}^{N-k}$. For $p\in (1,\infty)$ and $q \in (0,\infty),$ we consider the following Hardy-Sobolev type inequality: \begin{align} \int_Ω |g_1(y)g_2(z)| |u(y,z)|^q \, dy \, dz \leq C \left( \int_Ω | \nabla u(y,z) |^p \, dy \, dz \right)^{\frac{q}{p}}, \quad \forall \, u \in \mathcal{C}^1_c(Ω), \end{align} for some $C>0$. Depending on the values of $N,k,p,q,$ we have identified various pairs of Lorentz spaces, Lorentz-Zygmund spaces and weighted Lebesgue spaces for $(g_1, g_2)$ so that the above inequality holds. Furthermore, we give a sufficient condition on $g_1,g_2$ so that the best constant in the above inequality is attained in the Beppo-Levi space $\mathcal{D}^{1,p}_0(Ω)$-the completion of $\mathcal{C}^1_c(Ω)$ with respect to $\|\nabla u\|_{L^p(Ω)}$.
