Title: Fractional sublinear Sobolev inequality for $\mathcal{L}-$superharmonic functions Show Chinese title

Title: 分数次线性Sobolev不等式对于$\mathcal{L}-$超调和函数

Abstract: We establish a Sobolev-type inequality in Lorentz spaces for $\mathcal{L}$-superharmonic functions \[ \|u\|_{L^{\frac{nq}{n-\alpha q},t}(\mathbb{R}^n)} \leq c \left\| \frac{u(x) - u(y)}{|x-y|^{\frac{n}{q}+\alpha}} \right\|_{L^{q,t}(\mathbb{R}^n \times \mathbb{R}^n)} \] in the sublinear case $p-1 < q < 1$ and $p-1\leq t\leq \infty$. The nonlocal nonlinear elliptic operator $\mathcal{L}$ is modeled from the fractional $p$-Laplacian $(- \Delta_{p})^{\alpha} $ with $0 < \alpha < 1$ and $1<p<2$. Related Gagliardo-Nirenberg interpolation for $\mathcal{L}$-superharmonic functions is also derived. Show Chinese abstract

Abstract: 我们建立了一个在Lorentz空间中的Sobolev型不等式，针对子线性情况下$\mathcal{L}$-超调和函数\[ \|u\|_{L^{\frac{nq}{n-\alpha q},t}(\mathbb{R}^n)} \leq c \left\| \frac{u(x) - u(y)}{|x-y|^{\frac{n}{q}+\alpha}} \right\|_{L^{q,t}(\mathbb{R}^n \times \mathbb{R}^n)} \]在$p-1 < q < 1$和$p-1\leq t\leq \infty$中。 非局部非线性椭圆算子$\mathcal{L}$是从分数阶$p$-拉普拉斯算子$(- \Delta_{p})^{\alpha} $建模而来，具有$0 < \alpha < 1$和$1<p<2$。 相关的 Gagliardo-Nirenberg 插值用于$\mathcal{L}$-超调和函数也得到了推导。