Bulletin of the
Korean Mathematical Society BKMS

Let $d\in\mathbb{N}$ and ${\alpha}\in(0,\min\{2,d\})$. For any $a\in[a^\ast,\infty)$, the fractional Schr\"odinger operator $\mathcal{L}_a$ is defined by \begin{equation*} \mathcal{L}_a:=(-\Delta)^{{\alpha}/2}+a{|x|}^{-{\alpha}}, \end{equation*} where $a^*:=-{\frac{2^{\alpha}{\Gamma}((d+{\alpha})/4)^2}{{\Gamma}((d-{\alpha})/4)^2}}$. In this paper, we study two-weight Sobolev inequalities associated with $\mathcal{L}_a$ and two-weight norm estimates for several square functions associated with $\mathcal{L}_a$.

Keywords: Sobolev inequality, generalized Hardy operator, fractional Schr\"odinger operator, square function, weighted norm inequality

MSC numbers: 35R11, 42B35, 42B20, 47B38, 35B65

Supported by: This work is partially supported by the National Natural Science Foundation of China (Grant Nos.~11871254 and 12071431), the Key Project of Gansu Provincial National Science Foundation (Grant No.~23JRRA1022), the Fundamental Research Funds for the Central Universities (Grant No.~lzujbky-2021-ey18) and the Innovative Groups of Basic Research in Gansu Province (Grant No.~22JR5RA391).