Our aim in this paper is to obtain weighted $L^{\infty }$ estimates for Poisson problems involving Hardy–Leray operator $-\Delta +\frac{\mu }{{\left|x\right|}^2}$ or elliptic operators with critical singular gradients $-\Delta u+\frac{{\tau }_{+}}{{\left|x\right|}^2}x\cdot \nabla$. Our method overcomes the difficulty from the singular potentials. Moreover, these estimates could be applied to obtain global $W^{1,p}$ estimates for distributional solution of Hardy problem with the Radon source.

中文翻译：

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关于具有临界奇异势的泊松问题的全球估计

我们本文的目的是获得加权 ${\mathrm{大号}}^{\infty }$ Hardy-Leray算子的泊松问题的估计 $-\Delta +\frac{\mu }{{\left|X\right|}^{\mathrm{2个}}}$ 或具有临界奇异梯度的椭圆算子 $-\Delta ü+\frac{{\tau }_{+}}{{\left|X\right|}^{\mathrm{2个}}}X\cdot \nabla$。我们的方法克服了奇异电位带来的困难。此外，这些估计数可用于获取全球 ${\mathrm{w\; ^}}^{\mathrm{1个}，p}$ Radon源的Hardy问题的分布解的估计。