Uniqueness in the Calderón problem in dimension bigger than two was usually studied under the assumption that conductivity has bounded gradient. For conductivities with unbounded gradients uniqueness results have not been known until recent years. The latest result due to Haberman basically relies on the optimal $L^2$ restriction estimate for hypersurface which is known as the Tomas-Stein restriction theorem. In the course of developments of the Fourier restriction problem bilinear and multilinear generalizations of the (adjoint) restriction estimates under suitable transversality condition between surfaces have played important roles. Since such advanced machineries usually provide strengthened estimates, it seems natural to attempt to utilize these estimates to improve the known results. In this paper, we make use of the sharp bilinear restriction estimates, which is due to Tao, and relax the regularity assumption on conductivity. We also consider the inverse problem for the Schrödinger operator with potentials contained in the Sobolev spaces of negative orders.

通常在假设电导率有界梯度的情况下研究 Calderón 问题中维度大于 2 的唯一性。对于具有无界梯度的电导率唯一性结果直到最近几年才知道。由于 Haberman 的最新结果基本上依赖于最优${升}^2$超曲面的限制估计，称为 Tomas-Stein 限制定理。在傅里叶限制问题的发展过程中，表面之间合适的横向条件下（伴随）限制估计的双线性和多线性推广发挥了重要作用。由于这种先进的机器通常提供强化的估计，因此尝试利用这些估计来改进已知结果似乎很自然。在本文中，我们利用由陶引起的尖锐双线性限制估计，并放宽对电导率的规律性假设。我们还考虑了包含在负阶 Sobolev 空间中的势的薛定谔算子的逆问题。