SIAM Journal on Mathematical Analysis, Volume 52, Issue 4, Page 4068-4100, January 2020.
We prove the sharp regularizing estimates for the gain term of the Boltzmann collision operator, including hard sphere, hard potential, and Maxwell molecule models. Our new estimates characterize both the regularization and the convolution properties of the gain term and have the following features. The regularizing exponent is sharp both in the $L^2$ based inhomogeneous Sobolev spaces and the homogeneous Sobolev spaces, which is the exact exponent of the kinetic part of the collision kernel. The functions in these estimates belong to a wider scope of (weighted) Lebesgue spaces than the previous regularizing estimates. For the estimates in homogeneous Sobolev spaces, never seen before, we only need functions lying in Lebesgue spaces instead of weighted Lebesgue spaces; i.e., no loss of weight occurs in this case.

中文翻译：

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Boltzmann碰撞算子的增益项的尖锐正则估计

SIAM数学分析杂志，第52卷，第4期，第4068-4100页，2020年1月。
我们证明了玻耳兹曼碰撞算子的增益项的精确正则估计，包括硬球，硬势和麦克斯韦分子模型。我们的新估计值表征了增益项的正则化和卷积特性，并具有以下特征。在基于$ L ^ 2 $的非均匀Sobolev空间和均匀Sobolev空间中，正则化指数都非常尖锐，这是碰撞核动力学部分的精确指数。与先前的正则化估计相比，这些估计中的函数属于（加权）Lebesgue空间更大的范围。对于以前从未见过的齐次Sobolev空间中的估计，我们只需要位于Lebesgue空间中的函数，而不是加权Lebesgue空间；也就是说，在这种情况下不会出现体重减轻。