This paper may be viewed as a companion paper to Greenblatt (arXiv:1910.04547). In that paper, \(L^2\) Sobolev estimates derived from a Newton polyhedron-based resolution of singularities method are combined with interpolation arguments to prove \(L^p\) to \(L^q_s\) estimates, some sharp up to endpoints, for translation invariant Radon transforms over hypersurfaces and related operators. Here \(q \ge p\) and s can be positive, negative, or zero. In this paper, we instead use \(L^2\) Sobolev estimates derived from the resolution of singularities methods of Greenblatt (arXiv:1810.10507) and combine with analogous interpolation arguments, again resulting in \(L^p\) to \(L^q_s\) estimates for translation invariant Radon transforms for \(q \ge p\) which can be sharp up to endpoints. It will turn out that sometimes the results of this paper are stronger, and sometimes the results of Greenblatt (arXiv:1910.04547) are stronger. As in Greenblatt (submitted), some of the sharp estimates of this paper occur when \(s = 0\), thereby giving some new sharp \(L^p\) to \(L^q\) estimates for such operators, again up to endpoints. Our results lead to natural global analogues whose statements can be recast in terms of a hyperplane integrability condition analogous to that of Iosevich and Sawyer in their work (Adv Math 132(1):46–119, 1997) on the \(L^p\) boundedness of maximal averages over hypersurfaces.

可以将本文视为Greenblatt（arXiv：1910.04547）的配套文件。在那篇论文中，从基于牛顿多面体的奇异性解析方法得出的\（L ^ 2 \） Sobolev估计与内插参数相结合以证明\（L ^ p \）到\（L ^ q_s \）估计，有些敏锐直至端点，用于超曲面和相关算子上的平移不变Radon变换。这里\（Q \ GE p \）和小号可以是正，负或零。在本文中，我们改用\（L ^ 2 \） Sobolev估计值，这些估计值是从Greenblatt的奇异性方法的解析中得出的（arXiv：1810.10507），并与类似的插值参数组合，再次得出\（L ^ p \）到\（L ^ q_s \）估计\（q \ ge p \）的平移不变Radon变换，该估计可以精确到端点。结果表明，有时本文的结果更强，而有时Greenblatt（arXiv：1910.04547）的结果更强。就像在Greenblatt（提交）中一样，当\（s = 0 \）时，会发生一些本文的精确估计，从而为此类算子提供一些新的\（L ^ p \）至\（L ^ q \）估计，再次达到端点。我们的结果导致了自然的全局类似物，其陈述可以根据Iosevich和Sawyer在其（Adv Math 132（1）：46–119，1997）上的超平面可积性条件在\（L ^ p \） 超曲面上的最大平均值的有界性。