Abstract We extend the theorems of [M. Greenblatt, L p L^{p} Sobolev regularity of averaging operators over hypersurfaces and the Newton polyhedron, J. Funct. Anal. 276 2019, 5, 1510–1527] on L p {L^{p}} to L s p {L^{p}_{s}} Sobolev improvement for translation invariant Radon and fractional singular Radon transforms over hypersurfaces, proving L p {L^{p}} to L s q {L^{q}_{s}} boundedness results for such operators. Here q ≥ p {q\geq p} but s can be positive, negative, or zero. For many such operators we will have a triangle Z ⊂ ( 0 , 1 ) × ( 0 , 1 ) × ℝ {Z\subset(0,1)\times(0,1)\times{\mathbb{R}}} such that one has L p {L^{p}} to L s q {L^{q}_{s}} boundedness for ( 1 p , 1 q , s ) {({1\over p},{1\over q},s)} beneath Z, and in the case of Radon transforms one does not have L p {L^{p}} to L s q {L^{q}_{s}} boundedness for ( 1 p , 1 q , s ) {({1\over p},{1\over q},s)} above the plane containing Z, thereby providing a Sobolev space improvement result which is sharp up to endpoints for ( 1 p , 1 q ) {({1\over p},{1\over q})} below Z. This triangle Z intersects the plane { ( x 1 , x 2 , x 3 ) : x 3 = 0 } {\{(x_{1},x_{2},x_{3}):x_{3}=0\}} , and therefore we also have an L p {L^{p}} to L q {L^{q}} improvement result that is also sharp up to endpoints for certain ranges of p and q.

中文翻译：

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Radon 变换在超曲面和相关算子上的平滑定理

摘要 我们扩展了 [M. Greenblatt, L p L^{p} 超曲面和牛顿多面体上平均算子的 Sobolev 规律，J. Funct。肛门。276 2019, 5, 1510–1527] 在 L p {L^{p}} 到 L sp {L^{p}_{s}} Sobolev 改进用于平移不变 Radon 和分数奇异 Radon 变换在超曲面上，证明 L p {L^{p}} 到 L sq {L^{q}_{s}} 此类运算符的有界结果。这里 q ≥ p {q\geq p} 但 s 可以是正数、负数或零。对于许多这样的运算符，我们将有一个三角形 Z ⊂ ( 0 , 1 ) × ( 0 , 1 ) × ℝ {Z\subset(0,1)\times(0,1)\times{\mathbb{R}}}使得对于 ( 1 p , 1 q , s ) {({1\over p},{1\在 Z 下方的 q},s)} 上，并且在 Radon 变换的情况下，对于 ( 1 p , 1 q ,