超曲面上Radon变换的光滑定理及相关算子
Smoothing theorems for Radon transforms over hypersurfaces and related operators
摘要：
本文将[G 1]中关于超曲面上平移不变Radon变换和分数阶奇异Radon变换的$L^p$的定理推广到$L^p_s$ Sobolev改进，证明了这类算子的$L^p$到$L^q_s$有界性结果.这里，$q \geq p$但$s$可以为正、负或零。对于许多这样的算子，我们将有一个三角形$Z \subset (0,1) \times (0,1) \times {\mathbb R} $，使得对于$Z$之下的$({1 \over p}, {1 \over q}, s)$，具有$L^p$到$L^q_{s}$的有界性，而在Radon变换的情况下，对于包含$Z$的平面之上的$({1 \over p}, {1 \over q}, s)$，不具有$L^p$到$L^q_{s}$的有界性，从而提供了一个索伯列夫空间改进结果，该结果对于$Z$之下的$({1 \over p}, {1 \over q})$，直到端点都是尖锐的。该三角形$Z$与平面$\{(x_1,x_2,x_3): x_3 = 0\}$相交，因此我们也具有$L^p$到$L^q$的改进结果，该改进结果对于$p$和$q$的某些范围也是尖锐的直到端点。
英文摘要：
We extend the theorems of [G1] on $L^p$ to $L^p_s$ Sobolev improvement for translation invariant Radon and fractional singular Radon transforms over hypersurfaces, proving $L^p$ to $L^q_s$ boundedness results for such operators. Here $q \geq p$ but $s$ can be positive, negative, or zero. For many such operators we will have a triangle $Z \subset (0,1) \times (0,1) \times {\mathbb R} $ such that one has $L^p$ to $L^q_{s}$ boundedness for $({1 \over p}, {1 \over q}, s)$ beneath $Z$, and in the case of Radon transforms one does not have $L^p$ to $L^q_{s}$ boundedness for $({1 \over p}, {1 \over q}, s)$ above the plane containing $Z$, thereby providing a Sobolev space improvement result which is sharp up to endpoints for $({1 \over p}, {1 \over q})$ below $Z$. This triangle $Z$ intersects the plane $\{(x_1,x_2,x_3): x_3 = 0\}$, and therefore we also have an $L^p$ to $L^q$ improvement result that is also sharp up to endpoints for certain ranges of $p$ and $q$.
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