Smoothing theorems for Radon transforms over hypersurfaces and related operators

Abstract

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.