$L^p$ improving estimates for some classes of Radon transforms

Abstract

Abstract. In this paper, we give L p − L q estimates and the L p regularizing estimate of Radon transforms associated to real analytic functions, and we also give estimates of the decay rate of the L p operator norm of corresponding oscillatory integral operators. For L p −L q estimates and estimates of the decay rate of the L p operator norm we obtain sharp results except for extreme points; however, for L p regularity we allow some restrictions on the phase function.

Sharp $$L^{p}$$ L p -boundedness of oscillatory integral operators with polynomial phases

“…By interpolation, it is easily verified that these estimates imply the sharp L 2 decay rate obtained by Phong and Stein in [22]. We shall point out that the above estimates are sharp provided that (k, l) is a vertex of the reduced Newton polyhedron of S; see [31]. If (k, l) is not a vertex but lying on the boundary of N (S), the above sharp estimates were obtained in [31].…”

“…The sharp L p − L q estimates and L p Sobolev regularity were obtained by Phong and Stein in [21] for homogeneous polynomials S except endpoint estimates. For analytic phases, endpoint L p − L q estimates are previously known and sharp L p Sobolev regularities were obtained except extreme points; see [14] and [31] as well as [1] and [2] by imposing certain left and right finite type conditions. It is notable that endpoint L p Sobolev regularity may fail; see [5].…”

“…In the case of two sided fold singularities, Greenleaf and Seeger obtained in [10] the sharp L p estimates for a smooth phase S. The sharp L p estimates were obtained in [30] for homogeneous polynomial phases S under the assumption a 1 a n−1 = 0. For an analytic phase S, sharp L p estimates of T λ had been established in [31] except some extreme points of the reduced Newton polyhedron of S. Assume S is a real analytic function near the origin. Then the power expansion gives S(x, y) = k,l≥0 a k,l x k y l near the origin.…”

Uniform Sublevel Radon-like Inequalities

“…Related ideas have appeared in the work Sogge and Stein [50], Christ [10], and many others. Along these same lines, there have been many works to explicitly establish L p -improving estimates for Radon-like operators in various circumstances, including Littman [35]; Oberlin and Stein [38]; Oberlin [39], [40]; Phong and Stein [44]; Greenleaf and Seeger [23], [24]; Iosevich and Sawyer [30] Greenleaf, Seeger, and Wainger [26], [25]; Seeger [49]; Christ [12]; Bak [1]; Bak, Oberlin, and Seeger [2]; Tao and Wright [52]; Lee [33], [34]; and Yang [55]. This listing is not exhaustive and does not include the (even larger) body of work devoted to L 2 -and L p -Sobolev estimates, which would indirectly imply (usually non-sharp) L p -improving estimates as well.…”

Uniform Estimates for Damped Radon Transform on the Plane

“…for ∈ ∞ 0 (R 2 ). Mapping properties of such operators in various function spaces have been studied by many authors [1][2][3][4][5][6][7][8][9]. Sharper estimates are available in translation-invariant cases where ( 1 , ) = ( 1 − ) with a 2 function defined on an interval [10,11] and it is widely known that the so-called affine arclength measure introduced by Drury [12] is better suited in obtaining degeneracy independent results in many interesting cases.…”