A Bessel Function Multiplier

We obtain nearly sharp estimates for the L p (R 2) norms of certain convolution operators. For n 1 let n be the measure on R 2 obtained by multiplying normalized arclength measure on fjxj = 1g by the oscillating factor e inarg(x). For 1 p 1, let C(p; n) denoted the norm of the operator T n f : = n f on L p (R 2). The purpose of this note is to estimate the rate of decay of C(p; n) as n ! 1. By duality, it is enough to consider p 2. Examples below will show that C(p; n) C(p)n ? 1 6 ? 1 3p if 2 p 4; (1) and C(p; n) C(p)n ? 1 p if 4 p 1: (2) On the other hand, we will observe that C(2; n) Cn ? 1 3 C(1; n) C (3) and then prove the following result. Theorem. There is a positive number a such that C(4; n) Cn ? 1 4 (log(n)) a : (4) Interpolating (3) and (4) gives upper bounds for C(p; n) which diier only by a power of log(n) from the lower bounds of (1) and (2), thus providing nearly sharp estimates for C(p; n). The above question naturally arises when considering the L p (R 3) mapping properties of the operator T given by convolution with respect to a compact piece of arclength measure on the helix t ! (cos t; sin t; t) : T is an example of a folding Fourier integral operator in dimension 3, whose singular set is of dimension 1. The sharp L p ! L 2 mapping properties of T were established by the rst author in 1991 Mathematics Subject Classiication. 42B15, 42B20.