On plate decompositions of cone multipliers

An important inequality due to Wolff on plate decompositions of cone multipliers is known to have consequences for sharp L p results on cone multipliers, local smoothing for the wave equation, convolutions with radial kernels, Bergman projections in tubes over cones, averages over finite-type curves in ℝ 3 and associated maximal functions. We observe that the range of p in Wolff's inequality, for the conic and the spherical versions, can be improved by using bilinear restriction results. We also use this inequality to give some improved estimates on square functions associated to decompositions of cone multipliers in low dimensions. This gives a new L 4 bound for the cone multiplier operator in ℝ 3 .