On plate decompositions of cone multipliers

Abstract

Abstract. 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 .

Lebesgue space estimates for a class of Fourier integral operators associated with wave propagation

“…In the constant coefficient case one can recover from Theorem 1.1 the space time estimates of [6] which correspond to an endpoint version of Sogge's conjecture in the range given for ℓ = d − 1, see also §4 for other generalizations. For previous partial results on Sogge's conjecture, also in lower dimensions, see the groundbreaking paper of Wolff [20] and the subsequent papers [12], [5].…”

A mixed norm variant of Wolff’s inequality for paraboloids

“…1 decompositions of cone multipliers in R d+1 (see [13], [7], [5], [4]). Let α(p) := d( As before, the power α(p) is optimal for each p (except for ε > 0), and the inequality is conjectured to hold for all p > 2 + ).…”

“…Theorem 1.1 states that the stronger mixed norm inequality holds in the same range as the currently known range for the Wolff inequality (1.4) (cf. [5]), that is for p ≥ 2 + . We also remark that the resolution of the problem for the paraboloid is necessary for the corresponding problems for cones in R d+1 .…”

“…In dimensions d ≥ 3 it is conjectured, but not known, that (1.5) holds on L q 0 (R d ) with q 0 = 2d/(d − 1). Partial results in higher dimensions follow from the bilinear adjoint restriction theorem of Tao [11] using arguments in [8], [5]; indeed (1.5) is known to hold for q > 2 + 4/d. By Minkowski's inequality the square function bound (1.5) also implies the weaker (and possibly non optimal)…”

An endpoint estimate for the cone multiplier

“…But compared to the BochnerRiesz multiplier, little is known about the cone multiplier, and this conjecture still remains open for any d ≥ 2. There are some partial results for this conjecture (see [3,4,5,6,7,10]). In particular, in [10], the first author proved that T δ is an L p -bounded operator for 1 ≤ p ≤ 2(d − 1)/(d + 1), δ > δ(p) and d ≥ 4.…”