Some Sharp Bilinear Space–time Estimates for the Wave Equation

Abstract

Abstract. We prove a family of sharp bilinear space-time estimates for the half-wave propagator e it √ −∆ . As a consequence, for radially symmetric initial data, we establish sharp estimates of this kind for a range of exponents beyond the classical range.

Some sharp null-form type estimates for the Klein--Gordon equation

“…For the Schrödinger equation, in addition to the Ozawa-Tsutsumi estimates in [34], estimates resembling (1.1) may be found in work of Carneiro [13] and Planchon-Vega [35], with a unification of each of these results by Bennett et al in [5]. For the wave equation, Bez-Rogers [9] and Bez-Jeavons-Ozawa [11] have established estimates resembling (1.1). We also remark that the related literature on sharp Strichartz estimates is large.…”

On sharp bilinear Strichartz estimates of Ozawa–Tsutsumi type

“…It is interesting to contrast our observations with the case of the half-wave propagator e it √ −∆ where the situation is somewhat different. Sharp space-time estimates which are analogous to those in Theorems 1.1 and 1.4 have very recently been obtained in [4], in which case the class of extremisers is the same for both uv and uv and in each case the multiplier operator is a power of | | = |∂ 2 t − ∆|.…”

Stability of sharp Fourier restriction to spheres

“…For simplicity, we shall use the stated bounds in the final computation. 7 The case j " 0 will be used later on in the argument; see §7.3.2.…”

“…This was inspired by the classical work of Strichartz [49] in 1977, which in turn appeared shortly after Beckner's celebrated sharpening of the Hausdorff-Young inequality [3]. A non-exhaustive list of works in sharp Fourier restriction theory includes [5,10,17,22,26,27,30,32,45] for the paraboloid (Schrödinger equation), [7,8,22,33,42,44] for the cone (wave equation), [14,16,31,41,43] for the hyperboloid (Klein-Gordon equation), and [4,6,9,12,20,21,29,34,35,36,46] in other related settings. We refer the reader to the survey [24] for a more detailed account on the latest developments.…”