The Stein-Tomas inequality in trace ideals

Abstract

The Stein-Tomas inequality in trace ideals Séminaire Laurent Schwartz -EDP et applications (2015-2016

On the Strichartz estimates for orthonormal systems of initial data with regularity

“…which gives (1.2) with α = 1 without making use of the orthogonality; the pertinent point here is to raise α as far as possible by capitalising on the orthogonality of the f j . Of course, the above "trivial" argument in (1.4) can be used for a larger range of q than the range q ∈ [1, d+1 d−1 ) in Theorem 1.1, but interestingly, at the Keel-Tao endpoint (p, q) = (1, d d−2 ) where q is as large as possible (in this discussion, we are assuming d ≥ 3), the exponent α = 1 cannot be improved (see [15]). It follows that q = d+1 d−1 plays the role of an endpoint in the context of (1.2).…”

“…Thus, for d ≥ 3 and q ∈ ( d+1 d−1 , d d−2 ), the only remaining issue is the critical case α = p; such estimates would follow by interpolation if the following interesting conjecture (raised in [13]; see also [15]) were true. Conjecture 1.3.…”

“…(Here, as in Section 3, we are using the notation · (p,r),q = · L p,r t L q x and · p,q = · L p t L q x .) Estimates of this type were established by Frank-Sabin [15] by making use of interpolation along an analytic family of operators for which we only have a limited class of estimates. Here, we proceed more concretely and start by decomposing the operator W 1 U U * W 2 by writing…”

Strichartz estimates for orthonormal families of initial data and weighted oscillatory integral estimates

“…Also, the sharp value of reaches its maximum at the point ( , ) = ( 2( +1) , 2( +1) −1 ). In fact, as pointed out in [31], the estimates in ( ) follow from those in ( ) by interpolation between ( , ) = (2, 2 −2 ) and points arbitrarily close to ( , ) = ( 2( +1) , 2( +1) −1 ); in this sense, ( , ) = ( 2( +1) , 2( +1) −1 ) may be considered as an endpoint case of the estimate (1.11). It remains open whether one can establish a suitable estimate in weaker form with ( , ) = ( 2( +1) , 2( +1) −1 ) 1 so that other sharp estimates can be recovered from it by interpolation.…”

“…Other applications include consequences for the wave operator for time-dependent potentials, which may be found in [29]. In a somewhat different direction, one may obtain refined versions of the classical (single-function) Strichartz estimates for data in certain Besov spaces rather quickly from (1.1) via the Littlewood-Paley inequality; this observation may be found in [31] and provides an approach to refined Strichartz inequalities, which is distinct and rather simpler than the more well-known approach via bilinear Fourier (adjoint) restriction estimates. In addition to the papers cited already, we refer the reader forward to Section 7, where we present several applications of the nature described above; these applications will be consequences of the new estimates we obtain in the current paper and on which we now begin to focus.…”