Restriction estimates, sharp spectral multipliers and endpoint estimates for Bochner-Riesz means

Abstract

Abstract. We consider abstract non-negative self-adjoint operators on L 2 (X) which satisfy the finite speed propagation property for the corresponding wave equation. For such operators we introduce a restriction type condition which in the case of the standard Laplace operator is equivalent to (p, 2) restriction estimate of Stein and Tomas. Next we show that in the considered abstract setting our restriction type condition implies sharp spectral multipliers and endpoint estimates for the BochnerRiesz summability. We also observe that this restriction estimate holds for operators satisfying dispersive or Strichartz estimates. We obtain new spectral multiplier results for several second order differential operators and recover some known results. Our examples include Schrödinger operators with inverse square potentials on R n , the harmonic oscillator, elliptic operators on compact manifolds and Schrödinger operators on asymptotically conic manifolds.

Weighted restriction type estimates for Grushin operators and application to spectral multipliers and Bochner–Riesz summability

“…On the other hand, we proved restriction type estimates for the operator L in Section 3. Therefore we may follow ideas in [1], Sections 3 and 4, to prove spectral multiplier results as well as Bochner-Riesz summability results.…”

“…Starting from the result quoted above from [9] and [10], one can use complex interpolation between L 2 boundedness for any δ > 0 and L 1 boundedness for a fixed δ > (d 1 + d 2 )/2 − 1/2 to obtain that for δ > (d 1 …”

“…The sharpened result for the Laplacian, i.e., δ > max{n|1/p − 1/2| − 1/2, 0} for 1 ≤ p ≤ (2n + 2)/(n + 3), is obtained by the restriction theorem for the Fourier transform on the unit sphere. In an abstract setting, versions of the restriction estimate are introduced in [1] and we are tempted to follow [1] in order to prove boundedness of Bochner-Riesz means for L. There is however an obstacle. The restriction type estimate introduced in [1] leads to spectral multipliers using "the" homogeneous dimension Q = d 1 + 2d 2 rather than the topological one d 1 + d 2 .…”

“…In an abstract setting, versions of the restriction estimate are introduced in [1] and we are tempted to follow [1] in order to prove boundedness of Bochner-Riesz means for L. There is however an obstacle. The restriction type estimate introduced in [1] leads to spectral multipliers using "the" homogeneous dimension Q = d 1 + 2d 2 rather than the topological one d 1 + d 2 . The exponent we will get for the Bochner-Riesz means is then max{Q|1/p−1/2|−1/2, 0}.…”

Sobolev spaces adapted to the Schrödinger operator with inverse-square potential

“…In particular, the papers [6,10] explicitly verify that the multiplier theorems proved therein apply to the operators L a for a > − ( d−2 2 ) 2 and so verify Theorem 1.1 in these cases. In fact, [6, Theorem 1.1] also yields Theorem 1.1 in the endpoint case a = − ( d−2 2 ) 2 ; to see this, one need only observe that the known heat kernel estimates (reproduced below in Theorem 2.1) guarantee that L a obeys hypothesis GGE from [6] with exponent p o if and only if d σ < p o ≤ 2.…”