Harmonic analysis related to Schrödinger operators

Abstract

In this article we give an overview on some recent development of Littlewood-Paley theory for Schrödinger operators. We extend the Littlewood-Paley theory for special potentials considered in the authors' previous work. We elaborate our approach by considering potential in C ∞ 0 or Schwartz class in one dimension. In particular the low energy estimates are treated by establishing some new and refined asymptotics for the eigenfunctions and their Fourier transforms. We give maximal function characterization of the Besov spaces and Triebel-Lizorkin spaces associated with H. We then prove a spectral multiplier theorem on these spaces and derive Strichartz estimates for the wave equation with a potential. We also consider similar problem for the unbounded potentials in the Hermite and Laguerre cases, whose V = a|x| 2 + b|x| −2 are known to be critical in the study of perturbation of nonlinear dispersive equations. This improves upon the previous results when we apply the upper Gaussian bound for the heat kernel and its gradient.

Littlewood-Paley decompositions on manifolds with ends

“…In [7], the authors develop a dyadic Littlewood-Paley theory for tensors on compact surfaces with limited regularity (but in low dimension) which is of great interest for nonlinear applications. In [10], the L p equivalence of norms for dyadic square functions (including small frequencies) associated to Schrödinger operators are proved for a restricted range of p. See also the recent survey [9] for Schrödinger operators on R n . In the present paper, we shall use the analysis of ϕ(−h 2 ∆ g ) for h ∈ (0, 1], obtained in [1], to derive Littlewood-Paley inequalities on manifolds with ends.…”

Interpolation theorems for self-adjoint operators

“…This is the same condition assumed in [28,18] except that we drop the gradient estimate condition on the kernel. This is the case when L is a Schrödinger operator −Δ + V , V ≥ 0 belonging to L 1 loc (R n ) [14,19] or L is a uniformly elliptic operator in L 2 (R n ) [8,Theorem 3.4.10].…”

“…There are interesting discussions on interpolation theory in [20] and [26,25,22] for generalized Besov spaces associated with differential operators, which require certain Riesz summability for L that seems a nontrivial 81 condition to verify. Nevertheless, we would like to mention that the Riesz summability, the spectral multiplier theorem and the decay estimate in (1) are actually intimately related [10,18].…”

“…From [14], [28] or [18] we know that if the heat kernel of L satisfies the upper Gaussian bound |e −tL (x, y)| ≤ c n t −n/2 e −c|x−y| 2 /t ,…”

Spectral multipliers for Schrödinger operators

“…Under additional smoothness condition on V , one can identify F α,q p (H) = F 2α,q p (R n ), which allows us to obtain the boundedness of µ(H) on F α,q p and B α,q p spaces on R n according to Theorem 2.3, cf. [26,34].…”