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Sobolev Spaces Adapted to the Schr\" Odinger Operator with Inverse

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Sobolev Spaces Adapted to the Schr\ SOBOLEV SPACES ADAPTED TO THE SCHRODINGER¨ OPERATOR WITH INVERSE-SQUARE POTENTIAL R. KILLIP, C. MIAO, M. VISAN, J. ZHANG, AND J. ZHENG p Abstract. We study the L -theory for the Schr¨odinger operator La with inverse-square potential a|x|−2. Our main result describes when Lp-based s/2 Sobolev spaces defined in terms of the operator (La) agree with those de- fined via (−∆)s/2. We consider all regularities 0 <s< 2. In order to make the paper self-contained, we also review (with proofs) multiplier theorems, Littlewood–Paley theory, and Hardy-type inequalities as- sociated to the operator La. Key Words: Riesz transforms, inverse-square potential; Littlewood– Paley theory; Mikhlin Multiplier Theorem; heat kernel estimate. AMS Classification: 35P25, 35Q55. 1. Introduction In this paper, we will discuss several basic harmonic analysis questions related to the Schr¨odinger operator a d 2 2 a = ∆+ x 2 with a −2 (1.1) L − | | ≥− in dimensions d 3. More precisely, we interpret a as the Friedrichs extension ≥ Rd L of this operator defined initially on Cc∞( 0 ). We discuss this more fully in d 2 2 \{ } subsection 1.1. The restriction a ( −2 ) ensures that the operator a is positive semi-definite. ≥− L For much of what follows, it is convenient to introduce a different parameteriza- tion of the family of operators , namely, via La d 2 1 2 σ := − (d 2) +4a. (1.2) 2 − 2 − d 2 This has the opposite sign to a and rangesp over ( , −2 ] as a ranges from + arXiv:1503.02716v2 [math.AP] 12 Apr 2016 d 2 2 −∞ ∞ to ( − ) . The underlying significance of σ will become apparent below. − 2 The operator a arises frequently in mathematics and physics, commonly as a scaling limit of moreL complicated problems. Several instances where this occurs in physics are discussed in the mathematical papers [7, 8, 15, 28, 29]; they range from combustion theory to the Dirac equation with Coulomb potential, and to the study of perturbations of classic space-time metrics such as Schwarzschild and Reissner–Nordstr¨om. The appearance of as a scaling limit (both microscopically and astronomi- La cally) is a signal of one of its unique properties: a is scale invariant. In particular, the potential and Laplacian can be regarded as equallyL strong at every length scale. Correspondingly, problems involving a are seldom amenable to simple perturba- tive arguments. This is one of the reasonsL that we (and indeed many before us) have singled out this particular operator for in-depth study. 1 2 R.KILLIP,C.MIAO,M.VISAN,J.ZHANG,ANDJ.ZHENG Our main motivation for developing harmonic analysis tools for a, however, comes from problems in dispersive PDEs. Specifically, the study ofL large-data problems at the (scaling-)critical regularity and the development of sharp scattering thresholds for focusing problems. A key principle in attacking such problems in that one must first study the effective equations in various limiting regimes. Given the (possibly broken, but still intrinsic) scale and space-translation invariance of such problems, one must accept that solutions may live at any possible length scale, as well as at any spatial location. Naturally, taking a scaling limit results in a scale invariant problem. Thus, the broad goal of demonstrating well-posedness of critical problems in general geometries rests on a thorough understanding of limiting problems, such as that associated to the inverse-square potential. The harmonic analysis tools developed in this paper have been essential to [16], which considers global well-posedness and scattering for both the defocusing and focusing energy- critical NLS with inverse-square potential, and to [17], in which sharp thresholds for well-posedness and scattering are established for the focusing cubic NLS with inverse-square potential. The spectral theorem allows one to define functions of the operator a and provides a simple necessary and sufficient condition for their boundednessL on L2. A sufficient condition for functions of an operator to be Lp-bounded is a basic prerequisite for the modern approach to many PDE problems. In the setting of constant-coefficient differential operators, this role is played by the classical Mikhlin Multiplier Theorem. The analogue for our operator is known: Theorem 1.1 (Mikhlin Multipliers). Suppose m : [0, ) C satisfies ∞ → j j ∂ m(λ) . λ− for all j 0 (1.3) ≥ and that either a 0 and 1 <p< , or • ≥d 2 2 ∞ d ( − ) a< 0 and r <p<r′ := . • − 2 ≤ 0 0 σ Then m(√ ) extends to a bounded operator on Lp(Rd). La In the case a 0, the operator a obeys Gaussian heat kernel bounds and so the result follows from≥ general resultsL covering this class of operators; see, for example, d 2 2 [3, 14]. When ( −2 ) a < 0, the heat kernel for a no longer obeys Gaussian bounds; indeed,− the heat≤ kernel is singular at the originL (see Theorem 2.1). In fact, as has been observed previously, the singularity is sufficiently bad that Mikhlin- type multipliers are unbounded outside the range of p given in Theorem 1.1; we give the simple details in Section 4. Thus, the stated range of p is sharp, including the question of endpoints. Several authors have developed general multiplier theorems under weaker hy- potheses on the heat kernel. In particular, the papers [6, 10] explicitly verify that d 2 2 the multiplier theorems proved therein apply to the operators a for a> ( −2 ) and so verify Theorem 1.1 in these cases. In fact, [6, Theorem 1.1]L also yields− The- d 2 2 orem 1.1 in the endpoint case a = ( −2 ) ; to see this, one need only observe that the known heat kernel estimates (reproduced− below in Theorem 2.1) guarantee that d a obeys hypothesis GGE from [6] with exponent po if and only if σ <po 2. L Nevertheless, we have elected to include here a proof of Theorem 1.1 for the≤ more d 2 2 difficult regime ( −2 ) a < 0. We do this for two principal reasons: (i) This result is a key foundation− for≤ what is done in this paper. (ii) The precise formulation SCHRODINGER¨ OPERATOR WITH INVERSE-SQUARE POTENTIAL 3 of Theorem 1.1 permits us to give a simpler and more self-contained proof than those of the papers cited above. Indeed, we are dealing with a concrete operator acting on Rd. Moreover (and more significantly), for applications to dispersive PDE, the Mikhlin condition (1.3) suffices. By comparison, the papers cited above tackle the deep and difficult problem of finding minimal regularity H¨ormander-type conditions under which multiplier operators are bounded. The argument we give to verify Theorem 1.1 is strongly influenced by [2, 9, 23]. The argument only requires bounds on a concrete number of derivates of the multiplier (see Theorem 4.1); however, in the interests of clarity and simplicity, we have made no effort to optimize this number. The square function estimates of Littlewood–Paley follow as an immediate corol- lary of the multiplier theorem via the usual randomization argument. Such esti- mates are an invaluable tool since they allow one to analyze problems one length scale at a time and then reassemble the pieces. It is convenient for us to consider two kinds of Littlewood–Paley projections: one defined via a Cc∞((0, )) multiplier and another defined as a difference of heat kernels. See Section∞ 5 for more details. The former notion more closely matches modern expositions of the classical translation-invariant theory, while the heat kernel version allows one to exploit heat-kernel bounds (and the semigroup property) to prove estimates. There is no great cost associated to either choice, since the multiplier theorem permits one to readily pass back and forth between the two notions. An example of this in action can be found in the proof of Bernstein inequalities; see Lemma 5.1. One important application of the traditional Littlewood–Paley theory is the proof of Leibniz (=product) and chain rules for differential operators of non-integer order. For example, if 1 <p< and s> 0, then ∞ fg s,p Rd . f s,p1 Rd g p2 Rd + f p3 Rd g s,p4 Rd k kH ( ) k kH ( )k kL ( ) k kL ( )k kH ( ) whenever 1 = 1 + 1 = 1 + 1 . For a textbook presentation of these theorems p p1 p2 p3 p4 and original references, see [26]. Rather than pursue a direct proof of such inequalities, we will prove a result that allows one to deduce such results directly from their Euclidean counterparts. This is the main result of this paper: d 2 2 Theorem 1.2 (Equivalence of Sobolev norms). Suppose d 3, a ( −2 ) , and s+σ 1 d σ ≥ ≥− 0 <s< 2. If 1 <p< satisfies < < min 1, − , then ∞ d p { d } s s 2 d ( ∆) 2 f . a f for all f C∞(R ). (1.4) − Lp d,p,s L Lp ∈ c s σ 1 d σ If max , < < min 1, − , which ensures already that 1 <p< , then { d d } p { d } ∞ s s 2 d a f . ( ∆) 2 f for all f C∞(R ). (1.5) L Lp d,p,s − Lp ∈ c Note that for the applications described above, both inequalities (1.4) and (1.5) are equally important. The crucial role of this result in the study of nonlinear Schr¨odinger equations with inverse-square potential is documented in [16, 17]. Here again, (1.4) and (1.5) are equally important. In the case of [16] it is also essential that Theorem 1.2 covers both the regimes where s< 1 and s> 1.
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