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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
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
See also [4, 5] for prior partial results. Moreover, the earlier paper [13] showed that for s = 1, the range of p stated above cannot be enlarged, even if one restricts f to be spherically symmetric. d 2 2 When p = 2, the equivalence at s = 1 for a > ( −2 ) is an immediate conse- quence of the sharp Hardy inequality − 2 d 2 2 f(x) 2 −2 | 2| dx f(x) dx, (1.6) Rd x ≤ Rd |∇ | Z | | Z d 2 2 which itself is just a recapitulation of the fact that a 0 when a ( −2 ) . Sharpness of the constant in (1.6) also shows that (1.4)L must≥ fail when p≥= − 2, s = 1, d 2 2 and a = ( −2 ) . This scenario coincides precisely with an excluded endpoint in Theorem− 1.2. d 2 2 When p = 2 and a> ( −2 ) , equivalence of Sobolev norms for 0 s 1 follows by complex interpolation− from the endpoints s = 0 and s = 1.≤ This≤ argument appears as Proposition 1 in [7] and relies on bounds of imaginary powers of a. These follow from the spectral theorem in the L2 setting, but are highly non-L trivial for general p. In [23] it is shown that Gaussian heat kernel bounds imply suitable bounds on imaginary powers. This is used in [29] to prove an analogue of Theorem 1.2 for a 0, albeit with a smaller set of exponents. The proof of Theorem≥ 1.2 follows a path laid down in [18]: The classical Mikhlin Multiplier Theorem together with Theorem 1.1 allow us to reformulate the norms on either side of (1.4) and (1.5) in terms of Littlewood–Paley square functions (cf. Theorem 5.3). Here we use Littlewood–Paley projections formed in terms of the heat kernel, because this makes it easy to bound the difference between the kernel associated to the operator a and that associated to ∆. By bounding the difference between these operatorsL (see Lemma 6.1), we are− ultimately lead to
1 1 2 2 2s ˜ 2 2s ˜a 2 f(x) N PN f N PN f . s . (1.7) − p Rd x p Rd N 2Z N 2Z L ( ) | | L ( ) X∈ X∈
This assertion is the content of Proposition 6.2. The proof of Theor em 1.2 is then completed by invoking the classical Hardy inequality for the Laplacian, as well as an analogue of this for the operator a that we prove in Proposition 3.2. This completes our description ofL the main topics of the paper. The exact presentation is organized as follows. This section contains two subsections. In the first we review the definition of a and justify our choice of the Friedrichs extension. In the second, we review someL basic notation. In Section 2, we use the heat kernel estimate to derive bounds on the kernel of the Riesz potential. Section 3 is devoted to proving Hardy-type inequalities associated with a. We prove Theorem 1.1 in Section 4. In Section 5, we develop a Littlewood–PaleyL theory in our setting. We prove Theorem 1.2 in Section 6.
a 1.1. a as a self-adjoint operator. Let a◦ denote the natural action of ∆+ x 2 L Rd L − | | on Cc∞( 0 ). Recall that we define a as the Friedrichs extension of a◦. The purpose of\{ this} subsection is to elaborateL on the meaning and significanceL of this. As elsewhere in the paper, we restrict attention to d 3. d 2 2 ≥ When a ( −2 ) , the operator a◦ is easily seen to be a positive semi-definite symmetric operator.≥− For example, positivityL can seen via the factorization x x 1 a◦ = + σ x 2 + σ x 2 = ∆ σ(d 2 σ) x 2 , L −∇ | | · ∇ | | − − − − | | SCHRODINGER¨ OPERATOR WITH INVERSE-SQUARE POTENTIAL 5
d with σ is as in (1.2), which shows that for φ C∞(R 0 ) we have ∈ c \{ } x 2 φ, a◦φ = φ(x)+ σ x 2 φ(x) dx 0. h L i Rd ∇ ≥ Z | |
The general theory of self-adjoint extensions now guarantees that there is a unique self-adjoint extension a of a◦, whose form domain Q( a) = D(√ a) 2 d L L d L L ⊆ L (R ) is given by the completion of C∞(R 0 ) with respect to the norm c \{ } 2 2 a 2 σx 2 2 φ Q( a) = φ + 1+ x 2 φ dx = φ + x 2 φ + φ dx. || || L Rd ∇ Rd ∇ Z | | Z | | This extension is known as the Friedrichs extension; it is also positive s emi-definite. No other self-adjoint extension has domain contained inside Q( a). In this sense (see also below), the Friedrichs extension is singled out as havingL the smallest and hence least singular quadratic form domain. For further details, see [21, X.3]. d 2 2 § When a > ( −2 ) , the sharp Hardy inequality (1.6) shows that Q( a) = 1 d − d 2 2 L H (R ). When a = ( − ) , the Hardy inequality still guarantees Q( ) − 2 La ⊇ H1(Rd); however, the reverse inclusion fails. To see this failure, one need only consider a function u(x) that is compactly supported, smooth except at the origin, and obeys d−2 1 1 u(x)= x − 2 [log(1/ x )]− 2 for x < . | | | | | | 2 This is the first of many subtleties associated with the endpoint case. We are able to give a seamless treatment of the endpoint case in this paper for two main reasons. First, unlike the Green’s function, for example, the structure of the heat kernel is insensitive to a zero-energy resonance (in the form of an incipient eigenvalue). The difference between the Green’s function and the heat kernel is apparent even when looking at the case a = 0 as the dimension varies; indeed, the heat kernel varies coherently with dimension, while the Green’s function has logarithmic terms in two dimensions (a signal of the zero-energy resonance in that case). By basing our arguments around the heat kernel, we avoid these peculiarities. The second key factor in our treatment of the endpoint case is the fact that precise estimates for the heat kernel have already been verified in this case. This is a result of [20] that we reproduce here in Theorem 2.1. For further discussion of the peculiarities and subtleties of the heat equation in the endpoint case see [28], as well as the references therein. As the operator a◦ is spherically symmetric, we may decompose it as a direct sum over spherical harmonics.L This is very instructive for understanding the structure of solutions and the possibility of alternate self-adjoint extensions. To this end, we d 1 employ polar coordinates in the form x = rω with 0 r< and ω S − . ≤ ∞ ∈ Consider now functions of the form f(r)Y (ω) with f C∞((0, )) and Y a ℓ ∈ c ∞ ℓ spherical harmonic of degree ℓ 0. The action of a◦ on such functions is equivalent to the action of the Bessel operator≥ L
d2 d 1 d a+ℓ(ℓ+d 2) f(r) f(r) − f(r)+ − f(r) 7→ − dr2 − r dr r2 on the radial factor f(r). This is a symmetric positive semi-definite operator with 2 d 1 respect to the L (r − dr) inner product. An alternate form of the Bessel operator,
4ν2 1 2 d 2 2 g(r) g′′(r)+ − g(r) with ν = − + a + ℓ(ℓ + d 2), (1.8) 7→ − 4r2 2 − 6 R.KILLIP,C.MIAO,M.VISAN,J.ZHANG,ANDJ.ZHENG
(d 1)/2 arises by employing the Liouville transformation g(r) = r − f(r). This is a symmetric operator with respect to the L2(dr) inner product. Note also that the d 2 2 2 condition a ( −2 ) guarantees that ν 0. We choose ν as the positive root. The spectral≥− theory of the Bessel equation≥ has been extensively studied, typically in the form (1.8); see, for example, [1, 25]. The operator is always limit point at infinity (in the sense of Weyl); this is a trivial consequence of the fact that the potential is bounded near infinity. On the other hand, the operator is limit point at the origin if and only if ν 1. To see this, we note that at every energy z C there is a basis of (formal) eigenfunctions≥ whose leading asymptotics at the origin∈ are given by
1 1 : ν =0 1 log(r): ν =0 2 2 g1(r)= r ν and g2(r)= r ν · (r : ν > 0 · (r− : ν > 0, respectively. (These functions are actually exact zero-energy eigenfunctions.) No- tice that g1 is always square integrable near the origin, while this is true for g2 if and only if ν < 1. Thus, when ν 1, the operator (1.8) is essentially self-adjoint ≥ and formal eigenfunctions have the asymptotics of g1(r). When 0 ν < 1, the op- erator (1.8) admits a one-parameter family of self-adjoint extensions≤ corresponding to possible choices of boundary condition at r = 0. Demanding that eigenfunctions have the asymptotics proportional to those of g1(r) corresponds to the choice of the Friedrichs extension. Transferring the foregoing information back to our original operator a◦ leads to the following conclusions: L d 2 2 (a) If a 1 ( −2 ) , then a◦ is essentially self-adjoint and a denotes the unique≥ self-adjoint− extension.L Formal eigenfunctions thatL are spherically symmetric have asymptotics σ u(x)= c x − 1+ O( x ) (1.9) | | | | as x 0. Eigenfunctions at higher angular momentum decay more rapidly at| the| → origin. d 2 2 d 2 2 (b) If ( −2 ) a < 1 ( −2 ) , then a◦ has deficiency indices (1, 1) and so admits− a one-parameter≤ − family of self-adjointL extensions. Such extensions differ in their action on spherically symmetric functions, but agree on the orthogonal complement. The Friedrichs extension is characterized by the fact that spherically symmetric (formal) eigenfunctions have asymptotics (1.9) at the origin. As mentioned earlier, the inverse-square potential is important for its appear- ance as a scaling limit of less singular potentials. This also justifies our focus on the Friedrichs extension. Indeed, tedious but elementary ODE analysis shows the following:
d 2 3 Proposition 1.3. Let V C∞(R ) obey V (x)= a x − +O( x − ) as x with d 2 2 ∈ | | | | 2 | | → ∞ a ( −2 ) . Then the family of Schr¨odinger operators ∆+ λ V (λx) converges in≥− strong resolvent sense to as λ . − La → ∞ 1.2. Notation. If X, Y are nonnegative quantities, we write X . Y or X = O(Y ) whenever there exists a constant C such that X CY . We write X Y whenever X . Y . X. This notation suffices to suppress≤ most absolute constants,∼ with one notable exception, namely, bounds on the rate of Gaussian decay in the heat kernel. SCHRODINGER¨ OPERATOR WITH INVERSE-SQUARE POTENTIAL 7
For this (and subsequent estimates derived from it), we use the letter c to denote a positive constant, which may vary from line to line. For any 1 r , we write for the norm in Lr(Rd) where integration is ≤ ≤ ∞ k·kr with respect to Lebesgue measure and denote by r′ the conjugate exponent defined 1 1 via r + r′ = 1. We will also use the notations A B := max A, B and A B := min A, B . ∨ { } ∧ { } Acknowledgements. We are grateful to E. M. Ouhabaz and an anonymous ref- eree for references connected with Theorem 1.1. R. Killip was supported by NSF grant DMS-1265868. He is grateful for the hospitality of the Institute of Applied Physics and Computational Mathematics, Beijing, where this project was initi- ated. C. Miao was supported by NSFC grants 11171033 and 11231006. M. Visan was supported by NSF grant DMS-1161396. J. Zhang was supported by PFMEC (20121101120044), Beijing Natural Science Foundation (1144014), and NSFC grant 11401024. J. Zheng was partly supported by the ERC grant SCAPDE.
2. Heat and Riesz kernels We begin by recalling estimates on the heat kernel associated to the operator ; these were found by Liskevich–Sobol [19] and Milman–Semenov [20]. La d 2 2 Theorem 2.1 (Heat kernel bounds). Assume d 3 and a ( − ) . Then ≥ ≥ − 2 there exist positive constants C1, C2 and c1,c2 such that for all t> 0 and all x, y Rd 0 , ∈ \{ } |x−y|2 |x−y|2 √t σ √t σ d t √t σ √t σ d 2 − c1t a 2 − c2t C1 1 x 1 y t− e e− L (x, y) C2 1 x 1 y t− e . ∨ | | ∨ | | ≤ ≤ ∨ | | ∨ | | The bounds provided by Theorem 2.1 are an essential ingredient in the following Lemma 2.2 (Riesz kernels). Let d 3 and suppose 0
s ∞ s 2 1 t a dt −a (x, y) := e− L (x, y)t 2 L Γ(s/2) t Z0 satisfy s σ x y − 2 s d − a (x, y) x y − x| |y x| |y 1 . (2.1) L ∼ | − | | − | ∧ | − | ∧ Proof. By symmetry, we need only consider the case when x y . In view of this reduction and Theorem 2.1, the key equivalence to be proved| | ≤ is | | 2 σ σ σ ∞ s−d c|x−y| √τ √τ x 2 τ dτ s d − τ e− x 1 y 1 τ c x y − x| |y 1 (2.2) 0 | | ∨ | | ∨ ∼ | − | | − | ∧ Z for all x y and any c > 0. Making the change of variables t = x y 2/τ we deduce| that| ≤ | | | − | s d LHS(2.2) x y − I + I + I ∼ | − | 1 2 3 where |x−y|2 2 ∞ d−s x y σ |x| d−s−σ 2 ct dt 2 ct dt I := t e− , I := | − σ| t e− , 1 2 t 2 x 2 t |x−y| | | |x−y| Z |x|2 Z |y|2 |x−y|2 2 x y 2σ |y| d−s−2σ 2 ct dt and I3 := |x−σ y| σ t e− t . | | | | Z0 8 R.KILLIP,C.MIAO,M.VISAN,J.ZHANG,ANDJ.ZHENG
In estimating these integrals, we will repeatedly use our hypotheses that d s> 0 and d s 2σ> 0. These imply d s σ > 0. − Case− 1: −x y 4 x . As x −y ,− in this case we have x y . x y . Correspondingly,| − | ≤ | | | | ≤ | | | − | | | ∼ | |
|x−y|2 2 x y σ |x| d−s−σ dt x y d−s I 1, I . | − | t 2 | − | . 1, 1 ∼c 2 x σ t ∼ x d−s | | Z0 | | and |x−y|2 2 x y 2σ |y| d−s−2σ dt x y d−s | − | 2 | − | I3 . x σ y σ t t . x σ y d−s−σ . 1. | | | | Z0 | | | | As I2 and I3 are positive, these estimates verify (2.2) in this case. Case 2: x y 4 x . As x y , we have x y y x in this case. Clearly, | − | ≥ | | | | ≤ | | | − | ∼ | | ≥ | | σ x y 2σ x − I3 c |x−σ y| σ x| |y . ∼ | | | | ∼ | − | On the other hand, σ x y σ ∞ d−s−2σ dt x 2 ct − I2 . | −x σ| t e− . x| |y | | 0 t | − | Z and σ ∞ ct/2 x − I . e− dt . | | . 1 c 2 c x y |x−y| | − | Z |x|2 As I1 and I2 are positive, these estimates verify (2.2) in this case.
3. Hardy inequality
In this section, we establish Hardy-type inequalities adapted to the operator a. In the absence of a potential (i.e., for a = 0), these are well-known (cf. [27, II.16]):L § Lemma 3.1 (Hardy inequality for ∆). For 1 The range of s in Lemma 3.1 is sharp; indeed, failure for s = d/p can be seen when f is any Schwartz function that does not vanish at the origin. The main result of this section is the analogue of Lemma 3.1 in the presence of an inverse-square potential: Proposition 3.2 (Hardy inequality for a). Suppose d 3, 0 s + σ < d < d σ. (3.3) p − When a> 0 (or equivalently σ < 0), we see that the Hardy inequality holds for a d 2 2 larger range of pairs (s,p) than in the case a = 0; when ( −2 ) a< 0, the range is strictly smaller. Note that under the over-arching hypothesis− d≤ s 2σ> 0, we are guaranteed that s + σ < d σ. Nonetheless, it is possible that− −s + σ < 0 or that d σ > d, which trivialize− the corresponding part of (3.3). − SCHRODINGER¨ OPERATOR WITH INVERSE-SQUARE POTENTIAL 9 The proof of Proposition 3.2 is built on the following variant of Schur’s test; see [18] for historical references and the (trivial) proof. Lemma 3.3 (Schur’s test with weights). Suppose (X,dµ) and (Y,dν) are measure spaces and let w(x, y) be a positive measurable function defined on X Y . Let K(x, y): X Y C satisfy × × → 1 sup w(x, y) p K(x, y) dν(y)= C0 < , (3.4) x X Y ∞ ∈ Z 1 p′ sup w(x, y)− K(x, y) dµ(x)= C1 < , (3.5) y Y X ∞ ∈ Z for some 1 1 1 p′ p Tf C C f p . (3.6) Lp(X,dµ) ≤ 0 1 k kL (Y,dν) Proof of Proposition 3.2. The estimate (3.2) is equivalent to s s 2 x − −a g . g Lp(Rd). (3.7) | | L Lp(Rd) k k By Lemma 2.2, to prove (3.7) it suffices to show that the kernel σ s s d x y − K (x, y) := x − x y − | | | | 1 (3.8) σ | | | − | x y ∧ x y ∧ | − | | − | defines a bounded operator on Lp(Rd). To this end, we divide the kernel into three pieces. Case 1: x y 4( x y ). In this case, we have x y and the kernel becomes | − |≤ | |∧| | | | ∼ | | s s d K (x, y)= x − x y − . σ | | | − | It is easily seen that Kσ(x, y)dy + Kσ(x, y)dx . 1, x y 4 x x y 4 y Z| − |≤ | | Z| − |≤ | | and so Lp-boundness on this region follows immediately from Lemma 3.3 with w(x, y) 1. Case 2:≡4 x x y 4 y . In this region, we have x y y x and the kernel becomes| | ≤ | − |≤ | | | − | ∼ | | ≥ | | s σ σ+s d s σ σ+s d K (x, y) x − − x y − x − − y − . σ ∼ | | | − | ∼ | | | | Let the weight w(x, y) be defined by x α w(x, y)= | | with p(s + σ) <α 1 s σ+ α σ+s d α w(x, y) p Kσ(x, y) dy . x − − p y − − p dy . 1, 4 x x y 4 y | | y x | | Z | |≤| − |≤ | | Z| |≥| | 10 R.KILLIP,C.MIAO,M.VISAN,J.ZHANG,ANDJ.ZHENG the hypothesis (3.4) is satisfied in this region. The fact that hypothesis (3.5) is also satisfied follows from 1 α α ′ σ+s d+ ′ s σ ′ w(x, y)− p Kσ(x, y) dx . y − p x − − − p dx . 1. 4 x x y 4 y | | x y | | Z | |≤| − |≤ | | Z| |≤| | Thus, an application of Lemma 3.3 yields Lp-boundedness on this region. Case 3: 4 y x y 4 x . On this region, we have x y x y and the kernel becomes| | ≤ | − |≤ | | | − | ∼ | | ≥ | | s σ σ+s d σ d σ K (x, y) x − y − x y − x − y − . σ ∼ | | | | | − | ∼ | | | | Subcase 3(a): σ< 0. We note that σ d σ Kσ(x, y) dy . x − y − dy . 1 4 y x y 4 x | | | | y x | | Z | |≤| − |≤ | | Z| |≤| | and σ σ d Kσ(x, y) dx . y − x − dx . 1. 4 y x y 4 x | | | | x y | | Z | |≤| − |≤ | | Z| |≥| | Lp-boundness in this setting follows immediately from Lemma 3.3. Subcase 3(b): σ > 0. In this case, we use Lemma 3.3 with weight given by x α w(x, y)= | | with p′σ<α 1 σ d+ α σ α w(x, y) p Kσ(x, y) dy . x − p y − − p dy . 1, 4 y x y 4 x | | | | y x | | Z | |≤| − |≤ | | Z| |≤| | while hypothesis (3.5) is deduced from 1 α α ′ σ+ ′ σ d ′ w(x, y)− p Kσ(x, y) dx . y − p x − − p dx . 1. 4 y x y 4 x | | | | x y | | Z | |≤| − |≤ | | Z| |≥| | This completes the proof of (3.2). Next, we will prove the sharpness of this inequality by constructing counterexamples. Rd Failure of (3.2) for (s + σ)p d. Let x0 1 and let ϕ Cc∞( ) be non- s ≥ | | ≫ ∈ 2 negative, supported in B(x0, 1), satisfying ϕ 1 on B(x0, 1/2). Let f := −a ϕ. Then ≡ L s 2 a f = ϕ p . 1. L Lp k kL On the other hand, using Lemma 2.2 we see that for x 1, | |≤ s+σ d s s x y − − 2 − 2 σ f(x) = [ a ϕ](x)= a (x, y)ϕ(y)dy & | − | σ dy &x0 x − . L L y x 1 x | | Z Z| − 0|≤ 2 | | Hence, s & (s+σ) x − f(x) Lp(Rd) x0 x − Lp( x 1) = whenever (s + σ)p d. | | | | | | | |≤ ∞ ≥ d Failure of (3.2) for p d σ . Let ϕ be a non-negative bump function supported ≤ − s 2 in B(0, 1) satisfying ϕ 1 on B(0, 1/2). Let f := −a ϕ. Then ≡ L s 2 a f = ϕ p . 1. L Lp k kL SCHRODINGER¨ OPERATOR WITH INVERSE-SQUARE POTENTIAL 11 On the other hand, using Lemma 2.2 we see that for x 10, | |≥ s s 2 2 s+σ d σ s+σ d f(x) = [ a− ϕ](x)= −a (x, y)ϕ(y)dy & x − y − dy & x − . L L | | y 1 | | | | Z Z| |≤ 2 Hence, s σ d d x − f(x) Lp(Rd) & x − Lp( x 10) = whenever p d σ . | | | | | |≥ ∞ ≤ − This completes the proof of the lemma. 4. Multiplier theorem As discussed in the Introduction, the purpose of this section is to give a self- d 2 2 contained proof of Theorem 1.1 in the case ( − ) a< 0: − 2 ≤ d 2 2 Theorem 4.1 (Mikhlin Multipliers). Fix ( −2 ) a < 0 and suppose that m : [0, ) C satisfies − ≤ ∞ → j j d ∂ m(λ) . λ− for all 0 j 3 +3. (4.1) ≤ ≤ 4 2 Then m(√ a), which we define via the L functional calculus, extends uniquely p dL 2 d p d d from L (R ) L (R ) to a bounded operator on L (R ), for all r Proof of Theorem 4.1. By the spectral theorem, the operator T := m(√ a) is bounded on L2. Thus, using the Marcinkiewicz interpolation theorem and a dualityL argument, it suffices to show that T is of weak-type (q, q) whenever r0 1 ′ 1 q q f(x) dx . f q Q q . h Q . h − f(x) dx. (4.4) | | k kL (Qk )| k| | k| | | ZQk ZQk In the decomposition above, the ‘bad’ parts bk do not obey any cancellation condition. To remedy this, we further decompose bk = gk + ˜bk according to the following definition: µ r2 µ r2 µ νr2 ˜b := (1 e− kLa ) b and g := 1 (1 e− kLa ) b = c e− kLa b , (4.5) k − k k − − k ν k ν=1 X 12 R.KILLIP,C.MIAO,M.VISAN,J.ZHANG,ANDJ.ZHENG where r denotes the radius of Q and µ := d +1. As f = g + g + ˜b , k k ⌊ 4 ⌋ k k k k so Tf = Tg + Tg + T˜b , and then k k k k P P Tf >hP TgP> 1 h T g > 1 h T ˜b > 1 h . | | ⊆ | | 3 ∪ k 3 ∪ k 3 k k X X By the Chebyshev inequality, the boundedness of T in L 2, and (4.3), 1 2 2 2 2 q q q q Tg > h . h− Tg 2 . h− g 2 . h− g q . h− f q . | | 3 k kL k kL k kL k kL Thus, the contribution of this term to (4.2) is acceptable. Arguing similarly, we estimate 1 2 2 2 2 T gk > 3 h . h− T gk L2 . h− gk L2 . (4.6) k k k X X X Using the heat kernel estimate given by Theorem 2.1, we obtain 2 ′ 2 2 (νr +ν r ) a ′ k l gk L2 = cν cν bk,e− L bl (4.7) k ν,ν′ k,l X X X |x−y|2 d r σ cr2 r σ k − k k . rk− x 1 bk(x) e y 1 bl(y) dx dy. Ql Qk | | ∨ | | | | ∨ | | rk rl Z Z X≥ To proceed from here, we first freeze k and x Q and focus on ∈ k |x−y|2 |x−y|2 cr2 r σ cr2 − k k − k e y 1 bl(y) dy . e bl(y) dy (4.8) Ql | | ∨ | | Ql | | l:rl rk Z l:rl rk Z X≤ X≤ σ rk + y bl(y) dy. (4.9) Ql | | | | l:Ql B(0,2rk) Z ⊂X Here we used that Ql B(0, rk) = ∅ implies Ql B(0, 2rk) because rl rk. To estimate RHS(4.8)∩ we first6 observe that since⊆ diam(Q ) 2r , ≤ l ≤ k 2 1 2 2 x y x y′ 4r for all y,y′ Q . | − | ≥ 2 | − | − k ∈ l Using this and then (4.3) and (4.4), we see that |x−y′|2 1 2 − 2cr RHS(4.8) . bl L1 e k dy′ k k Ql l:r r Ql Xl≤ k | | Z |x−y′|2 2cr2 . h e− k dy′ l:r r Ql Xl≤ k Z d . hrk. On the other hand, applying H¨older (in l and y) and then (4.3) yields 1/q′ 1/q ′ rk σq q RHS(4.9) . y dy bl(y) dy " Ql | | # " Ql | | # l:Ql B(0,2rk) Z l:Ql B(0,2rk) Z ⊂X ⊂X 1/q′ 1/q ′ rk σq q . y dy h Ql " B(0,2rk) | | # " | |# Z l:Ql B(0,2rk) ⊂X d . hrk. Note that it was important here that q < r0 since this guarantees σq′ < d. SCHRODINGER¨ OPERATOR WITH INVERSE-SQUARE POTENTIAL 13 Plugging this new information back into (4.7) and then using H¨older’s inequality and (4.3) we deduce that 2 σ rk gk L2 . h x 1 bk(x) dx | | ∨ | | k k ZQk X X ′ 1/q 1/q σq′ rk q . h x 1 dx bk(x) dx " Qk | | ∨ # " Qk | | # Xk Z Xk Z 1/q′ . h Q f q | k| k kL " k # 2 Xq q . h − f q k kL In view of (4.6), this shows that the contribution from the functions gk is acceptable. ˜ 1 It remains to estimate the contribution of T k bk > 3 h . Let Qk∗ be the 2√d dilate of Q . As k P 1 d 1 T ˜b > h Q∗ x R Q∗ : T ˜b > h , k 3 ⊂ ∪j j ∪ ∈ \ ∪j j k 3 k k X X by Chebyshev’s inequality and (4.3), ˜ 1 1 ˜ T bk > 3 h . Qj∗ + h− T bk L1(Rd Q∗) | | k k j k \ X X X q q 1 ˜ . h− f Lq + h− T bk L1(Rd Q∗). k k k k \ X Therefore, to complete the proof of the theorem we need only show that ˜ 1 q q T bk L1(Rd Q∗) . h − bk Lq . (4.10) \ k k k Rd To this end, we divide the region Qk∗ into dyadic annuli of the form R < Z \ dist x, Q 2R for r R 2 . We will prove the following L2 estimate: { k}≤ k ≤ ∈ 1 1 rk 2µ d( ′ ) ˜ − 2 − q q T bk L2(dist x,Q >R) . R R bk L , (4.11) { k} k k with the implicit constant independent of k and R r > 0. Claim (4.10) follows k immediately from this, H¨older, (4.3), and (4.4): ≥ ˜ ˜ T bk 1 Rd ∗ = T bk 1 L ( Qk ) L (R r2 µ (T˜b )(x)= m( )(1 e− kLa ) (x, y)b (y)dy. k La − k ZQk p 14 R.KILLIP,C.MIAO,M.VISAN,J.ZHANG,ANDJ.ZHENG r2 λ2 µ Let a(λ) := m(λ)(1 e− k ) , which we extend to all of R as an even function. Using (4.1), it is easy to− check that j j 2µ d ∂ a(λ) . λ − (1 r λ ) for all 0 j 3 +3. (4.12) | | ∧ k| | ≤ ≤ ⌊ 4 ⌋ Now let ϕ be a smooth, positive, even function, supported on [ 1 , 1 ], and such − 2 2 that ϕ(τ)=1 for τ < 1 . Defineϕ ˇ andϕ ˇ via | | 4 R 1 ϕˇ (λ) := Rϕˇ(Rλ) := eiλτ ϕ τ dτ. R 2π R Z Since a and ϕ are even, 1 ∞ τ a (λ) := (a ϕˇ )(λ)= π− cos(λτ)ˆa(τ)ϕ dτ. 1 ∗ R R Z0 The wave equation with inverse-square potential utt + au = 0 obeys finite speed of propagation; see, for example, [22] or [11, 3]. NotingL that ϕ(τ/R) is supported § on the set τ : τ R , we therefore obtain { | |≤ 2 } supp a δ supp cos τ δ B(y, 1 R). 1 La y ⊆ La y ⊆ 2 τ R p [≤ 2 p Thus, this part of the multiplier a does not contribute to LHS(4.11). We now consider the remaining part of the multiplier a, namely, a (λ) := a (λ) a(λ)= [a(θ) a(λ)]ϕ ˇ (λ θ) dθ. 2 1 − − R − Z 1 When λ R− , using (4.12) and the rapid decay ofϕ ˇ, we get | |≤ 2µ 2µ 2µ a(λ)ˇϕ (λ θ) dθ .(1 r λ ) . (1 r λ ) ( λ R)− , R − ∧ k| | ∧ k| | | | Z and 2µ 2µ a(θ)ˇϕ (λ θ) dθ . (1 r λ ) ( λ R)− . R − ∧ k| | | | Z Thus 2µ 2µ 1 a (λ) . (1 r λ ) ( λ R)− when λ R− . (4.13) | 2 | ∧ k| | | | | |≤ 1 d When λ R− , expanding a(θ) in a Taylor series to order j 1=3 4 + 2, we write | | ≥ − ⌊ ⌋ j 1 − a(ℓ)(λ) a(θ) a(λ)= P (θ)+ (θ) where P (θ) := (θ λ)ℓ − j E j ℓ! − Xℓ=1 and denotes the error, which we estimate using (4.12) as follows: E 2µ θ λ j 1 (θ) a(θ) + a(λ) + P (θ) . (1 r λ ) − when θ λ > λ |E | ≤ | | | | | j | ∧ k| | λ | − | 2 | | (note 2µ 3µ = j) and ≤ (j) j 2µ θ λ j 1 (θ) a L∞([ λ , 3λ ]) θ λ . (1 rk λ ) −λ when θ λ 2 λ . |E |≤ 2 2 | − | ∧ | | | − |≤ | | For any ℓ 1, ≥ ℓ ℓ (θ λ) ϕˇ (θ λ) dθ = (∂ ϕ )∨(0) = 0. − R − R Z SCHRODINGER¨ OPERATOR WITH INVERSE-SQUARE POTENTIAL 15 Thus Pj (θ) makes no contribution to the convolution defining a2(λ) and so 2µ j 1 a (λ) . (θ) ϕˇ (λ θ) dθ . (1 r λ ) ( λ R)− when λ R− . (4.14) | 2 | |E || R − | ∧ k| | | | | |≥ Z Combining (4.13) with (4.14) and the fact that R r , we deduce that ≥ k 2µ j−2µ 1 r λ 2µ−j ∞ 2 2 k 2 2 2 rk 2µ t 2 t/R tλ dt a2(λ) . ∧λ R| | (1 + R λ ) . R ) R2 e− e− t | | | | 0 Z and so, by the spectral theorem followed by the triangle inequality, j−2µ 2 r 2µ ∞ t 2 t/R t dt a ( )b . k ) e− e− La b . (4.15) 2 La k L2(Rd) R R2 k L2 t Z0 p To proceed from here we need the following estimate, which we will justify later: d d t a 2 2q′ e− L b . t− 4 t + r b q . (4.16) k L2 k k kkL Substituting this estimate into (4.15) yields 1 1 ∞ j−2µ d d t rk 2µ d( ′ ) t 2 4 t 2q′ dt − 2 − q q − − R2 a2( a)bk L2(Rd) . R ) R bk L R2 1+ R2 e t L k k 0 1 1 Z p rk 2µ d( ′ ) . ) R− 2 − q b q , R k kkL j 2µ d for any R rk. (Note that we also rely on the fact that −2 4 > 0.) Excepting the fact that≥ it remains to justify (4.16), this completes the− proof of (4.11) and with it, the proof of Theorem 4.1. In view of the heat kernel bounds described in Theorem 2.1, (4.16) reduces to |x−y|2 d √ σ √ σ d ′ t t ct 4 2 2q 1 x 1 y e− bk(y)dy . t t + rk bk Lq . (4.17) Rd ∨ | | ∨ | | L2 k k Z We consider four cases: Case 1: x < √t, y < √t. Using H¨older’s inequality and recalling that q > r0, we estimate| | the contribution| | of this region to LHS(4.17) by σ σ σ σ σ σ q t x − y − bk(y)dy . t x − 2 y − q′ bk L 2 L ( x <√t) L ( y <√t) | | y <√t | | L ( x <√t) | | | | | | | | k k Z| | | | d 1 1 2 ( 2 + ′ ) . t q bk Lq . k k Case 2: x √t, y < √t. Using the Minkowski inequality and arguing as in Case 1, we| estimate| ≥ the| | contribution of this region to LHS(4.17) by 2 σ σ |x−y| σ + d σ 2 ct . 2 4 ′ q t y − bk(y) e− L2 dy t y − Lq ( y <√t) bk L y <√t | | | | x | | | | k k Z| | d 1 1 ( + ′ ) . t 2 2 q b q . k kkL Case 3: x < √t, y √t. We estimate the contribution of this region to LHS(4.17)| by| | | ≥ 2 σ |y| d ( 1 + 1 ) σ 2 2 q′ 2 ct ′ q . q t x − L2( x <√t) e− Lq bk L t bk L . | | | | k k k k √ √ Case 4: x t, y t. Using the Minkowski and H¨older inequalities, we estimate the| | contribution ≥ | | ≥ of this region to LHS(4.17) by |x−y|2 |x|2 d d d q′ ct ct 4 4 q bk(y)e− dy . bk(y) e− 2 dy . t bk L1 . t r bk L . 2 Lx k y √t Lx Rd | | k k k k Z| |≥ Z 16 R.KILLIP,C.MIAO,M.VISAN,J.ZHANG,ANDJ.ZHENG This completes the proof of (4.17), and with it, the proof of Theorem 4.1. We end this section with an example which shows that one cannot improve upon the range of exponents p in Theorem 4.1, even if one assumes (4.1) holds for all d j 0. By self-adjointness, it suffices to prove this failure for p σ when σ> 0. ≥ λ2 ≥ To this end, let m(λ) := e− ; note that m satisfies (4.1) for all j 0. Let ϕ be a smooth radial cutoff supported in B(0, 1) such that ϕ(x) = 1 for≥x 1/2. Using the heat kernel estimate provided by Theorem 2.1, for x 1 we| have| ≤ | |≤ 2 a σ σ c x y σ [m(√ a)ϕ](x) = [e−L ϕ](x) & x − y − e− | − | dy & x − . L | | y 1/2 | | | | Z| |≤ Therefore, m(√ )ϕ / Lp for any p d . La ∈ ≥ σ 5. Littlewood–Paley theory In this section, we develop basic Littlewood–Paley theory, such as Bernstein and square function inequalities, adapted to the operator a. Our results in this section rely on Theorem 1.1. L Let φ : [0, ) [0, 1] be a smooth function such that ∞ → φ(λ)=1 for 0 λ 1 and φ(λ)=0 for λ 2. ≤ ≤ ≥ For each dyadic number N 2Z, we define ∈ φ (λ) := φ(λ/N) and ψ (λ) := φ (λ) φ (λ). N N N − N/2 Clearly, ψN (λ) N 2Z forms a partition of unity for λ (0, ). We define the Littlewood–Paley{ } projections∈ as follows: ∈ ∞ a a a a P N := φN a , PN := ψN ( a , and P>N := I P N . ≤ L L − ≤ We also define another p family of Littlewood–Paleyp projections via the heat ker- nel, as follows: 2 2 2 ˜a a/N ˜a a/N 4 a/N ˜a ˜a P N := e−L , PN := e−L e− L , and P>N := I P N . ≤ − − ≤ In what follows, we will write PN , P˜N , and so forth, to represent the analogous operators associated to the Euclidean Laplacian ∆. − Lemma 5.1 (Bernstein estimates). Let 1 < p q when a 0 and let d d 2 2 ≤ ≤ ∞ ≥ r0 2 2 2 d d a/N d cN x e−L f q Rd . N e− | | f p Rd . N p − q f p Rd , k kL ( ) Lr(Rd)k kL ( ) k kL ( ) where r is determined by 1 + 1 = 1 + 1 . q r p SCHRODINGER¨ OPERATOR WITH INVERSE-SQUARE POTENTIAL 17 d 2 2 We now turn to the case ( −2 ) a < 0 (or equivalently, σ > 0). By Theo- rem 2.1, − ≤ 2 2 2 a/N d 1 σ cN x y 1 σ e−L f . N 1 N x e− | − | 1 N y f(y) dy . Lq (Rd) ∨ Rd ∨ Lq(Rd) | | Z | | (5.1) To estimate RHS(5.1), we divide into four cases. 1 1 Case 1: x N − , y N − . Using H¨older and recalling that r0 < p q < d | | ≤ | | ≤ ≤ r0′ = σ , we estimate the contribution of this region to RHS(5.1) by d 2σ σ σ N − x − y − f(y) dy q −1 | | y N −1 | | L ( x N ) Z| |≤ | |≤ d d d 2σ σ σ p q ′ p − p . N − x − Lq( x N −1) y − Lp ( y N −1) f L . N f L . | | | |≤ | | | |≤ k k k k 1 1 Case 2: x N − , y > N − . Using H¨older’s inequality, we estimate the contribution| | of ≤ this region| | to RHS(5.1) by d σ σ cN 2 x y 2 N − x − e− | − | f(y) dy | | Rd Lq( x N −1) | |≤ Z 2 2 d d d σ σ cN y p q . N x e ′ f p . N − f p . − − Lq( x N −1) − | | Lp L L | | | |≤ k k k k 1 1 Case 3: x >N − , y N − . Using the Minkowski and H¨older inequalities, we estimate the| | contribution| |≤ of this region to RHS(5.1) by d σ σ cN 2 x y 2 N − y − e− | − | f(y) dy q y N −1 | | L Z| |≤ 2 2 d d d σ cN x σ p q ′ p − p . N − e− | | Lq y − Lp ( y N −1) f L . N f L . | | | |≤ k k k k 1 1 Case 4: x > N − , y > N − . Using Young’s inequality, we estimate the contribution| | of this region| | to RHS(5.1) by d cN 2 x y 2 d cN 2 x 2 d d p p − q p N e− | − | f(y) dy . N e− | | Lr f L . N f L , Rd Lq k k k k Z 1 1 1 where 1 + q = r + p . This completes the proof of the lemma. Lemma 5.2 (Expansion of the identity). For any 1 As a direct consequence of Theorem 1.1, we obtain two-sided square functions estimates; see [24, IV.5] for the requisite argument. § 18 R.KILLIP,C.MIAO,M.VISAN,J.ZHANG,ANDJ.ZHENG Theorem 5.3 (Square function estimates). Fix s 0. Let 1 6. Generalized Riesz transforms In this section we prove Theorem 1.2. We start by estimating the difference between the kernels of the Littlewood–Paley projection operators adapted to the Euclidean Laplacian and , respectively. La ˜ ˜a Lemma 6.1 (Difference of kernels). Let KN (x, y) := PN PN (x, y). (1) For a 0, there exists c> 0 such that − ≥ d 2 cN 2 x y 2 K (x, y) . N max 1,N( x + y ) − e− | − | , (6.1) N | | | | uniformly for x, y Rd 0 . d 2 2 ∈ \{ } (2) For ( − ) a< 0, there exists c> 0 such that for any M > 0, − 2 ≤ d 2σ σ 1 N − ( x y )− , x , y N − , d | ||σ | M | | | |≤ 1 N (N x )− (N y )− , 2 x N − y , KN (x, y) . | | | | | |≤ ≤ | | (6.2) N d(N y ) σ(N x ) M , 2 y N 1 x , − − − d 2 | | | 2| cN 2 x y 2 | |≤ 1 ≤1 | | N − ( x + y )− e− | − | , x , y N − , | | | | | | | |≥ 2 uniformly for x, y Rd 0 . ∈ \{ } Proof. By the definition of the Littlewood–Paley projections, we may write ∆/N 2 4∆/N 2 /N 2 4 /N 2 K (x, y)= e e (x, y) e−La e− La (x, y) (6.3) N − − − ∆/N 2 /N 2 4∆/N 2 4 /N 2 = e e−La (x, y) e e− La (x, y). − − − t Let us begin with the case a 0. By the maximum principle, 0 e− La (x, y) et∆(x, y) and consequently, ≥ ≤ ≤ ∆/N 2 4∆/N 2 d N 2 x y 2/16 K (x, y) e (x, y)+ e (x, y) . N e− | − | . | N |≤ 1 This suffices to verify (6.1) when x + y N − or x 2 y or y 2 x . Thus, | | | |≤ 1 | |≥ | | | |≥ | | we need only prove (6.1) when x + y N − with x y . By Duhamel’s formula, | | | |≥ | | ∼ | | t t∆ t a (t s)∆ 1 s a e e− L (x, y)= a e − (x, z) z 2 e− L (z,y) dz ds. (6.4) − Rd Z0 Z | | Thus, by exploiting the maximum principle again and using the fact that x z 2 z y 2 x z 2+ z y 2 2 z x+y 2 x y 2 x y 2 | − 2 | | t− s| + | −s | | − | t | − | = t + | −2t | | −2t | − ≥ ≥ SCHRODINGER¨ OPERATOR WITH INVERSE-SQUARE POTENTIAL 19 we can deduce that t∆ t e (x, y) e− La (x, y) − t 2 2 d |x−z| d |z−y| 2 4(t−s) 1 2 4s . (t s)− e− z 2 s− e− dz ds 0 Rd − | | Z Z t d d 2 2 t 2 d |x−y| 2 s− 2 |z| (t s)− 2 |z| 2 16t 8s − 8(t−s) . t− e− 2 e− dz ds + − 2 e dz ds 0 Rd z + y t Rd z + x Z Z | | Z 2 Z | | h 1 1 i 2 1 2 1 2 d |x−y| 2 s− |z| 2 s− |z| 2 16t 8 8 . t− e− y 2 e− dzds + x 2 e− dzds . 0 Rd z + 0 Rd z + h Z Z | √ts | Z Z | √ts | i Using the fact that for d 3 ≥ |z|2 8 1 2 e− 2 dz . x − for x 1, z x0 0 0 Rd | | | |≥ Z | − | we obtain 2 t∆ t d |x−y| t e (x, y) e− La (x, y) . t− 2 e− 16t for all x , y √t. − ( x + y )2 | | | |≥ | | | | 2 2 Taking t = N − and t =4N − , the remaining case of (6.1) now follows. d 2 2 Next we consider the case when ( −2 ) a < 0. By the maximum principle and Theorem 2.1, we have − ≤ 2 2 2 2 a/N 4 a/N d cN x y 1 σ 1 σ KN (x, y) e−L (x, y)+e− L (x, y) . N e− | − | 1 N x 1 N y ) . | |≤ ∨ | | ∨ | | 1 1 This directly justifies (6.2), except when x , y 2 N − and x y . Let us now focus on this remaining scenario. In view| of| Theorem| |≥ 2.1, the| analysis| ∼ | | of this case via (6.4) follows as in the case a 0, except in the regime where z is small. More precisely, to finish the proof, it suffices≥ to show t c x y 2/t e− | − | (t s)∆ 1 s a . e − (x, z) z 2 e− L (z,y) dz ds (d 2)/2 2 2 (6.5) 0 z 2< s | | t − ( x + y ) Z Z| | 4 | | | | when x , y √t and x y . Under| | | these|≥ conditions| | ∼on |x|, y, and z, we have x z 2 y z 2 ( x 2 + y 2). Thus invoking Theorem 2.1 and using 0 s t, we| − deduce| ∼ |that− | ∼ | | | | ≤ ≤ t tσ/2 d/2 ct 2 2 LHS(6.5) . z 2+σ dz [s(t s)]− exp s(t s) ( x + y ) ds. 0 z 2 d/2 ct( x 2+ y 2) 2 2 d/2 c 2 2 sup [s(t s)]− exp |s(|t s|) | . [t( x + y )]− exp t ( x + y ) , 0 Proposition 6.2 (Difference of square functions). Let d 3 and 0 have 2, we ℓ1 ֒ ℓ By the triangle inequality and the embedding Proof. → 1 2 2s ˜ ˜a 2 LHS(6.6) . N PN PN f − p Rd N 2Z L ( ) ∈ X 1 2 2 2s . N KN (x, y)f(y) dy p Rd N 2Z Z L ( ) X∈ s . N KN (x, y) f(y) dy . | | | | p Rd Z N 2Z L ( ) X∈ where KN is as in Lemma 6.1. Case 1: a 0. We decompose the sum into the contribution of high frequencies ≥ 1 1 N ( x + y )− and that of low frequencies N ( x + y )− . We first discuss the≥ contribution| | | | of high frequencies. ≤ | | | | 1 For N ( x + y )− , using Lemma 6.1 we find that for any M,L > 0, ≥ | | | | M − d N( x + y ) , x 2 y or y 2 x , KN (x, y) . N | | | | 2 L | |≥ | | | |≥ | | | | ( N( x + y ) − N x y − , x y . | | | | | − | | | ∼ | | Choosing M > d+s and d+s 2 s N KN (x, y) f(y) dy | || | p Z N ( x + y )−1 L ≥ |X| | | d+s N M f(y) dy p ≤ 2 y x −1 N( x + y ) | | L Z | |≤| | N ( x + y ) ≥ |X| | | | | | | d+s N + M f(y) dy p 2 x y −1 N( x + y ) | | L Z | |≤| | N ( x + y ) ≥ |X| | | | | | | d+s N + 2 L f(y) dy p x y −1 N( x + y ) N x y | | L Z| |∼| | N ( x + y ) ≥ |X| | | | | | | | − | s s y f(y) y f(y) . | | d+s | s | dy + | | d+s | s | dy 2 y x ( x + y ) y Lp 2 x y ( x + y ) y Lp Z | |≤| | | | | | | | Z | |≤| | | | | | | | s y f(y ) + d|+|s L L | s | dy x y ( x + y ) − x y y Lp Z| |∼| | | | | | | − | | | , I + I + I . 1 2 3 Using Lemma 3.3, we will estimate these three integrals below in terms of RHS(6.6). It turns out that these three integrals also control the contribution SCHRODINGER¨ OPERATOR WITH INVERSE-SQUARE POTENTIAL 21 1 from the low frequencies. Indeed, for N ( x + y )− , Lemma 6.1 guarantees that K (x, y) . N d and consequently, ≤ | | | | | N | s s y f(y) N KN (x, y) f(y) dy . | | d+s | s | dy | | | | p ( x + y ) y p Z N ( x + y )−1 L Z | | | | | | L ≤ |X| | | . I1 + I2 + I3. We first consider the contribution of I1. On the corresponding region of integra- tion, the kernel becomes s y s (d+s) K(x, y)= | | y x − . ( x + y )d+s ∼ | | | | | | | | As s> 0, a simple computation shows that sup K(x, y) dy + sup K(x, y) dx . 1. x Rd 2 y x y Rd 2 y x ∈ Z | |≤| | ∈ Z | |≤| | Thus, Schur’s test shows the contribution of I1 is acceptable. We now consider the contribution of I2. On the corresponding region of integra- tion, the kernel takes the form s y d K(x, y)= | | y − . ( x + y )d+s ∼ | | | | | | We will apply Lemma 3.3 with weight given by x α w(x, y) := |y| for 0 <α 1 α d α w(x, y) p K(x, y) dy . x p y − − p dy . 1, 2 x y | | 2 x y | | Z | |≤| | Z | |≤| | while hypothesis (3.5) follows from 1 d+ α α w(x, y)− p′ K(x, y) dx . y − p′ x − p′ dx . 1. 2 x y | | 2 x y | | Z | |≤| | Z | |≤| | Thus, Lemma 3.3 shows that the contribution of I2 is acceptable. Finally, we consider the contribution of I3. On the corresponding region of integration, the kernel becomes s y d+L L d+L L K(x, y)= | | x − x y − y − x y − . ( x + y )d+s L x y L ∼ | | | − | ∼ | | | − | | | | | − | − | Recalling that L < d, we estimate sup K(x, y) dy + sup K(x, y) dx x Rd x y y Rd x y ∈ Z| |∼| | ∈ Z| |∼| | d+L L d+L L . sup x − x y − dy + sup y − x y − dx . 1. x Rd | | x y . x | − | y Rd | | x y . y | − | ∈ Z| − | | | ∈ Z| − | | | Thus, Schur’s test ensures that the contribution of I3 is acceptable. This completes the proof of the proposition in the case a 0. ≥ d 2 2 Case 2: ( −2 ) a< 0. As before, we decompose the sum into the contribution − ≤ 1 1 of high frequencies N ( x y )− and that of low frequencies N ( x y )− . ≥ | | ∨ | | ≤ | | ∨ | | 22 R.KILLIP,C.MIAO,M.VISAN,J.ZHANG,ANDJ.ZHENG We consider first the low frequencies. Using Lemma 6.1 and the fact that d+s 2σ, we estimate ≥ s N KN (x, y) f(y) dy Rd | || | p Z N ( x y )−1 L ≤ |X|∨| | d+s 2σ s σ N − y − f(y) . σ σ f(y) dy . | |d+s 2σ σ | s | dy Rd x y | | p Rd ( x + y ) − x y p Z N ( x y )−1 | | | | L Z | | | | | | | | L ≤ |X|∨| | s σ y − f(y) 1 f(y) . | d|+s σ | s | dy + d σ σ | s | dy 2 y x x − y Lp 2 x y y − x y Lp Z | |≤| | Z | |≤| | | | | | | | | | | | 1 f(y) + d | s | dy =: II1 + II2 + II3. x y x y Lp Z| |∼| | | | | | Before turning our attention to the estimation of these three integrals, let us pause to consider the high frequency contribution. To this end, with x fixed, we divide the integral in the y variable into three regions: 2 y x , 2 x y , and x /2 y 2 x . { | | ≤ | |} { | | ≤ | |} If{| 2 y| ≤x |,|≤ then| choosing|} M > d + s σ, Lemma 6.1 yields | | ≤ | | − s d+s σ M N K (x, y) . N (N y )− (N x )− | N | | | | | N ( x y )−1 (2 y )−1 N ( x y )−1 ≥ |X|∨| | | | ≥ X≥ | |∨| | d+s M σ + N (N x )− − | | N (2 y )−1 ( x y )−1 ≥ | | X≥ | |∨| | σ σ s d . y − x − − . | | | | The contribution of this region is therefore bounded by II1. Arguing similarly, we see that when 2 x y and M > d + s σ, | | ≤ | | − s d+s σ M N K (x, y) . N (N x )− (N y )− | N | | | | | N ( x y )−1 (2 x )−1 N ( x y )−1 ≥ |X|∨| | | | ≥ X≥ | |∨| | d+s M σ + N (N y )− − , | | N (2 x )−1 ( x y )−1 ≥ | | X≥ | |∨| | σ σ s d . x − y − − . | | | | whose contribution is controlled by II2. The third and final high-frequency contribution comes from the region where x /2 y 2 x . Choosing d + s 2 Our earlier analysis of I3 then shows that the resulting contribution is acceptable. To complete the proof of Proposition 6.2 it remains only to estimate the three integrals II1, II2, and II3. We begin with II1, applying Lemma 3.3 with weight x α w(x, y)= |y| with (σ s)p′ <α That hypothesis (3.4) is satisfied in this setting follows from s σ 1 y − d s+σ+ α s σ α p − − p − − p w(x, y) | d|+s σ dy . x y dy . 1, 2 y x x − | | 2 y x | | Z | |≤| | | | Z | |≤| | while hypothesis (3.5) follows from s σ 1 y − s σ+ α (d+s σ+ α ) − p′ − p′ − − p′ w(x, y) | d|+s σ dx . y x dx . 1. 2 y x x − | | 2 y x | | Z | |≤| | | | Z | |≤| | Thus, Lemma 3.3 shows that the contribution of II1 is acceptable. We now consider the contribution of II2. We use Lemma 3.3 with weight x α w(x, y)= |y| with σp<α 1 1 σ+ α d α +σ p − p − − p w(x, y) d σ σ dy . x y dy . 1, 2 x y y − x | | 2 x y | | Z | |≤| | | | | | Z | |≤| | while hypothesis (3.5) is derived from 1 1 d+σ+ α σ α − p′ − p′ − − p′ w(x, y) d σ σ dx . y x dx . 1. 2 x y y − x | | 2 x y | | Z | |≤| | | | | | Z | |≤| | Thus by Lemma 3.3, the contribution of this term to RHS(6.6) is acceptable. It remains only to control the contribution of II3. Noting that 1 1 sup d dy + sup d dx . 1, x Rd x y x x Rd x y x ∈ Z| |∼| | | | ∈ Z| |∼| | | | Schur’s Lemma ensures that this integral is acceptable. d 2 2 This completes the proof of the proposition in the case ( − ) a< 0. − 2 ≤ We now have all the necessary ingredients to complete the proof of Theorem 1.2. d 2 2 s+σ Proof of Theorem 1.2. 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Zhang and J. Zheng, Scattering theory for nonlinear Schr¨odinger with inverse-square potential. J. Funct. Anal. 267 (2014), 2907–2932. Department of Mathematics, UCLA E-mail address: [email protected] Institute of Applied Physics and Computational Mathematics, Beijing 100088 E-mail address: miao [email protected] Department of Mathematics, UCLA E-mail address: [email protected] Department of Mathematics, Beijing Institute of Technology, Beijing 100081 E-mail address: [email protected] Universite´ Nice Sophia-Antipolis, 06108 Nice Cedex 02, France E-mail address: [email protected], [email protected]