ON COMMUTATORS OF FRACTIONAL INTEGRALS 1. Introduction

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 132, Number 12, Pages 3549–3557 S 0002-9939(04)07437-4 Article electronically published on July 14, 2004

ON COMMUTATORS OF FRACTIONAL INTEGRALS XUAN THINH DUONG AND LI XIN YAN (Communicated by Andreas Seeger)

Abstract. Let L be the infinitesimal generator of an analytic semigroup on L2 (Rn ) with Gaussian kernel bounds, and let L−α/2 be the fractional integrals of L for 0 < α < n. For a BMO function b(x) on Rn , we show boundedness of the commutators [b, L−α/2 ](f )(x) = b(x)L−α/2 (f )(x) − L−α/2 (bf )(x) from n 1 Lp (Rn ) to Lq (Rn ), where 1 < p < α , q = 1p − α . Our result of this boundedn n ness still holds when R is replaced by a Lipschitz domain of Rn with infinite measure. We give applications to large classes of differential operators such as the magnetic Schr¨ odinger operators and second-order elliptic operators of divergence form.

1. Introduction Suppose that L is a linear operator on L2 (Rn ) which generates an analytic semigroup e−tL with a kernel pt (x, y) satisfying a Gaussian upper bound, that is, (G)

|pt (x, y)| ≤

C tn/2

e−c

|x−y|2 t

for x, y ∈ Rn and all t > 0. For 0 < α < n, the fractional integrals L−α/2 of the operator L is defined by Z ∞ 1 dt e−tL (f ) −α/2+1 (x). L−α/2 f (x) = Γ(α/2) 0 t Let b be a BMO function on Rn . The commutator of b and L−α/2 is defined by [b, L−α/2 ](f )(x) = b(x)L−α/2 (f )(x) − L−α/2 (bf )(x). Note that if L = −4 is the Laplacian on Rn , then L−α/2 is the classical fractional integral Iα . See, for example, Chapter 5 in [St1]. It is well known that when b(x) ∈ BMO(Rn ), the commutator [b, Iα ] is bounded from Lp (Rn ) to Lq (Rn ), 1 < p < n/α, 1/q = 1/p − α/n. See [Ch]. The aim of this paper is to prove the following theorem. Received by the editors January 3, 2003. 2000 Mathematics Subject Classification. Primary 42B20, 47B38. Key words and phrases. Gaussian bound, fractional integrals,BMO, commutator. Both authors were supported by a grant from Australia Research Council, and the second author was also partially supported by the NNSF of China (Grant No. 10371134). c

2004 American Mathematical Society

3549

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XUAN THINH DUONG AND LI XIN YAN

Theorem 1.1. Assume condition (G) and let b(x) ∈ BMO(Rn ). Then for 0 < α < n, the commutator [b, L−α/2 ](f ) satisfies ||[b, L−α/2 ](f )||q ≤ C||b||∗ ||f ||p ,

1 1. Then for every ball Q we have |bQ − bMQ | ≤ Ckbk∗ logM. (ii) (John-Nirenberg Lemma) Let 1 ≤ p < ∞. Then b ∈ BM O if and only if Z 1 |b − bQ |p dx ≤ Ckbkp∗ . |Q| Q (iii) If 1 < r < p < n/α and 1/q = 1/p − α/n, then kMα,r (f )kq ≤ Ckf kp . Proof. For the proof of this lemma, see [Ch]. See also [CRW] and [Ja].

Lemma 2.2. For all 0 < α < n, we have kL−α/2 (f )kq ≤ Ckf kp , where 1 < p < n/α and 1/q = 1/p − α/n. Proof. We recall that the classical fractional integral Iα is defined by Z f (y) dy, 0 < α < n. Iα (f )(x) = |x − y|n−α n R Since the semigroup e−tL has a kernel pt (x, y) which satisfies the upper bound (G), it is easy to check that |L−α/2 (f )(x)| ≤ CIα (|f |)(x) for all x ∈ Rn . Using the Lp -boundedness of Iα , we have kL−α/2 (f )kq ≤ CkIα (|f |)kq ≤ Ckf kp , where 1 < p < n/α and 1/q = 1/p − α/n. See [St1, Chapter 5]. Hence Lemma 2.2 is proved. Lemma 2.3. Assume that the semigroup e−tL has a kernel pt (x, y) which satisfies the upper bound (G), and let b ∈ BMO. Then, for every function f ∈ Lp (Rn ), p > 1, x ∈ Rn and 1 < r < ∞ we have Z 1 1 |e−tQ L (b − bQ )f (y)|dy ≤ Ckbk∗ (M (|f |r )) r (x), sup |Q| x∈Q Q 2 . where tQ = rQ

For the proof of this lemma, see Lemma 2.3 in [DY].

Now, we have the following analogy of the classical Fefferman-Stein inequality [St2, Chapter IV] for the sharp maximal function ML# f . Lemma 2.4. Assume that the semigroup e−tL has a kernel pt (x, y) which satisfies the upper bound (G). Let λ > 0 and f ∈ Lp (Rn ) for some 1 < p < ∞. Then for every 0 < η < 1, we can find γ > 0 independent of λ, f in such a way that |{x ∈ Rn : M f (x) > Aλ, ML# f (x) ≤ γλ}| ≤ η|{x ∈ Rn : M f (x) > λ}|, where A > 1 is a fixed constant which depends only on n. As a consequence, we have the following estimate: (2.3)

kf kp ≤ kM f kp ≤ CkML# f kp

for every f ∈ Lp (Rn ), 1 < p < ∞. Proof. See Proposition 4.1 in [M].


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3. A kernel estimate on fractional integrals We now prove an important lemma which gives an estimate on the kernel of the difference operator L−α/2 − e−tL L−α/2 . Lemma 3.1. Assume that the semigroup e−tL has a kernel pt (x, y) which satisfies the upper bound (G). Then for 0 < α < 1, the difference operator L−α/2 − e−tL L−α/2 has an associated kernel Kα,t (x, y) which satisfies C t . |x − y|n−α |x − y|2

|Kα,t (x, y)| ≤

(3.1)

Proof. Let gα,t (z) = z −α/2 (1 − e−tz ). We first represent the operator gα,t (L) by using the semigroup e−zL . As in [DM], gα,t (L) (acting on L2 (Rn )) is given by gα,t (L) =

1 2πi

Z

(L − λI)−1 gα,t (λ)dλ,

γ

where the contour γ = γ+ ∪γ− is given by γ+ (t) = teiν for t ≥ 0 and γ− (t) = −te−iν for t < 0, and ν > π/2. For λ ∈ γ, substitute Z ∞ −1 eλs e−sL ds. (L − λI) = 0

Changing the order of integration gives Z gα,t (L) =

∞

e−sL n(s)ds,

0

where 1 n(s) = 2πi

Z eλs gα,t (λ)dλ. γ

Consequently, the kernel Kα,t (x, y) of gα,t (L) is given by Z

∞

Kα,t (x, y) =

ps (x, y)n(s)ds. 0

It follows from (G) that Z

∞

|Kα,t (x, y)| ≤ C

s

−n/2 −c |x−y| s

Z

2

e

0

∞

|esλ λ−α/2 (1 − e−tλ )|d|λ| ds.

0

Observe that |1 − e−tλ | ≤ c when t|λ| ≥ 1 and |1 − e−tλ | ≤ ct|λ| when t|λ| ≤ 1. We then split the integral on the right-hand side into parts I and II, corresponding to the integration over t|λ| > 1 and t|λ| ≤ 1. Then, Z I≤C 0

∞

s−n/2 e−c

|x−y|2 s

Z

∞

e−βsν ν −α/2 dνds

1/t


ON COMMUTATORS OF FRACTIONAL INTEGRALS

3553

with β > 0. By changing variables tν → ν and s/t → s, we have Z ∞ Z ∞ |x−y|2 s−n/2 e−c ts e−βsν ν −α/2 dνds I ≤ Ct(α−n)/2 1 Z ∞ Z0 ∞ 2 1 1 −βsν (α−n)/2 (α−n)/2 −c |x−y| ts s e e (sν)(4−α)/2 dν ds = Ct 2 2 s ν 1 Z0 ∞ |x−y|2 ds s(α−n−2)/2 e−c ts ≤ Ct(α−n)/2 s 0 t C −βsν (since e (sν)(4−α)/2 is bounded). ≤ |x − y|n−α |x − y|2 Similarly, Z II

≤ C

∞

s−n/2 e−c

0

Z

≤ Ct

(α−n)/2

≤ Ct

(α−n)/2

|x−y|2 s

Z

1/t

e−βsν ν −α/2 (tν)dνds

0 ∞

2

Z

1

s

−n/2 −c |x−y| ts

s

(α−n−2)/2 −c |x−y| ts

Z0 ∞

e

0

e

0

≤

e−βsν ν (2−α)/2 dνds 2

ds s

t C . n−α |x − y| |x − y|2

The estimates on I and II complete the proof of Lemma 3.1.

4. Proof of the main result We first prove Theorem 1.1 in the case 0 < α < 1. Choose two real numbers r and s greater than 1 such that rs < p < n/α. We will prove that there exists a constant C such that for all x ∈ Rn and for all Q 3 x, (4.1) Z 1 |(I − e−tQ L )[b, L−α/2 ]f (y)|dy ≤ Ckbk∗ [Mr (L−α/2 f )(x) + Mα,rs (f )(x)], |Q| Q 2 , and rQ is the radius of Q. where tQ = rQ From (4.1) and (2.3), Theorem 1.1 follows by Lemma 2.2 and the continuity of the maximal function Mα,r f ,

||[b, L−α/2 ]f ||q

≤

C||ML# ([b, L−α/2 ])f ||q

≤

Ckbk∗ kMr (L−α/2 f )kq + Ckbk∗ kMα,rs (f )kq

≤

C||b||∗ ||f ||p ,

n where 1q = p1 − α n and 1 < p < α . We now prove (4.1). For an arbitrary fixed x ∈ Rn , choose a ball Q = Q(x0 ; r) = {y ∈ Rn : |x0 − y| < r} which contains x. Let f1 = f χ2Q and f2 = f − f1 . We write

[b, L−α/2 ]f = (b − bQ )L−α/2 f − L−α/2 (b − bQ )f1 − L−α/2 (b − bQ )f2 and

e−tQ L ([b, L−α/2 ]f ) = e−tQ L (b − bQ )L−α/2 f − L−α/2 (b − bQ )f1 − L−α/2 (b − bQ )f2 .


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Then, Z 1 |[b, L−α/2 ]f (y) − e−tQ L [b, L−α/2 ]f (y)|dy |Q| Q Z Z 1 1 −α/2 |(b(y) − bQ )L f (y)|dy + |L−α/2 ((b(y) − bQ )f1 )(y)|dy ≤ |Q| Q |Q| Q Z 1 |e−tQ L ((b(y) − bQ )L−α/2 f )(y)|dy + |Q| Q Z 1 |e−tQ L L−α/2 ((b(y) − bQ )f1 )(y)|dy + |Q| Q Z 1 |(L−α/2 − e−tQ L L−α/2 )((b − bQ )f2 )(y)|dy + |Q| Q = I + II + III + IV + V. older’s inequality and Lemma Let r0 be the dual of r such that 1/r + 1/r0 = 1. By H¨ 2.1, Z Z 1 r0 1/r 0 1 |b(y) − bQ | dy) ( |L−α/2 f (y)|r dy)1/r I ≤ ( |Q| Q |Q| Q ≤

Ckbk∗ Mr (L−α/2 (f ))(x).

By Lemmas 2.1 and 2.2 again, we have Z 1 |L−α/2 ((b(y) − bQ )f1 )(y)|w dy)1/w II ≤ ( |Q| Rn Z 1 ≤ C ( |(b(y) − bQ )f (y)|s dy)1/s 1 |Q| w Q Z Z 0 0 1 1 ≤ C( |b(y) − bQ |sr dy)1/sr ( 1− αsr |f (y)|sr dy)1/sr |Q| Q |Q| n Q 1 α 1 ≤ Ckbk∗ Mα,rs f (x) ( = − ). w s n Similarly, we obtain by using Lemmas 2.1, 2.2 and 2.3, III + IV ≤ Ckbk∗ Mr (L−α/2 f )(x) + Ckbk∗ Mα,rs (f )(x). Let us see what happens with the term V. Using Lemmas 2.1 and 3.1, we have Z Z 1 |Kα,tQ (y, z)||(b(z) − bQ )f (z)|dzdy V ≤ |Q| Q (2Q)c ∞ Z X 1 rQ |(b(z) − bQ )f (z)|dz ≤ C n−α |x − z| |x − z| k k+1 0 0 rQ k=1 2 rQ ≤|x0 −z| 0 and f ∈ L2 (Rn ), |e−tA f | ≤ e−t4 |f |, which implies that the semigroup e−tA has a kernel pt (x, y) which satisfies the upper bound (G). Note that unless ~a and V satisfy additional conditions, the heat kernel can be a discontinuous function of the space variables and the H¨ older continuous estimates may fail to hold (see [Sh] and [Si]). (b) Let A = ((aij (x)) P1≤i,j≤n be an n × n matrix of complex with entries aij ∈ L∞ (Rn ) satisfying Re aij (x)ξi ξj ≥ λ|ξ|2 for all x ∈ Rn , ξ = (ξ1 , ξ2 , . . . , ξn ) ∈ Cn and some λ > 0. We define a divergence form operator Lf ≡ −div(A∇f ), which we interpret in the usual weak sense via a sesquilinear form. It is known that the Gaussian bound (G) on the heat kernel e−tL is true when A has real entries (see, for example, [AE]), or when n = 1, 2 in the case of complex entries; see [AT, Chapter 1]. Acknowledgment The authors thank T. Coulhon for useful suggestions. References [AE]

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ON COMMUTATORS OF FRACTIONAL INTEGRALS

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Department of Mathematics, Macquarie University, New South Wales 2109, Australia E-mail address: [email protected] Department of Mathematics, Zhongshan University, Guangzhou, 510275, People’s Republic of China E-mail address: [email protected] Current address: Department of Mathematics, Macquarie University, New South Wales 2109, Australia
