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Volume 30, Issue 3

Issues

Commutators of the fractional integrals for second-order elliptic operators on Morrey spaces

Yanping Chen
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  • Department of Applied Mathematics, School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, P. R. China
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/ Yong Ding
  • School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems (BNU), Ministry of Education, Beijing 100875, P. R. China
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Published Online: 2017-08-26 | DOI: https://doi.org/10.1515/forum-2017-0062

Abstract

Let L=-div(A) be a second-order divergence form elliptic operator and let A be an accretive, n×n matrix with bounded measurable complex coefficients in n. Let L-α2 be the fractional integral associated to L for 0<α<n. For bLloc(n) and k, the k-th order commutator of b and L-α2 is given by

(L-α2)b,kf(x)=L-α2((b(x)-b)kf)(x).

In the paper, we mainly show that if bBMO(n), 0<λ<n and 0<α<n-λ, then (L-α2)b,k is bounded from Lp,λ to Lq,λ for p-(L)<p<q<p+(L)n-λn and 1q=1p-αn-λ, where p-(L) and p+(L) are the two critical exponents for the Lp uniform boundedness of the semigroup {e-tL}t>0. Also, we establish the boundedness of the commutator of the fractional integral with Lipschitz function on Morrey spaces. The results encompass what is known for the classical Riesz potentials and elliptic operators with Gaussian domination by the classical heat operator.

Keywords: Commutator; elliptic operator; fractional integral; Morrey space

MSC 2010: 42B20; 42B25

1 Introduction

For ξ=(ξ1,,ξn)n, denote its complex conjugate (ξ¯1,,ξ¯n) by ξ¯. Let A=A(x) be an n×n matrix of complex L coefficients defined on n and satisfying the ellipticity condition

λ|ξ|2ReAξξ¯and|Aξζ¯|Λ|ξ||ζ|

for all ξ,ζn and for some λ, Λ such that 0<λΛ<. Here, we use the inner product notation uv=u1v1++unvn. Therefore, Aξζ¯j,kaj,k(x)ξkζ¯j. Associated with such a matrix A, we define a second-order divergence form operator

Lf=-div(Af)=-j=1nj((Af)j),

which is understood in the standard weak sense as a maximal-accretive operator on L2(n) with domain 𝒟(L) by means of a sesquilinear form. The maximal accretivity condition yields the existence of an analytic contraction semigroup on L2(n) generated by -L. The operator -L generates a semigroup {e-tL}t>0. Auscher [4] established an almost complete study of Lp-Lq off-diagonal estimates with pq for the semigroup {e-tL}t>0, which can be stated as follows:

e-tLfχELq(F)Ct-n2(1p-1q)e-cd(E,F)2tfLp(E)(1.1)

whenever p-(L)<p<p+(L), where p-(L) and p+(L) are the two critical exponents for the Lp uniform boundedness of the semigroup {e-tL}t>0. According to the results proved or cited in [4], p+(L)= and p-(L)=1 if n=1,2, and p+(L)>2nn-2 and p-(L)<2nn+2 if n3. In specific situations, much more can be said. For example, p-(-)=1, p+(-)= by the well-known formula for the heat kernel, where is the Laplacian. From the maximum principle for real parabolic equations, one has p-(L)=1, p+(L)= if L has real coefficients or under Gaussian domination. One can consult [6, 5, 4, 7, 21, 38] among numerous references, for the development and applications of elliptic operators.

Let bLloc(n). For any k we define the k-th order commutator by

Tb,kf(x)=T((b(x)-b)kf)(x),

fLc(n). Note that Tb,0=T. Commutators are usually considered for linear operators T, in which case they can be alternatively defined by recurrence: the first order commutator is

Tb,1f(x)=[b,T]f(x)=b(x)Tf(x)-T(bf)(x),

and for k2 the k-th order commutator is given by Tb,k=[b,Tb,k-1]. For 0<α<n, the fractional integral associated to L is defined by

L-α2f=1Γ(α2)0tα2e-tL(f)dtt

for all fL2(n), where the integrals εR converge in L2 as ε0 and R. Auscher [4] proved that for p-(L)<p<q<p+(L), the integral L-α2 is bounded from Lp(n) into Lq(n) provided that 1q=1p-αn. Let bLloc(n) and k; the k-th order commutator of b and L-α2 is given by

(L-α2)b,kf(x)=L-α2((b(x)-b)kf)(x)

for fLc(n). Note that (L-α2)b,0=L-α2 and (L-α2)b,1=[b,L-α2]. If L=-, then L-α2 is the classical fractional integral Iα. See, for example, [39, Chapter 5].

Let bBMO(n). That is,

b*:=supQ1|Q|Q|b(y)-bQ|𝑑y<,

where the supremum is taken over all cubes Q in n and

bQ=1|Q|Qb(x)𝑑x.

It is well known that if bBMO(n), the commutator [b,Iα] is bounded from Lp(n) to Lq(n), 1<p<nα, 1q=1p-αn (see [9]). In 2004, by using the sharp maximal function introduced in [26], Duong and Yan [20] extended the result of [9] from - to a general operator L and proved that for 0<α<n, 1<p<nα and 1q=1p-αn, one has [b,L-α2]fLqCbBMOfLp, provided that L is a linear operator on L2(n) which generates an analytic semigroup e-tL with a kernel pt(x,y) satisfying a Gaussian upper bound estimate, that is,

|pt(x,y)|C1tn/2e-c|x-y|2t(1.2)

for all x,yn and t>0. Using a good-λ method, the (Lp,Lq) boundedness of the commutator of the fractional integral [b,L-α2] was obtained in [7] without the extra hypothesis (1.2). It is said that bLipβ(n) if there exists a constant C such that for all x,yn and 0<β1 we have |b(x)-b(y)|C|y|β. Paluszynski [31] showed that bLipβ if and only if [b,Iα] is bounded from Lp(n) to Lq(n), where 0<β1, 1<p<nα+β and 1q=1p-α+βn. Many authors also made important progress on the fractional integral, one can consult [1, 10, 12, 18, 41] among numerous references, for its development and applications.

On the other hand, the Morrey space Lp,λ(n) (see the definition below), which was introduced by Morrey in 1938, is in connection with certain problems in elliptic PDE [28]. Later, for the Morrey spaces were found many important applications to the Navier–Stokes equations (see [27, 23, 40]), the Schrödinger equations (see [32, 35, 34, 37, 36]), the elliptic equations with discontinuous coefficients (see [8, 13, 15, 17, 22, 25, 30]) and some nonlinear potential analysis (see [3, 2]). The Morrey spaces associated with the heat kernel were studied (see [14, 20, 19, 42]). Recently, the authors of this paper and Wang studied the compactness in the Morrey spaces (see [10, 11]).

For 1p<, n1 and 0λn, the Morrey space Lp,λ(n) is defined by

Lp,λ(n)={fLlocp:fp,λ=supyn,r>0(1rλB(y,r)|f(x)|p𝑑x)1p<},

where B(y,r) denotes the ball centered at y and with radius r>0. The spaces Lp,λ(n) becomes a Banach space with norm p,λ. Moreover, for λ=0 and λ=n, the Morrey spaces Lp,0(n) and Lp,n(n) coincide (with equality of norms) with the space Lp(n) and L(n), respectively.

Moreover, for any tn, r>0 and fLc(n), without loss of generality, we may assume that suppfQ, where Q is a cube, and

(1rλB(t,r)|f(x)|p𝑑x)1pfL|B(t,r)Q|1prλp{Crn-λpC(Q)n-λpif 0<r(Q),C(Q)nprλpC(Q)n-λpif r>(Q)>0.

Thus, fLp,λC(Q)(n-λ)/p, where C is independent of r and t.

In 1975, Adams [1, Theorem 3.1] established the classical Sobolev imbedding to Morrey spaces. Namely, let 0<λ<n and 0<α<n-λ; then there is a constant C>0, such that IαfLq,λCfLp,λ for 1<p,q< and 1q=1p-αn-λ. Also (see [16]), that the commutator [b,Iα] which is generated by the potential operator Iα and a BMO function b is bounded from Lp,λ(n) to Lq,λ(n) if 0<λ<n, 0<α<n-λ, 1<p,q< and 1q=1p-αn-λ. See also [16, 24, 33, 29] and the references therein.

Our contribution to this direction is to give the Morrey boundedness of the k-th order commutator of the fractional integral (L-α2)b,k, which is a new theory for the second-order elliptic operators on n. Now, let us formulate the main results in this paper as follows.

Theorem 1.1.

Let 0λ<n, 0<α<n-λ, kN and bBMO(Rn). Let L be defined as in (1.2). Then for p-(L)<p<q<p+(L)n-λn and 1q=1p-αn-λ, we have for fLc(Rn),

(L-α2)b,kfLq,λCbkfLp,λ,

where C is independent of f and b.

Taking k=0 in Theorem 1.1, we have the following corollary.

Corollary 1.2.

Let 0λ<n, let 0<α<n-λ and let kN. Further, let L be defined as in (1.2). Then for p-(L)<p<q<p+(L)n-λn and 1q=1p-αn-λ, we have for fLc(Rn),

L-α2fLq,λCfLp,λ,

where C is independent of f.

Taking k=1 in Theorem 1.1, we have the following corollary.

Corollary 1.3.

Let 0λ<n, let 0<α<n-λ and let bBMO(Rn). Further, let L be defined as in (1.2). Then for p-(L)<p<q<p+(L)n-λn and 1q=1p-αn-λ, we have for fLc(Rn),

[b,L-α2]fLq,λCbfLp,λ,

where C is independent of f and b.

Remark 1.4.

In specific situations, much more can be said. For example, from the well-known formula for the heat kernel we have p-(-)=1 and p+(-)=. When L=-, then L-α2 is just the classical fractional integral Iα. Our results not only extend the result of [1, 16] from - to a general operator L, but also without any extra assumption such as (1.2). The standard techniques of singular integrals as in [1, 16] are not applicable in the proof of Theorem 1.1. We overcome this difficulty by using the properties of the Morrey space and the Marcinkiewicz interpolating theorem related to the Morrey space; this method is fundamentally different from [7].

Our second result is related to bLipβ(n), and is stated as follows.

Theorem 1.5.

Let 0λ<n, 0<β1, 0<α<n-λ, bLipβ(Rn). Let L be defined as in (1.2). Then for p-(L)<p<q<p+(L)n-λn and 1q=1p-α+βn-λ we have for fLc(Rn),

[b,L-α2]fLq,λCbLipβfLp,λ,

where C is independent of f and b.

Throughout this paper, the Lebesgue measure of a measurable subset Sn is denoted by |S|, while its indicator function 1S is defined by 1S(x)=1 if xS and 1S(x)=0 if xnS. For any cube Q in n, we denote by |Q| and (Q) its measure and its side length, respectively. We also use the notation CQ to denote the concentric cube with Q having side length C(Q). For p1, we denote by p the dual exponent of p, that is, p=pp-1. Throughout this paper, the letter C will stand for a positive constant which is independent of the essential variables and not necessarily the same one in each occurrence. The remainder of this paper is organized as follows: in Section 2, we give the proof of Theorem 1.1, and in Section 3 we give the proof of Theorem 1.5.

2 The fractional integral and BMO function

2.1 Some lemmas

Let us begin with some lemmas, which will be used in the proof of Theorem 1.1.

Lemma 2.1.

Let E and F be two closed sets of Rn and set d=dist(E,F), the distance between E and F. Fix t>0. Let 1p0<p1. Suppose that Tt satisfies

(F|Ttf(x)|q~𝑑x)1q~Ct12(nq~-np~)e-d2ct(E|f(x)|p~𝑑x)1p~(2.1)

for all p0<p~q~<p1. Let bBMO and kN and let Tt;b,k denote the k-th order commutator of Tt. Let 0λ<n. Then we get for any fixed p0<pq<p1n-λn and fLc,

Tt;b,kfLq,λCt-n-λ2(1p-1q)bkfLp,λ,

where C is independent of t, b and f.

Proof.

Let B be any fixed ball with radius r. For any l, we denote

Sl(B)=2l+1B2lBif l1  and  S0(B)=2B.

We write

(1rλB|Tt;b,kf(x)|q𝑑x)1q=(1rλB|Tt;b,kl0f1Sl(B)(x)|q𝑑x)1q(2.2)l0(1rλB|Tt;b,kf1Sl(B)(x)|q𝑑x)1q.

Let bB=1|B|Bb(x)𝑑x. Note that

Tt;b,kf1Sl(B)(x)=Tt((b(x)-b)kf1Sl(B))(x)=Tt((b(x)-bB-(b-bB))kf1Sl(B))(x)=Tt;b-bB,kf1Sl(B)(x).

We take p0<pq<p1n-λn; then

(1rλB|Tt;b,kf1Sl(B)(x)|q𝑑x)1q=(1rλB|Tt;b-bB,kf1Sl(B)(x)|q)1qj=0kCkj(1rλB|(b(x)-bB)jTt((b-bB)k-jf1Sl(B))(x)|q𝑑x)1q.

Taking s(s<nλ) such that p0<pqs<p1, by Hölder’s inequality we get

(1rλB|Tt;b,kf1Sl(B)(x)|q𝑑x)1qj=0kCkj[(B|(b(x)-bB)|qjs𝑑x)1sr-λ(B|Tt((b-bB)k-jf1Sl(B))(x)|qs𝑑x)1s]1q.

For any fixed ball B and any fixed 1<w<, we have

(1|B|B|b(y)-bB|wj𝑑y)1wCbj.

Thus we get

(1rλB|Tt;b,kf1Sl(B)(x)|q𝑑x)1qCj=0kCkjbj[|B|1sr-λ(B|Tt((b-bB)k-jf1Sl(B))(x)|qs𝑑x)1s]1q.

We take θ such that p0<θ<pqs<p1 in (2.1) and use this to estimate

(1rλB|Tt;b,kf1Sl(B)(x)|q𝑑x)1qCt-n2(1θ-1qs)j=0kCkjbj[|B|1sr-λe-dist(Sl(B),B)2ct(Sl(B)|(b(x)-bB)k-jf1Sl(B)(x)|θ𝑑x)qθ]1q.

By Hölder’s inequality and |b2lB-bB|Clb, we get

(1rλB|Tt;b,kf1Sl(B)(x)|q𝑑x)1qCt-n2(1θ-1qs)rnqsj=0kCkjbj[r-λe-4lr2ct(2l+1B|(b(x)-bB)|(k-j)pθp-θ𝑑x)(p-θ)qpθ(2l+1B|f(x)|p𝑑x)qp]1qCt-n2(1θ-1qs)rnqsj=0kCkjbjbk-jr-λqe-4lr2ctql|2l+1B|(p-θ)pθ(2l+1B|f(x)|p𝑑x)1pCt-n2(1θ-1qs)rnqsrn(p-θ)pθbkr-λqe-4lr2ctql2(p-θ)lnpθ(2l+1B|f(x)|p𝑑x)1pCt-n2(1θ-1qs)rnqsrn(p-θ)pθbkr-λqe-4lr2ctql2(p-θ)lnpθ2lλprλp(1(2l+1r)λ2l+1B|f(x)|p𝑑x)1pCl2((p-θ)npθ+λp)le-4lr2ctqt-n2(1θ-1qs)rnqs+n(p-θ)pθ+λ(1p-1q)bkfLp,λ.(2.3)

Then by (2.2) and (2.3) we get

(1rλB|Tt;b,kf(x)|q𝑑x)1qCl0(1rλB|Tt;b,kf1Sl(B)(x)|q𝑑x)1qC(l0l2((p-θ)npθ+λp)le-4lr2ctq)t-n2(1θ-1qs)rnqs+n(p-θ)pθ+λ(1p-1q)bkfLp,λ.

To proceed with the estimates we claim that for s>0 and β0 if 0<c1<c2, then

k=0k2kβe-c24ks2{Ce-c1s2if s1,Cs-βln1sif 0<s<1.(2.4)

In fact, if s1, since e-c24ks2Ce-c~4ke-c1s2 for some c~>0, we get

k=0k2kβe-c24ks2Ck=0k2kβe-c~4ke-c1s2Ce-c1s2.

If 0<s<1, then there is k0 such that 2-k0s<2-k0+1. We obtain that

k=0k2kβe-c24ks2Ck=0k2kβe-c24k-k0Ck=0k0k2kβe-c24k-k0+Ck=k0+1(k+k0)2kβe-c24k-k0Ck02k0β+Ck=1(k+2k0)2(k+k0)βe-c24kCk02k0βCs-βln1s.

Therefore, we established (2.4). Applying (2.4), we get for 0<c<c1,

l0l2((p-θ)npθ+λp)le-4lr2ctq{Ce-r2c1tqif rt,C(rt)-((p-θ)npθ+λp)lntrif r<t.(2.5)

Therefore, we get by (2.5) for rt,

(1rλB|Tt;b,kf(x)|q𝑑x)1qCe-r2c1tqt-n2(1θ-1qs)rnqs+n(p-θ)pθ+λ(1p-1q)bkfLp,λCt-n-λ2(1p-1q)[e-r2c1tqt-n2(1θ-1qs)+n-λ2(1p-1q)rnqs+n(p-θ)pθ+λ(1p-1q)]bkfLp,λCt-n-λ2(1p-1q)[e-r2c1tq(rt)nθ+λ(1p-1q)]bkfLp,λ.

Since

e-r2c1tq(rt)nθ+λ(1p-1q)C

for rt, we get

(1rλB|Tt;b,kf(x)|q𝑑x)1qCt-n-λ2(1p-1q)bkfLp,λ.(2.6)

For r<t, by (2.5) we have

(1rλB|Tt;b,kf(x)|q𝑑x)1qClntr(rt)-((p-θ)npθ+λp)t-n2(1θ-1qs)rnqs+n(p-θ)pθ+λ(1p-1q)bkfLp,λCt-n-λ2(1p-1q)[lntr(rt)-((p-θ)npθ+λp)t-n2(1θ-1qs)+n-λ2(1p-1q)rnqs+n(p-θ)pθ+λ(1p-1q)]bkfLp,λCt-n-λ2(1p-1q)[lntrt(p-θ)n2pθ+λ2pt-n2(1θ-1qs)+n-λ2(1p-1q)rnqs-λq]bkfLp,λCt-n-λ2(1p-1q)[lntrt-n-λ2ptn2qs+n-λ2(1p-1q)rnqs-λq]bkfLp,λCt-n-λ2(1p-1q)[lntrtn2qs-n-λ2qrnqs-λq]bkfLp,λCt-n-λ2(1p-1q)[lntrt-n2qs+λ2qrnqs-λq]bkfLp,λCt-n-λ2(1p-1q)[lntr(rt)nqs-λq]bkfLp,λ.

Since s<nλ, we get

(1rλB|Tt;b,kf(x)|q𝑑x)1qCt-n-λ2(1p-1q)bkfLp,λ.(2.7)

Combining the estimates of (2.6) and (2.7), we finish the proof of Lemma 2.1. ∎

2.2 Proof of Theorem 1.1

Proof.

For 0<α<n-λ, bBMO(n) and k0, for all fLc(n) we write

(L-α2)b,kf=1Γ(α2)0(tα2e-tL)b,k(f)dtt.

Fix α=n-λp-n-λq. Let B be any fixed ball with radius r and let p-(L)<p<d0<q<d1<p+(L)n-λn. Applying Lemma 2.1 to (1.1), we get

(e-sL)b,kfLq,λCbks12(n-λq-n-λp)fLp,λ,p-(L)<pq<p+(L)n-λn.(2.8)

Then we have

(1rλB|1Γ(α2)0s(tα2e-tL)b,kf(x)dtt|d0𝑑x)1d0C0stα2(1rλB|(e-tL)b,kf(x)|d0𝑑x)1d0dttCbkfLp,λ0stn-λ2(1d0-1p)tα2dtt=CbkfLp,λ0stn-λ2(1d0-1q)dttCbks12(n-λd0-n-λq)fLp,λ.(2.9)

On the other hand, by (2.8) we have

(1rλB|1Γ(α2)s(tα2e-tL)b,kf(x)dtt|d1𝑑x)1d1Cstα2(1rλB|(e-tL)b,kf(x)|d1𝑑x)1d1dttCbkfLp,λstα2tn-λ2(1d1-1p)dtt=CbkfLp,λstn-λ2(1d1-1q)dttCbks12(n-λd1-n-λq)fLp,λ.(2.10)

Hence, we have for μ>0,

|{xB:|(L-α2)b,kf(x)|>2μ}|=|{xB:|1Γ(α2)0(tα2e-tL)b,kf(x)dtt|>2μ}||{xB:|1Γ(α2)0s(tα2e-tL)b,kf(x)dtt|>μ}|+|{xB:|1Γ(α2)s(tα2e-tL)b,kf(x)dtt|>μ}|1μd0B|1Γ(α2)0s(tα2e-tL)b,kf(x)dtt|d0𝑑x+1μd1B|1Γ(α2)s(tα2e-tL)b,kf(x)dtt|d1𝑑x.

Then by (2.9) and (2.10) we get

|{xB:|(L-α2)b,kf(x)|>2μ}|Crλ(bkd0μd0sd02(n-λd0-n-λq)fLp,λd0+bkd1μd1sd12(n-λd1-n-λq)fLp,λd1).

Choosing

μbkfLp,λ=s-n-λ2q,

for p-(L)<p<q<p+(L)n-λn and 1q=1p-αn-λ we get

|{xB:|(L-α2)b,kf(x)|>2μ}|CrλbkqμqfLp,λq.(2.11)

Now, let us use (2.11) to estimate

1rλB|(L-α2)b,kf(x)|q𝑑x

for p-(L)<q<p+(L)n-λn. Let p-(L)<q0<q<q1<p+(L)n-λn, let p-(L)<p0<p<p1<p+(L)n-λn and let 1q=1p-αn-λ, 1qi=1pi-αn-λ, i=0,1. It is easy to verify that

qp=q-q1p-p1p1q1=q-q0p-p0p0q0.

For any μ>0, take β=qp and 1γ=bk. Let

fμ(x)={f(x)if |f(x)|γμβ,0if |f(x)|>γμβ,

and

fμ(x)=f(x)-fμ(x).

Then, using (2.11), we get

1rλB|(L-α2)b,kf(x)|q𝑑x=qrλ0μq-1|{xB:|(L-α2)b,kf(x)|>2μ}|𝑑μ=qrλ0μq-1|{xB:|(L-α2)b,k(fμ+fμ)(x)|>2μ}|𝑑μqrλ0μq-1|{xB:|(L-α2)b,kfμ(x)|>μ}|𝑑μ+qrλ0μq-1|{xB:|(L-α2)b,kfμ(x)|>μ}|𝑑μCqbkq10μq-q1-1fμLp1,λq1𝑑μ+Cqbkq00μq-q0-1fμLp0,λq0𝑑μ:=I+II.

For I, we have

ICqbkq10μq-q1-1(supxn,r>01rλB(x,r)|fμ(y)|p1𝑑y)q1p1𝑑μ.

Since there exist an x0n and r0>0 such that

supxn,r>01rλB(x,r)|fμ(y)|p1𝑑y2r0λB(x0,r0)|fμ(y)|p1𝑑y,

by the above inequality we get

Ip1q1(Cqbkq10μq-q1-1(1r0λB(x0,r0)|fμ(y)|p1𝑑y)q1p1𝑑μ)p1q1.

Then by the Minkowski inequality we get

Ip1q1Cqp1q1bkp11r0λB(x0,r0)|f(y)|p1((|f(y)|γ)1βμq-q1-1𝑑μ)p1q1𝑑yCbkp1γ-q-q1βp1q11r0λB(x0,r0)|f(y)|p1|f(y)|q-q1βp1q1𝑑y.

Note that β=q-q1p-p1p1q1=q-q0p-p0p0q0 and 1γ=bk. We get

Ip1q1Cbkp1γ-(p-p1)1r0λB(x0,r0)|f(y)|p𝑑yCbkpfLp,λp.

Similarly, we can also get

IIp0q0CbkpfLp,λp.

Combining the estimates of I and II, we get

(L-α2)b,kfLq,λqC(bkfLp,λ)pq0p0+(bkfLp,λ)pq1p1.

If b*=1 and fp,λ=1, we can write

(L-α2)b,kfLq,λCbkfLp,λ.(2.12)

In fact, (2.12) also holds for any bBMO and fLp,λ by taking

b~=bbandf~=ffLp,λ.

3 The fractional integral and Lipschitz function

Lemma 3.1.

Let E and F be two closed sets of Rn and set d=dist(E,F), the distance between E and F. Fix t>0. Let 1p0<p1. Suppose that Tt satisfies

(F|Ttf(x)|q~𝑑x)1q~Ct12(nq~-np~)e-d2ct(E|f(x)|p~𝑑x)1p~

for all p0<p~q~<p1. Let bLipβ, 0<β1, and let [b,Tt] denote the commutator of Tt. Let 0λ<n. Then we get for any fixed p0<pq<p1n-λn and fLc,

[b,t-β2Tt]fLq,λCt-n-λ2(1p-1q)bLipβfLp,λ,

where C is independent of t, b and f.

Proof.

Let B be any fixed ball with radius r. For any l, we denote

Sl(B)=2l+1B2lBif l1  and  S0(B)=2B.

Set bB=1|B|Bb and write

(1rλB|[b,t-β2Tt]f(x)|q𝑑x)1ql0(1rλB|[b-bB,t-β2Tt]f1Sl(B)(x)|q𝑑x)1q.(3.1)

Take p0<pq<p1n-λn and s<nλ such that p0<pqs<p1. Then by |b(x)-bB|CrβbLipβ if xB, and Hölder’s inequality,

(1rλB|[b-bB,t-β2Tt]f1Sl(B)(x)|q𝑑x)1qj=01(1rλB|(b(x)-bB)jt-β2Tt((b-bB)1-jf1Sl(B))(x)|q𝑑x)1qCj=01bLipβjr-λq+βjrnsq(B|t-β2Tt((b-bB)1-jf1Sl(B))(x)|sq𝑑x)1sq.

We use that |b(x)-bB|C(2lr)βbLipβ if xSl(B) to estimate

(1rλB|[b-bB,t-β2Tt]f1Sl(B)(x)|q𝑑x)1qCj=01bLipβjr-λq+βjrnsqt-n2(1p-1sq)-β2e-dist(Sl(B),B)2ct(Sl(B)|(b(x)-bB)1-jf1Sl(B)(x)|p𝑑x)1pCj=01bLipβr-λq+βjrnsq(2lr)β(1-j)t-n2(1p-1sq)-β2e-dist(Sl(B),B)2ct(Sl(B)|f1Sl(B)(x)|p𝑑x)1pCbLipβr-λqrnsq(2lr)βt-n2(1p-1sq)-β2e-4lr2ct2lλprλp(1(2l+1r)λ2l+1B|f(x)|p𝑑x)1pC2(β+λp)le-4lr2ctt-n2(1p-1q)-β2-nsqrβ+λ(1p-1q)+nsqbLipβfLp,λ.(3.2)

Then by (3.1) and (3.2) we get

(1rλB|Tt;b,kf(x)|q𝑑x)1qC(l02(β+λp)le-4lr2ct)t-n2(1p-1q)-β2-nsqrβ+λ(1p-1q)+nsqbLipβfLp,λ.

Applying (2.4), we get for 0<c<c1,

l02(β+λp)le-4lr2ct{Ce-r2c1tif rt,C(rt)-(β+λp)lntrif r<t.(3.3)

Therefore, we get by (3.3) for rt,

(1rλB|Tt;b,kf(x)|q𝑑x)1qCe-r2c1tt-n2(1p-1q)-β2-nsqrβ+λ(1p-1q)+nsqbLipβfLp,λCt-n-λ2(1p-1q)[e-r2c1t(rt)β+λ(1p-1q)+nsq]bLipβfLp,λ.

Since

e-r2c1t(rt)β+λ(1p-1q)+nsqC

for rt, we get

(1rλB|Tt;b,kf(x)|q𝑑x)1qCt-n-λ2(1p-1q)bLipβfLp,λ.(3.4)

For r<t, by (3.3) we have

(1rλB|Tt;b,kf(x)|q𝑑x)1qC(rt)-(β+λp)lntrt-n2(1p-1q)-β2-nsqrβ+λ(1p-1q)+nsqbLipβfLp,λCt-n-λ2(1p-1q)[lntr(rt)-λq+nsq]bLipβfLp,λ.

Since s<nλ, we get

(1rλB|Tt;b,kf(x)|q𝑑x)1qCt-n-λ2(1p-1q)bLipβfLp,λ.(3.5)

Combining the estimates of (3.4) and (3.5), we finish the proof of Lemma 3.1. ∎

Proof of Theorem 1.5.

Define

[b,L-α2]f=1Γ(α2)0[b,tα2e-tL](f)dtt

for all fLc(n). Let fLp,λ=1. Fix

α+β=12(n-λp-n-λq).

For any fixed s>0 and p-(L)<p<q0<q<q1<p+(L)n-λn, by Lemma 3.1 we have

(1rλB|1Γ(α2)0s[b,tα2e-tL]f(x)dtt|q0𝑑x)1q0C0s[b,tα2e-tL](f)Lq0,λdtt=C0s[b,t-β2e-tL](f)Lq0,λtα+β2dttCbLipβ0st12(n-λq0-n-λp+α+β)dtt=CbLipβ0st12(n-λq0-n-λq)dtt=CbLipβs12(n-λq0-n-λq),

and similarly

(1rλB|1Γ(α2)s[b,tα2e-tL]f(x)dtt|q1𝑑x)1q1CbLipβs12(n-λq1-n-λq).

Then, repeating the argument of Theorem 1.1, we get the desired result. ∎

Acknowledgements

The authors would like to express their deep gratitude to the referee for giving many valuable suggestions.

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About the article


Received: 2017-03-27

Revised: 2017-08-02

Published Online: 2017-08-26

Published in Print: 2018-05-01


Funding Source: National Natural Science Foundation of China

Award identifier / Grant number: 11471033

Award identifier / Grant number: 11371057

Award identifier / Grant number: 11571160

Funding Source: Program for New Century Excellent Talents in University

Award identifier / Grant number: NCET-11-0574

Funding Source: University of Science and Technology Beijing

Award identifier / Grant number: FRF-BR-16-011A

Funding Source: Ministry of Education of the People’s Republic of China

Award identifier / Grant number: 20130003110003

The first author is supported by NSF of China (grant 11471033), NCET of China (grant NCET-11-0574) and the Fundamental Research Funds for the Central Universities (grant FRF-BR-16-011A). The second author is supported by NSF of China (grants 11371057, 11571160) and SRFDP of China (grant 20130003110003).


Citation Information: Forum Mathematicum, Volume 30, Issue 3, Pages 617–629, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum-2017-0062.

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