# Commutators of Square Functions Related to Fractional Differentiation for Second-Order Elliptic Operators

Authors: Xiongtao Wu; Wenyu Tao; Yanping Chen; Kai Zhu
Journal: Journal of Function Spaces
 (2018)

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## Abstract

LetL=-div(A∇) be a second-order divergence form elliptic operator, where A is an accretive n×n matrix with bounded measurable complex coefficients in Rn. In this paper, we mainly establish the Lp boundedness for the commutators generated by b∈Iα(BMO) and the square function related to fractional differentiation for second-order elliptic operators.


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## Body

## 1. Introduction

LetT be a linear operator in a measurable function space; then, the commutator formed by T and b∈Lloc(Rn) is defined by (1)b,Tfx≔bxTfx-Tbfx. For b∈Lloc(Rn), set (2)Mb,Q=1Q∫Qby-bQdy, where Q is a cube in Rn and bQ=Q-1∫Qb(y)dy. Then, the BMO space is defined as (3)BMO=b∈LlocRn:b∗≔supQ⊂RnMb,Q<∞.Let0<α<1, and consider the fractional differentiation operators of even and odd parities, defined for tempered distributions f∈S′(Rn), by Dαf^(ξ)=ξαf^(ξ). Let Iα be the Riesz potential operator of order α and be defined in the space of tempered distribution modulo polynomials by setting Iαf^(ξ)=ξ-αf^(ξ). The BMO Sobolev space Iα(BMO) is the image of BMO under Iα. Equivalently, b∈Iα(BMO) if and only if Dαb∈BMO. Let b∈Lipα, subsequently, if b satisfies bLipα=sup⁡bx-by/x-yα<∞, in which the supremum is taken over all x,y∈Rn and x≠y. For α∈(0,1), Iα(BMO) is a space of functions modulo constants that is properly contained in Lipα (see [1, 2]).Before presenting our main theorem, we introduce the second-order elliptic operatorL as follows: For ξ=(ξ1,…,ξn)∈Cn, 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 Rn that satisfy the ellipticity condition: (4)λξ2≤Re⁡Aξ·ξ¯,Aξ·ζ¯≤Λξζ, for ξ,ζ∈Cn and for some λ,Λ such that 0<λ≤Λ<∞. Here, the inner product notation u·v=u1v1+⋯+unvn. Therefore, Aξ·ζ¯≡∑j,kaj,k(x)ξkζ¯j. Associated with such a matrix A, we define a second-order divergence form operator:(5)Lf=-div⁡A∇f=-∑j=1n∂jA∇fj,which we interpret in the usual weak sense via a sesquilinear form. The operator -L generates a semigroup (e-tL)t>0 and the gradient of the semigroup (∇t1/2e-tL)t>0.In this paper, we first present a general criterion for weak-type(p,p) boundedness of commutators with square functions and b∈Iα(BMO).Theorem 1.
Let1<p<2, 0<α≤1, b∈Iα(BMO), and m be an integer greater than 1. Let E and F be two closed subsets of Rn with a Euclidean distance d(E,F) between each other, and let {Ts}s>0 be a family of sublinear operators acting on L2(Rn). Assume that, for t>0 and 2<p1<∞,(6)∫0∞b,TsI-e-tLmf2dss1/2Lp1F≤CtηdE,F2t-m-ηfLpE,where η=(1/2)(n/p1-n/p). Furthermore, if(7)∫0∞Tsf2dss1/2L2≤CDαfL2,(8)∫0∞b,Tsf2dss1/2L2≤CfL2.Then, we have (9)x∈Rn:∫0∞b,Tsfx2dss1/2>λ≤Cλ-pfLpp.We recall a square function, which is representative of larger classes of square functions associated withL, given as follows [3]: (10)gLfx=∫0∞t1/2∇e-tLfx2dtt1/2. In 2007, Aushcer [3] proved that gLfLp~fLp for qL<p<q~L. The interval (qL,q~L) is the maximal open interval required for the semigroup (L∇e-tL)t>0 to be Lp bounded. We recall that (pL,p~L) is the maximal open interval required for the semigroup (e-tL)t>0 to be Lp bounded. In [3], the author has shown in general that pL=1, p~L=∞ if n=1,2; pL<2n/(n+2), p~L>2n/(n-2) if n≥3 and pL=qL.Many researchers have contributed to the commutators associated with the second-order elliptic operator, and among the numerous studies, some related to development and applications have been cited herein [4–9]. In particular, commutators with fractional differentiations associated with L play an important role in the theory of linear partial differential equations and harmonic analysis [10–14]. Naturally, the case of the commutators of square functions being related to fractional differentials associated with L is worth studying.In this paper, we define a square function related to the fractional differential operator associated withL as follows: (11)GLαfx=∫0∞t1/2∇e-tLfx2dtt1+α1/2,0<α<1. Moreover, for 0<α<1 and b∈Iα(BMO), the commutator of GLα can be defined by (12)GL;bαfx=∫0∞b,t1/2∇e-tLfx2dtt1+α1/2. In this paper, we also establish the Lp boundedness for GL;bα.Theorem 2.
LetL be a second-order elliptic operator in divergence form defined by (5), 0<α<1, and b∈Iα(BMO). Then, for pL<p≤2, we have (13)GL;bαfLp≤CDαbBMOfLp.The remainder of this paper is organised as follows: in Section2, we present some lemmas that play an important role in the proof of the main results; in Section 3, we prove Theorem 1; in Section 4, we prove Theorem 2. For p≥1, p′ denotes the dual exponent of p, i.e., p′=p/(p-1). Throughout this paper, the letter “C” will stand for a positive constant that is independent of the essential variables but will not necessarily have the same value for each occurrence.


## 2. Preliminary Lemmas

The second-order elliptic operatorL in divergence form is defined by (5) and has the following off-diagonal estimates (see [3, 4, 9] and references therein).Lemma 3 (see [3, 4, 9]).
LetL be a second-order elliptic operator defined by (5), let E and F be two closed sets of Rn, and let set d(E,F) denote the distance between E and F. Then, for t>0, the complex-valued function f, and vector-valued functions f→, the following statements hold:
(i) LetqL<p≤q<q~L and f∈Lp(Rn) supported in E,(14)t1/2∇e-tLfLqF≤Ct-n/21/p-1/qe-cdE,F2/tfLpE. (ii) Let pL<p≤q<p~L, f∈Lp(Rn) supported in E and k∈N,(15)tLke-tLfLqF≤Ct-n/21/p-1/qe-cdE,F2/tfLpE. (iii) Let pL<p≤q<p~L and f→∈L˙1p(Rn) supported in E,(16)t∇e-tLdiv⁡f→LqF≤Ct-n/21/p-1/qe-cdE,F2/tf→LpE. In particular, if we choose E=F=Rn, the abovementioned Lp-Lq off-diagonal estimates become Lp-Lq estimates.Another very useful and well-known lemma for off-diagonal estimates is introduced here, which could be proved by using a similar argument for the proof of a previous lemma [9, lemma 2.3].Lemma 4 ([3, 9]).
Let1<p≤q<∞ and E,F be arbitrary closed subsets of Rn. Assume that the two families of the operators {At}t>0 and {Bs}s>0 satisfy the following off-diagonal estimates: (17)AtgχELqF≤Ce-dE,F2/ctgLqE,BsgχELqF≤Csn/21/q-1/pe-dE,F2/csgLpE. Then, for t,s>0 and f∈Lp(Rn) supported in E, we have (18)AtBsfLqF≤Csn/21/q-1/pe-dE,F2/cmax⁡t,sfLpE.Next, let us introduce a criterion that deals with the boundedness of the commutators of the operators satisfyingLp→Lq off-diagonal estimates, which can be proved in [15].Lemma 5 (see [15]).
LetE and F be two closed subsets of Rn with a Euclidean distance d(E,F), and let {Ts}s>0 be a family of sublinear operators acting on L2(Rn). Assume that, for 1≤p≤q≤∞ and f∈C0∞(Rn) with supp⁡f⊂E,(19)TsfLqE≤Cs1/2n/q-n/pe-dE,F2/csfLpE.If max{diamE,diamF}≤Cd(E,F), then for 0<α<1 and b∈Lipα(Rn), we have(20)b,s-α/2TsfLqF≤Cs1/2n/q-n/pe-dE,F2/csbLipαfLpE,where C is independent of s,b, and f.The following two lemmas are about theLp-Lq off-diagonal estimates related to some commutators of the Lipschitz function and semigroups for second-order elliptic operators.Lemma 6 (see [16]).
LetE and F be two closed subsets of Rn with a Euclidean distance d(E,F). Assume that 0<α<1 and b∈Lipα(Rn) and 0≤γ≤1. If max{diamE,diamF}≤Cd(E,F), then for s>0 and f∈Lp(Rn) supported in E, we obtain the following for some ε>0:(21)s-α/2b,sγ/2Lγ/2e-sLfLqF≤Cs1/2n/q-n/pe-dE,F2/csbLipαfLpE,pL<p≤q<2+ε.Lemma 7 (see [16]).
LetL be the second-order elliptic operator in divergence form defined by (5), E and F be two closed sets of Rn, and d(E,F) express the distance between E and F. Assume that 0<α<1, b∈Lipα(Rn), and 0≤γ≤1. Then, for s>0, f∈Lp(Rn) supported in E, we obtain(22)b,s1-α/2∇e-tLfLqF≤CbLipαs-n/21/p-1/qe-dE,F2/csfLpE,pL<p≤q<p~L, where C is independent of s,b,f.


## 3. Proof of Theorem1

For any fixedλ>0, without loss of generality, we may assume that f∈Lp(Rn) is nonnegative. Let us write M for the Hardy-Littlewood maximal function. We use the Calderón-Zygmund decomposition for f(x)p at height λp. Then, there exists a collection of pairwise disjoint cubes {Qj}j such that (23)x∈Rn:Mfpx1/p>λ=⋃jQj, and they satisfy the following property:(24)λ≤1Qj∫Qjfxpdx1/p≤Cλ.Then, we write f=g+h=g+∑jhj, where (25)gx=fxχRn∖⋃jQj+∑j1Qj∫QjfydyχQjx,hjx=fx-1Qj∫QjfydyχQjx. After estimating (24), p>1 and the standard arguments yield 0≤g(x)≤Cλ for almost every x∈Rn. Then, (26)∫Qjhjxdx=0,and1Qj∫Qjhjxpdx1/p≤Cλ,hencex∈Rn:∫0∞b,Tsfx2dss1/2>3λ≤x∈Rn:∫0∞b,Tsgx2dss1/2>λ+x∈Rn:∫0∞b,Tshx2dss1/2>2λ≔I+II. We estimate every term separately. For I, we use (8) and the properties of g to obtain (27)I≤1λ2∫Rn∫0∞b,Tsgx2dssdx≤Cλ2∫Rngx2dx≤Cλ2-pλ2∫Rngxpdx≤Cλ-pfLpp. Now, we proceed with II. Let us fix an integer m≥1. We write tj=l(Qj)2, where l(Qj) stands for the side length of the cube Qj. We use the notation Qj∗=2Qj, where, in general, we write ρQ for the ρ-dilated Q, i.e., for the cube with the same centre as Q and with the side length ρl(Q). Let E∗=Rn\⋃jQj∗. Because (28)∫0∞b,Tshx2dss1/2=∫0∞∑jb,Tshjx2dss1/2≤∫0∞∑jb,TsI-e-tjLmhjx2dss1/2+∫0∞∑jb,TsI-I-e-tjLmhjx2dss1/2,we obtain (29)II≤x:∫0∞∑jb,TsI-e-tjLmhjx2dss1/2>λ+x:∫0∞∑jb,TsI-I-e-tjLmhjx2dss1/2>λ≤⋃jQj∗+x∈E∗:∫0∞∑jb,TsI-e-tjLmhjx2dss1/2>λ+x:∫0∞∑jb,TsI-I-e-tjLmhjx2dss1/2>λ≔II1+II2+II3.The first term can be estimated as follows: (30)II1≤∑jQj∗≤CλpfLpp. Now, we complete the estimate of II2. By Chebychev’s inequality, we obtain(31)II2≤1λp1∫0∞∑jb,TsI-e-tjLmhjx2dss1/2Lp1E∗p1=1λp1supv∫Rn∫0∞∑jb,TsI-e-tjLmhjx2dssvxdxp1/2, where the supremum is taken over all the functions v∈Lp1/2′(E∗) with vLp1/2′(E∗)=1. We set (32)S0Qj=2Qj;SlQj=2l+1Qj∖2lQj,l=1,2,⋯.Let us recall that E∗=Rn\⋃jQj∗. Because supp v⊂(2Qj)c, we have (33)∫Rn∫0∞∑jb,TsI-e-tjLmhjx2dssvxdx1/2≤∫Rn∫0∞∑jb,TsI-e-tjLmhjxvx1/22dssdx1/2=∑j∑l=1∞∫SlQj∫0∞b,TsI-e-tjLmhjxvx1/22dssdx1/2≤C∑j∑l=1∞∫0∞b,TsI-e-tjLmhj2dss1/2Lp1SlQj2vLp1/2′SlQj1/2≤C∑j∑l=1∞tj2ηdSlQj,Qj2tj-2m-ηhjLpQj2vLp1/2′SlQj1/2,Where, in the last inequality, we used (6). Because tj=l2(Qj), for l≥1, we obtain d(Sl(Qj),Qj)≥2l-2l(Qj). Recall that η=(1/2)(n/p1-n/p). Subsequently, we obtain (34)
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.Then, using Qj~λ-pfLp(Qj)p, we obtain (35)∫Rn∫0∞∑jb,TsI-e-tjLmhjx2dssvxdx1/2≤Cλ1+1/2pfLp-p/2∑jQjess⁡infy∈Qj⁡Mvp1/2′y1/2p1/2′≤Cλ1+1/2pfLp-p/2∑j∫QjMvp1/2′x1/2p1/2′dx≤Cλ1+1/2pfLp-p/2∫⋃jQjMvp1/2′x1/2p1/2′dx.Then, because the Hardy-Littlewood maximal function is of weak-type (1,1), we use vL2=1 and Kolmogorov’s lemma to obtain (36)∫Rn∫0∞∑jb,TsI-e-tjLmhjx2dssvxdx1/2≤Cλ1+1/2pfLpQj-p/2⋃jQj1-1/2p1/2′vp1/2′11/2p1/2′≤Cλ1+1/2pfLp-p/2⋃jQj1/2+1/p1.Then, we plug the estimate into (31) to obtain (37)II2≤Cλpp1/2fLp-pp1/2⋃jQjp11/2+1/p1≤Cλpp1/2fLp-pp1/2x∈Rn:Mfpx1/p>λp1/2+1.Applying M of the weak-type (1,1), we obtain (38)II2≤Cλpp1/2fLp-pp1/2fLppp1/2+1λ-pp1/2+1≤Cλ-pfLpp. We now examine II3. Recall that (39)II3=x:∫0∞∑jb,TsI-I-e-tjLmhjx2dss1/2>λ. Then, (40)I-I-e-tjLm=I-∑k=0mCmke-ktjL=-∑k=1mCmke-ktjL. Thus, from Chebychev’s inequality, (41)II3≤Cλ2∑k=1m∫0∞b,Ts∑je-ktjLhj2dss1/2L22≤Cλ2∑k=1m∫0∞b,Ts∑je-ktjLhj2dss1/2L22+Cλ2∑k=1m∫0∞Tsb,∑je-ktjLhj2dss1/2L22≔II3,1+II3,2.We fix 1≤k≤m. Then, for II3,1, by (8), we obtain (42)II3,1≤Cλ2∑k=1m∑je-ktjLhjL22. Hence, by (see [9]) (43)∑je-ktjLhjL22≤Cλ2-pfLpp, we obtain (44)II3,1≤CfLppλp. For II3,2, by (7) and Lα/2fL2~DαfL2 (see [3]), we obtain (45)II3,2≤Cλ2∑k=1mDα∑jb,e-ktjLhjL22≤Cλ2∑k=1mLα/2∑jb,e-ktjLhjL22≤Cλ2∑k=1mb,Lα/2∑je-ktjLhjL22+∑k=1m∑jb,Lα/2e-ktjLhjL22. Then, (46)II3,2≤Cλ2∑k=1m∑je-ktjLhjL22+∑k=1m∑jb,Lα/2e-ktjLhjL22, where we use the fact that [b,Lα/2] is bounded on L2 with the bound (-Δ)α/2bBMO (see [16]). Next, we estimate the abovementioned two norms, ∑je-ktjLhjL2 and ∑jb,Lα/2e-ktjLhjL2. Now, taking v∈L2(Rn) with vL2=1,(47)∫Rn∑je-ktjLhjxvx¯dx=∑j∑l=0∞∫SlQje-ktjLhjxvx¯dx≤∑j∑l=0∞∫SlQje-ktjLhjx2dx1/2∫SlQjvx2dx1/2. Note that, for all pL<r≤2, e-ktjL satisfies the Lr-L2 off-diagonal estimates (see Lemma 3(ii)); let tj=l(Qj)2, and we obtain (47), which is controlled by(48)∑j∑l=0∞e-4ltj1/2n/2-n/rhjLrQj2l+1Qj1/212l+1Qj∫SlQjvx2dx1/2≤Cλ∑j∑l=0∞e-4llQjn/2-n/rQj1/r2l+1Qj1/212l+1Qj∫SlQjvx2dx1/2≤Cλ∑j∑l=0∞e-4l2lnQj12l+1Qj∫2l+1Qjvx2dx1/2≤Cλ∑jQjessinfy∈Qj⁡Mv21/2y≤Cλ∫∪jQjMv21/2xdx.≤Cλ⋃jQj1/2v2L11/2≤Cλx∈Rn:Mfpx1/p>λ1/2Thus, from (47), (48), and the fact that M is of weak-type (1,1), we obtain(49)∑je-ktjLhjL2≤CλfLpλp/2.For the second term of II3,2, ∑jb,Lα/2e-ktjLhjL2, by Lemma 6 and the same procedures performed previously (48): (50)∫Rn∑jb,Lα/2e-ktjLhjxvx¯dx=∑j∑l=0∞∫SlQjb,Lα/2e-ktjLhjxvx¯dx≤∑j∑l=0∞∫SlQjtj-α/2b,tjα/2Lα/2e-ktjLhjx2dx1/2∫SlQjvx2dx1/2≤C∑j∑l=0∞e-4ltj1/2n/2-n/rhjLr2l+1Qj1/212l+1Qj∫SlQjvx2dx1/2≤Cλx∈Rn:Mfpx1/p>λ1/2.Then, we have(51)∑jb,Lα/2e-ktjLhjL2≤CλfLpλp/2.Combining (49) and (51), we obtain(52)II3,2≤Cλ-pfLpp.Combining the estimates of II3,1 and II3,2, we obtain (53)II3≤CfLppλp. The proof of Theorem 1 is now completed by combining the estimates of I, II1, and II2 with II3.


## 4. Proof of Theorem2

First, we introduce a lemma that will be used to prove Theorem2.Lemma 8 (see [16]).
For0<α<1, let b∈Iα(BMO). Suppose that {θs}s>0 is a family of operators satisfying(54)θsf1SkQL2Q2≤C2-nks2klQ2m+2fL2SkQ2for an m≥1 and all 0<s≤Cl(Q). Assume that(55)sups>0θsL2→L2≤C,(56)θs1=0in the sense of Lloc2 for all s>0. Then, we have(57)∫Rn∫0∞s-α/2θsfx2dxdss≤CDαfL22.If, in addition,(58)θsdivL2→L2≤Cs,then(59)∫Rn∫0∞b,s-α/2θsfx2dxdss≤CDαbBMO2fL22.Recall that(60)GL;bαfx=∫0∞b,s1-α/2∇e-sLfx2dss1/2.The proof of Theorem 2 is presented in two steps.Step 1 (theL2 boundedness).
Letθs=s1/2∇e-sL. By Lemma 8, we only need to prove that s1/2∇e-sL satisfies (54)-(56) and (58). Because s1/2∇e-sLdivL2→L2≤C/s, it is easy to verify that s1/2∇e-sL satisfies (54), (56), and (58). Moreover, it is easy to verify that s1/2∇e-sL satisfies (55) by the L2 off-diagonal estimate of the operator s1/2∇e-sL (see Lemma 3(i, iii)). We obtain(61)GLαfxL2≤CDαfL2,(62)GL;bαfxL2≤CDαbBMOfL2.Step 2 (weak-type(p,p) boundedness for pL<p<2).
We first prove that the commutator is of the weak-type(p,p) for pL<p<2. We apply Theorem 1 with Ts=s(1-α)/2∇e-sL to prove this result. Because GLα and GL;bα are bounded on L2(Rn), verification of {s(1-α)/2∇e-sL}s>0 satisfies (6). Taking p<2<p1, by the Minkowski inequality, we obtain (63)∫0∞b,s1/2∇e-sLI-e-tLmf2dss1+α1/2Lp1F≤∫0∞b,s1/2∇e-sLI-e-tLmfLp1F2dss1+α1/2=C∫0t⋯+∫t∞⋯1/2=CIt+IIt1/2.Now, we study each operator separately. For the first operator, we have (64)I-e-tLm=∑k=0mCmk-1ke-ktL=I+∑k=1mcke-ktL, and, then, (65)It≤∫0tb,s1/2∇e-sLfLp1F2dss1+α+∑k=1mck∫0tb,s1/2∇e-sLe-ktLfLp1F2dss1+α=It,0+∑k=1mckIt,k.For It,0, by Lemma 7(i), we have (66)It,0≤∫0tb,s1-α2∇e-sLfLp1F2dss≤CbLipα2fLpE2∫0te-dE,F2/cssn/p1-n/pdss=CbLipα2fLpE2∫0te-dE,F2/css2ηdss=CbLipα2fLpE2t2η∫1∞e-dE,F2/ctss-2ηdss≤CbLipα2fLpE2t2η∫1∞dE,F2cts-2m-ηs-2ηdss≤CbLipα2fLpE2t2ηdE,F2t-2m-η,where η=(1/2)(n/p1-n/p). Now, with 1≤k≤m, by the commutative property of the semigroup and Lemma 7(i), we obtain (67)It,k≤∫0tb,s+kt1-α/2∇e-s+ktLfLp1F2ss+ktα-1dss1+α≤CbLipα2fLpE2∫0te-dE,F2/cs+kts+ktn/p1-n/ps+ktα-1sdss1+α≤CbLipα2fLpE2e-dE,F2/cttn/p1-n/p+α-1∫0ts-αds≤CbLipα2fLpE2e-dE,F2/cttn/p1-n/p≤CbLipα2fLpE2dE,F2t-2m-ηt2η.Collecting this estimate and the one proved for It,0, we obtain (68)It≤It,0+∑k=1mckIt,k≤CbLipα2t2ηdE,F2t-2m-ηfLpE2. Next, we proceed with the estimate of IIt: (69)IIt=∫t∞b,s1/2∇e-sLI-e-tLmfLp1F2dss1+α=C∫t∞b,s1/2∇e-m+1sLI-e-tLmfLp1F2dss1+α=C∫t∞b,s1/2∇e-sLe-sL-e-s+tLmfLp1F2dss1+α.Let E,F be two closed sets and gbe such that supp g⊂E. For t≤s, by a similar previous argument [9], we can prove that (70)ste-sL-es+tLgLp1F≤Cs1/2n/p1-n/p0e-dE,F2/csgLpE uniformly on t. Then, by Lemma 4, we have(71)e-sL-es+tLmgLp1F≤Ctsms1/2n/p1-n/p0e-dE,F2/csgLpE.We write IIt as follows: (72)IIt≤C∫t∞b,s1/2∇e-sLe-sL-e-s+tLmfLp1F2dss1+α+C∫t∞s1/2∇e-sLb,e-sL-e-s+tLmfLp1F2dss1+α≔G1+G2.First, we consider G1: (73)G1=∫t∞b,s1-α/2∇e-sL∘e-sL-e-s+tLmfLp1F2dss.Let us observe that, because of (71) and Lemma 7(i) in the composition of the operators, each of them verifies an off-diagonal estimate. This fact allows us to employ Lemma 4 to obtain (74)G1≤CbLipα2fLpE2∫t∞e-dE,F2/csts2msn/p1-n/pdss=CbLipα2fLpE2∫t∞e-dE,F2/csts2ms2ηdss=CbLipα2fLpE2t2η∫01e-dE,F2/ctss2m-2ηdss≤CbLipα2fLpE2t2η∫0∞e-dE,F2/ctss2m-2ηdss=CbLipα2fLpE2t2ηdE,F2t-2m-η∫0∞e-ss2m-ηdss≤CbLipα2fLpE2t2ηdE,F2t-2m-η,where η=(1/2)(n/p1-n/p). From Lemma 5 and (71), it is easy to obtain(75)b,s-α/2e-sL-es+tLmgLp1F≤CbLipαtsms1/2n/p1-n/p0e-dE,F2/csgLpE.Finally, for G2,(76)G2=∫t∞s1/2∇e-sL∘b,s-α/2e-sL-e-s+tLmfLp1F2dss.We know that the composition of the operators above (75) or Lemma 3 verifies an off-diagonal estimate. It follows from Lemma 4 that (77)G2≤CbLipα2fLpE2∫t∞e-dE,F2/csts2ms2ηdss≤CbLipα2fLpE2t2ηdE,F2t-2m-η.Collecting the estimates for G1 and G2, we obtain (78)IIt≤CbLipα2t2ηdE,F2t-2m-ηfLpE2,which combined with the estimate of It verify that {s(1-α)/2∇e-sL}s>0 satisfies (6). Thus, we have proven Theorem 2.



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*Source: 2624057-2018-12-02.xml*