# Global Hölder Estimates via Morrey Norms for Hypoelliptic Operators with Drift

Authors: Yuexia Hou; Pengcheng Niu
Journal: Journal of Function Spaces
 (2017)

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## Abstract

Suppose thatX0,X1,…,Xm are left invariant real vector fields on the homogeneous group G with X0 being homogeneous of degree two and X1,…,Xm homogeneous of degree one. In the paper we study the hypoelliptic operator with drift of the kind L=∑i,j=1maijXiXj+a0X0, where a0≠0 and (aij) is a constant matrix satisfying the elliptic condition on Rm. By proving the boundedness of two integral operators on the Morrey spaces with two weights, we obtain global Hölder estimates for L.


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## Body

## 1. Introduction and the Main Results

LetG be a homogeneous group on RN and let X0,X1,…,Xm(m<N) be left invariant real vector fields on G, where X0 is homogeneous of degree two and X1,…,Xm are homogeneous of degree one satisfying Hörmander’s condition (1)rankLX0,X1,…,Xmx=N,x∈G,L(X0,X1,…,Xm) denotes the Lie algebra generated by X0,X1,…,Xm. The purpose of this paper is to study the following hypoelliptic operator with drift:(2)L=∑i,j=1maijXiXj+a0X0,where a0≠0 and (aij)i,j=1m is a constant coefficients matrix and there exists a constant μ>0 such that(3)μ-1ξ2≤∑i,j=1maijξiξj≤μξ2,ξ∈Rmμ-1≤a0≤μ.Since Hörmander put forward the operator of sum of squares in [1], many authors paid attention to regularity of hypoelliptic operators constructed by Hörmander’s vector fields. Folland [2] concluded that any left invariant homogeneous differential operator of second order possesses a unique homogeneous fundamental solution. Bramanti and Brandolini [3] investigated further the related properties of the fundamental solutions. Recently, the a priori estimates for the operator L in (2) have been considered by several researchers. A priori Lp estimates, Cα estimates, and Sobolev-Morrey estimates for L especially were proved in [3–5], respectively. We mention that the operator L contains Laplacian and parabolic operators in the Euclidean space. When X0=∑i,j=1nbijxi∂xj-∂t, Xi=∂xi,i=1,…,m,m<N, the operator L becomes(4)L1u=∑i,j=1maij∂xixj2u+∑i,j=1nbijxi∂xju-∂tu,where (x,t)∈Rn+1, (aij)i,j=1m is a positive definite matrix in Rm, and (bij) is a constant coefficients matrix with a suitable upper triangular structure. Clearly, L1 is a class of Kolmogorov-Fokker-Planck ultraparabolic operators and appears in many research ranges, for example, stochastic processes and kinetic models [6, 7] and mathematical finance theory [8, 9]. After the previous study on L1 in [10, 11], the authors of [12–14] established an invariant Harnack inequality for the nonnegative solution of L1u=0 by using the mean value formula. Based on the theory of singular integral, Polidoro and Ragusa in [15] demonstrated Morrey-type imbedding results and gave a local Hölder continuity of the solution.Komori and Shirai in [16] defined weighted Morrey spaces in the Euclidean space, which are the extension of Morrey spaces (see [17]) and showed the boundedness in these spaces of some important operators in harmonic analysis. The authors of [18] established the boundedness of commutators of fractional integral operators with BMO functions on Morrey spaces with two weights. In the framework of homogeneous groups, we proved in [19] similar results to [16] and a priori estimates for L on Morrey spaces with two weights. In this paper, we try to study the global Hölder estimates for L. More precisely, motivated by [3, 16], we will establish the boundedness of two integral operators on the Morrey spaces with two weights and then prove global Hölder estimates for L.Before stating the main results, we first introduceAp,q classes and the Morrey space with two weights on the homogeneous group G. Let us recall that a weight is a nonnegative locally integrable function on G. Given a weight w and a measurable set E⊂G, we set (5)wE=∫Ewydy.For1<p<q<∞ and the weight w, if there exists c>1 such that, for any ball B in G,(6)1B∫Bwxqdx1/q1B∫Bwx-p′dx1/p′≤c,where 1/p+1/p′=1, then w is said to be in the class Ap,q. The infimum of these constants is denoted by [w]Ap,q.Forp∈(1,∞) and κ∈(0,1), the Morrey space with two weights μ and ν on G is defined by(7)Lp,κμ,ν,G=g∈Llocpw,G:gLp,κμ,ν,G<∞,where(8)gLp,κμ,ν,G=supB1νBκ∫Bgypμydy1/p,and the supremum is taken over all balls in G.Observe that ifμ=ν=1 and κ=λ/Q in (7), 0<λ<Q, Q is the homogeneous dimension of G, thus Lp,κ(μ,ν,G)=Lp,λ(G) which is the usual Morrey space.We next state the main results of this paper.Theorem 1.
If1<p<q<∞, 1/Q<1/p-1/q<2/Q, w∈Ap,q, and Lu∈Lp,p/q(wp,wq,G), then there exists c>0 such that, for any test function u and every x,z∈G, x≠z,(9)ux-uzz-1∘xθ≤cLuLp,p/qwp,wq,G,where θ=2-Q1/p-1/q.Theorem 2.
If1<p<q<∞, 0<1/p-1/q<1/Q, w∈Ap,q, and Lu∈Lp,p/q(wp,wq,G), then there exists c>0 such that, for any test function u and every x,z∈G, x≠z,(10)Xiux-Xiuzz-1∘xδ≤cLuLp,p/qwp,wq,G,i=1,…,m,where δ=1-Q1/p-1/q.Remark 3.
Authors in [19] have proved Morrey estimates with two weights for L: if 1<p<∞, 1/q=1/p-1/Q, 0<κ<p/q, and w∈Ap,q, there exists a constant c>0 such that, for every Lu∈Lp,κ(wp,wq,G), we have (11)XiuLq,κq/pwq,G≤cLuLp,κwp,wq,G,i=1,2,…,m.Our results reflect the relations between the weighted Morrey norms ofLu and Hölder exponents of u and Xiu. These statements are new even to elliptic operators, parabolic operators, and some ultraparabolic operators.Since the second and higher order vector fields derivatives of a test functionu are determined by Calderón-Zygmund operators (see [3]), we cannot use the method here to give Hölder estimates for higher order derivatives of u.The paper is organized as follows. In Section2 we present some preliminaries about homogeneous groups and fundamental solutions of L. Furthermore, we establish pointwise estimates for the two integral operators on the Morrey spaces with two weights. Section 3 is devoted to the proofs of the main results.


## 2. Preliminaries and Two Integral Operators

Given two smooth mappings(12)x,y⟼x∘y:RN×RN⟼RN;x⟼x-1:RN⟼RN,the space RN with these mappings forms a group and the identity is the origin. If there exist 0<ω1≤ω2≤⋯≤ωN, such that the dilations(13)Dλ:x1,…,xN⟼λω1x1,…,λωNxN,λ>0are group automorphisms, then the group with this structure is called a homogeneous group denoted by G. Homogeneous groups include the Euclidean space, the Heisenberg group, and the Carnot group; see [20, 21].Definition 4.
A homogeneous norm∥·∥ on G is defined as follows: for any x∈G, x≠0,(14)x=ρ⟺τ1ρx=1,where · is the Euclidean norm. Also, define 0=0.It is not difficult to verify that the following properties hold for the homogeneous norm:(1)
τ(λ)x=λx, x∈G, λ>0;(2)
there existsc1,c2≥1, such that, for x,y∈G,(15)x-1≤c1x,x∘y≤c2x+y.By virtue of these properties, it is natural to define the quasi-distanced by (16)dx,y=y-1∘x.Furthermore, the ball in G with respect to d is defined by(17)Bx,r≡Brx=y∈G:dx,y<r.Observe that B(0,r)=τ(r)B(0,1); then(18)Bx,r=rQB0,1,x∈G,r>0,where(19)Q=ω1+⋯+ωNis the homogeneous dimension of G. By (18) the doubling condition holds on G; that is,(20)Bx,2r≤cBx,r,forx∈G,r>0,where c is a positive constant, and therefore G,dx,d is a space of homogeneous type.Definition 5.
A differential operatorY on G is said to be homogeneous operator of degree β(β>0), if, for every test function φ,(21)YφDλx=λβYφDλx,λ>0,x∈G;a function f is said to be homogeneous operator of degree α, if(22)fDλx=λαfx,λ>0,x∈G.Obviously, ifY is a homogeneous differential operator of degree β and f is a homogeneous function of degree α, then Yf is homogeneous operator of degree α-β.Lemma 6 (see [3]).
The operatorL is a homogeneous left invariant differential operator of degree two on G and there is a unique fundamental solution Γ(·) such that, for any test function u and every x∈G,(1)
Γ(·)∈C∞G∖{0};(2)
Γ(·) is homogeneous operator of degree 2-Q;(3)
u(x)=(Lu∗Γ)(x)=∫GΓ(y-1∘x)Lu(y)dy;(4)
Xiu(x)=∫GXiΓ(y-1∘x)Lu(y)dy.If we setΓi=XiΓ, i=1,…,m, then it is obvious from Definition 5 that Γi(·) is homogeneous of degree 1-Q.Lemma 7 (see [22]).
For anyx,y,z∈G, the following hold:(1)
there exists a constantc>0, such that(23)Γy-1∘x≤cy-1∘xQ-2,Γiy-1∘x≤cy-1∘xQ-1;(2)
there exist two constantsc>0 and M>1, such that if y-1∘x≥Mx-1∘z, then(24)Γy-1∘x-Γy-1∘z≤cx-1∘zy-1∘xQ-1,Γiy-1∘x-Γiy-1∘z≤cx-1∘zy-1∘xQ.Now we introduce two integral operators. For1<p<q<∞ and σ>0, fixed z∈G, we define for every g∈Lp,p/q(wp,wq,G), x∈G, and x≠z,(25)Tαgx=∫y-1∘x≥σz-1∘xgyy-1∘xQ-αdy,α∈0,Q,Tβgx=∫y-1∘x<σz-1∘xgyy-1∘xQ-βdy,β∈0,Q.Lemma 8.
For1<p<q<∞, w∈Ap,q, x,z∈G, and x≠z, if α/Q<1/p-1/q, then there exists c>0 such that(26)Tαgx≤cgLp,p/qwp,wq,Gz-1∘xα-Q1/p-1/q;if β/Q>1/p-1/q, then there exists c>0 such that(27)Tβgx≤cgLp,p/qwp,wq,Gz-1∘xβ-Q1/p-1/q.Proof.
By decomposing the domain of integration and applying the Hölder inequality, it is shown that(28)Tαgx≤∑k=1∞∫2k-1σz-1∘x≤y-1∘x<2kσz-1∘xgyy-1∘xQ-αdy≤∑k=1∞12k-1σx-1∘x0Q-α∫B2kσz-1∘xxgydy≤∑k=1∞12k-1σx-1∘x0Q-α∫B2kσz-1∘xxgypwypdy1/p·∫B2kσz-1∘xxwy-p′dy1/p′.Due to w∈Ap,q, we get(29)∫Bwx-p′dx1/p′≤cB1/p′+1/qwqB1/q;then(30)Tαgx≤∑k=1∞12k-1σz-1∘xQ-α∫B2kσz-1∘xxgypwypdy1/p·B2kσz-1∘xx1/p′+1/qwqB2kσz-1∘xx1/q=c∑k=1∞1wqB2kσz-1∘xxp/q∫B2kσz-1∘xxgypwypdy1/p·2kσz-1∘xQ1/p′+1/q2k-1σz-1∘xQ-α≤cgLp,p/qwp,wq,Gz-1∘xα-Q1/p-1/q∑k=1∞2α-Q1/p-1/qk.If α/Q<1/p-1/q, then the series in the right hand side in (31) is convergent, and (26) is proved. Analogously, it yields(31)Tβgx≤∑k=1∞∫2-kσz-1∘x≤y-1∘x<21-kσz-1∘xgyy-1∘xQ-βdy≤∑k=1∞12-kσx-1∘x0Q-β∫B21-kσz-1∘xxgydy≤∑k=1∞12-kσx-1∘x0Q-β∫B21-kσz-1∘xxgypwypdy1/p·∫B21-kσz-1∘xxwy-p′dy1/p′.It follows from (29) that(32)Tβgx≤∑k=1∞12-kσz-1∘xQ-β∫B21-kσz-1∘xxgypwypdy1/p·B21-kσz-1∘xx1/p′+1/qwqB21-kσx-1∘x0x1/q=c∑k=1∞1wqB21-kσz-1∘xxp/q∫B21-kσz-1∘xxgypwypdy1/p·21-kσz-1∘xQ1/p′+1/q2-kσz-1∘xQ-β≤cgLp,p/qwp,wq,Gz-1∘xβ-Q1/p-1/q∑k=1∞2β-Q1/p-1/q-k.If β/Q>1/p-1/q, the series in (32) is convergent and the proof of (27) is ended.


## 3. Proof of the Main Theorems

Proof of Theorem1.
For any test functionu and every x,z∈G, x≠z, applying Lemma 6, there exists M>1 such that (33)ux-uz≤∫GΓy-1∘x-Γy-1∘zLuydy=∫y-1∘x≥Mx-1∘zΓy-1∘x-Γy-1∘zLuydy+∫y-1∘x<Mx-1∘zΓy-1∘x-Γy-1∘zLuydy≤∫y-1∘x≥Mx-1∘zΓy-1∘x-Γy-1∘zLuydy+∫y-1∘x<Mx-1∘zΓy-1∘xLuydy+∫y-1∘x<Mx-1∘zΓy-1∘zLuydy.It follows by Lemma 7 that(34)ux-uz≤∫y-1∘x≥Mx-1∘zcx-1∘zy-1∘xQ-1Luydy+∫y-1∘x<Mx-1∘zcy-1∘xQ-2Luydy+∫y-1∘x<Mx-1∘zcy-1∘zQ-2Luydy.Note that if y-1∘x≥Mx-1∘z, then(35)y-1∘x≥Mx-1∘z≥Mc1z-1∘x;if y-1∘x<Mx-1∘z, then(36)y-1∘x<Mx-1∘z<Mc1z-1∘x,y-1∘z<c2y-1∘x+x-1∘z<c2Mx-1∘z+x-1∘z=c21+Mx-1∘z.It immediately derives by (34) that(37)ux-uz≤∫y-1∘x≥M/c1z-1∘xcx-1∘zy-1∘xQ-1Luydy+∫y-1∘x<Mc1z-1∘xcy-1∘xQ-2Luydy+∫y-1∘z<c21+Mz-1∘xcy-1∘zQ-2Luydy≐I1+I2+I3.If 1/Q<1/p-1/q, it is shown by choosing α=1, σ=M/c1 in Lemma 8 that there exists c=c(p,λ,σ)>0 such that(38)I1≤cz-1∘xLuLp,p/qwp,wq,Gz-1∘x1-Q1/p-1/q=cLuLp,p/qwp,wq,Gz-1∘x2-Q1/p-1/q;if 2/Q>1/p-1/q, we get from Lemma 8 (β=2, σ=Mc1 and β=2, σ=c2(1+M), resp.) that(39)I2≤cLuLp,p/qwp,wq,Gz-1∘x2-Q1/p-1/q;I3≤cLuLp,p/qwp,wq,Gz-1∘x2-Q1/p-1/q.Putting (38) and (39) in (37), we have (9) and this finishes the proof.Proof of Theorem2.
Fori=1,…,m, we have from Lemma 6 that there exists M>1 such that for any test function u and every x,z∈G, x≠z,(40)Xiux-Xiuz≤∫GΓiy-1∘x-Γiy-1∘zLuydy=∫y-1∘x≥Mx-1∘zΓiy-1∘x-Γiy-1∘zLuydy+∫y-1∘x<Mx-1∘zΓiy-1∘x-Γiy-1∘zLuydy≤∫y-1∘x≥Mx-1∘zΓiy-1∘x-Γiy-1∘zLuydy+∫y-1∘x<Mx-1∘zΓiy-1∘xLuydy+∫y-1∘x<Mx-1∘zΓiy-1∘zLuydy.Summarizing (35) and (36) and using Lemma 7, it yields(41)Xiux-Xiuz≤∫y-1∘x≥Mx-1∘zcx-1∘zy-1∘xQLuydy+∫y-1∘x<Mx-1∘zcy-1∘xQ-1Luydy+∫y-1∘x<Mx-1∘zcy-1∘zQ-1Luydy≤∫y-1∘x≥M/c1z-1∘xcx-1∘zy-1∘xQLuydy+∫y-1∘x<Mc1z-1∘xcy-1∘xQ-1Luydy+∫y-1∘z<c21+Mz-1∘xcy-1∘zQ-1Luydy≐I4+I5+I6.If 0<1/p-1/q, we have by using Lemma 8 with α=0 and σ=M/c1 that there exists c=c(p,λ,σ)>0 such that(42)I4≤cz-1∘xgLp,p/qwp,wq,Gz-1∘x-Q1/p-1/q=cgLp,p/qwp,wq,Gz-1∘x1-Q1/p-1/q;if 1/Q>1/p-1/q, it is shown by applying Lemma 8 (β=1, σ=Mc1 and β=1, σ=c2(1+M), resp.) that(43)I5≤cgLp,p/qwp,wq,Gz-1∘x1-Q1/p-1/q;I6≤cgLp,p/qwp,wq,Gz-1∘x1-Q1/p-1/q.Substituting (42) and (43) into (41), we get (10).



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