Global Hölder Estimates via Morrey Norms for Hypoelliptic Operators with Drift

Hou, Yuexia; Niu, Pengcheng

doi:https://doi.org/10.1155/2017/2805457

Journal of Function Spaces

On this page

Abstract Introduction Preliminaries Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2017 | Article ID 2805457 | https://doi.org/10.1155/2017/2805457

Global Hölder Estimates via Morrey Norms for Hypoelliptic Operators with Drift

Yuexia Hou¹and Pengcheng Niu¹

Academic Editor: Guozhen Lu

Received25 Oct 2016

Accepted19 Jan 2017

Published19 Feb 2017

Abstract

Suppose that are left invariant real vector fields on the homogeneous group with being homogeneous of degree two and homogeneous of degree one. In the paper we study the hypoelliptic operator with drift of the kind where and is a constant matrix satisfying the elliptic condition on . By proving the boundedness of two integral operators on the Morrey spaces with two weights, we obtain global Hölder estimates for .

1. Introduction and the Main Results

Let be a homogeneous group on and let be left invariant real vector fields on , where is homogeneous of degree two and are homogeneous of degree one satisfying Hörmander’s condition denotes the Lie algebra generated by The purpose of this paper is to study the following hypoelliptic operator with drift:where and is a constant coefficients matrix and there exists a constant such that

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 in (2) have been considered by several researchers. A priori estimates, estimates, and Sobolev-Morrey estimates for especially were proved in [3–5], respectively. We mention that the operator contains Laplacian and parabolic operators in the Euclidean space. When , , the operator becomeswhere , is a positive definite matrix in , and is a constant coefficients matrix with a suitable upper triangular structure. Clearly, 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 in [10, 11], the authors of [12–14] established an invariant Harnack inequality for the nonnegative solution of 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 on Morrey spaces with two weights. In this paper, we try to study the global Hölder estimates for . 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 .

Before stating the main results, we first introduce classes and the Morrey space with two weights on the homogeneous group . Let us recall that a weight is a nonnegative locally integrable function on . Given a weight and a measurable set , we set

For and the weight , if there exists such that, for any ball in ,where , then is said to be in the class . The infimum of these constants is denoted by .

For and , the Morrey space with two weights and on is defined bywhereand the supremum is taken over all balls in .

Observe that if and in (7), , is the homogeneous dimension of , thus which is the usual Morrey space.

We next state the main results of this paper.

Theorem 1. If , , , and , then there exists such that, for any test function and every , ,where .

Theorem 2. If , , , and , then there exists such that, for any test function and every , ,where .

Remark 3. Authors in [19] have proved Morrey estimates with two weights for : if , , , and , there exists a constant such that, for every , we have

Our results reflect the relations between the weighted Morrey norms of and Hölder exponents of and . 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 function are determined by Calderón-Zygmund operators (see [3]), we cannot use the method here to give Hölder estimates for higher order derivatives of .

The paper is organized as follows. In Section 2 we present some preliminaries about homogeneous groups and fundamental solutions of . 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 mappingsthe space with these mappings forms a group and the identity is the origin. If there exist , such that the dilationsare group automorphisms, then the group with this structure is called a homogeneous group denoted by . Homogeneous groups include the Euclidean space, the Heisenberg group, and the Carnot group; see [20, 21].

Definition 4. A homogeneous norm on is defined as follows: for any , ,where is the Euclidean norm. Also, define .

It is not difficult to verify that the following properties hold for the homogeneous norm:(1), , ;(2)there exists , such that, for ,

By virtue of these properties, it is natural to define the quasi-distance by Furthermore, the ball in with respect to is defined byObserve that ; thenwhereis the homogeneous dimension of . By (18) the doubling condition holds on ; that is,where is a positive constant, and therefore is a space of homogeneous type.

Definition 5. A differential operator on is said to be homogeneous operator of degree , if, for every test function ,a function is said to be homogeneous operator of degree , if

Obviously, if is a homogeneous differential operator of degree and is a homogeneous function of degree , then is homogeneous operator of degree .

Lemma 6 (see [3]). The operator is a homogeneous left invariant differential operator of degree two on and there is a unique fundamental solution such that, for any test function and every ,(1);(2) is homogeneous operator of degree ;(3);(4).

If we set , , then it is obvious from Definition 5 that is homogeneous of degree .

Lemma 7 (see [22]). For any , the following hold:(1)there exists a constant , such that(2)there exist two constants and , such that if , then

Now we introduce two integral operators. For and , fixed , we define for every , , and ,

Lemma 8. For , , , and , if , then there exists such thatif , then there exists such that

Proof. By decomposing the domain of integration and applying the Hölder inequality, it is shown thatDue to , we getthenIf , then the series in the right hand side in (31) is convergent, and (26) is proved. Analogously, it yieldsIt follows from (29) thatIf , the series in (32) is convergent and the proof of (27) is ended.

3. Proof of the Main Theorems

Proof of Theorem 1. For any test function and every , , applying Lemma 6, there exists such that It follows by Lemma 7 thatNote that if , thenif , thenIt immediately derives by (34) thatIf , it is shown by choosing , in Lemma 8 that there exists such thatif , we get from Lemma 8 (, and , , resp.) thatPutting (38) and (39) in (37), we have (9) and this finishes the proof.

Proof of Theorem 2. For , we have from Lemma 6 that there exists such that for any test function and every , ,Summarizing (35) and (36) and using Lemma 7, it yieldsIf , we have by using Lemma 8 with and that there exists such thatif , it is shown by applying Lemma 8 (, and , , resp.) thatSubstituting (42) and (43) into (41), we get (10).

Competing Interests

The authors declare that there is no conflict of interests regarding the publication of this article.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant no. 11271299) and the Fundamental Research Funds for the Central Universities (Grant no. 3102015ZY069). The authors are very grateful to the anonymous referee who read carefully the manuscript and offered valuable suggestions.

References

L. Hörmander, “Hypoelliptic second order differential equations,” Acta Mathematica, vol. 119, no. 1, pp. 147–171, 1967.
View at: Publisher Site | Google Scholar | MathSciNet
G. B. Folland, “Subelliptic estimates and function spaces on nilpotent Lie groups,” Arkiv för Matematik, vol. 13, no. 2, pp. 161–207, 1975.
View at: Publisher Site | Google Scholar | MathSciNet
M. Bramanti and L. Brandolini, “ $L^{P}$ estimates for uniformly hypoelliptic operators with discontiuous coecients on homogeneous groups,” Rendiconti del Seminario Matematico Università e Politecnico di Torino, vol. 58, pp. 389–433, 2000.
View at: Google Scholar
C. E. Gutiérrez and E. Lanconelli, “Schauder estimates for sub-elliptic equations,” Journal of Evolution Equations, vol. 9, no. 4, pp. 707–726, 2009.
View at: Publisher Site | Google Scholar | MathSciNet
P. C. Niu and X. J. Feng, “Global Sobolev-Morrey estimates for hypoelliptic operators with drift on homogeneous groups,” Scientia Sinica Mathematica, vol. 42, no. 9, pp. 905–920, 2012 (Chinese).
View at: Publisher Site | Google Scholar
S. Chandresekhar, “Stochastic problems in physics and astronomy,” Reviews of Modern Physics, vol. 15, pp. 1–89, 1943.
View at: Publisher Site | Google Scholar | MathSciNet
S. Chapman and T. G. Cowling, The mathematical theory of nonuniform gases, Cambridge Mathematical Library, Cambridge University Press, Cambridge, Third edition, 1990.
View at: MathSciNet
E. Barucci, S. Polidoro, and V. Vespri, “Some results on partial differential equations and Asian options,” Mathematical Models and Methods in Applied Sciences, vol. 11, no. 3, pp. 475–497, 2001.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
A. Pascucci, “Hölder regularity for a Kolmogorov equation,” Transactions of the American Mathematical Society, vol. 355, no. 3, pp. 901–924, 2003.
View at: Publisher Site | Google Scholar | MathSciNet
L. P. Kuptsov, “Fundamental solutions for a class of second-order elliptic-parabolic equations,” Differentcial'nye Uravnenija, vol. 8, pp. 1649–1660, 1972.
View at: Google Scholar
L. P. Kuptsov, “Mean value theorem and a maximum principle for Kolmogorov's equation,” Matematicheskie Zametki, vol. 15, no. 3, pp. 479–489, 1974.
View at: Google Scholar
N. Garofalo and E. Lanconelli, “Level sets of the fundamental solution and Harnack inequality for degenerate equations of Kolmogorov type,” Transactions of the American Mathematical Society, vol. 321, no. 2, pp. 775–792, 1990.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
E. Lanconelli and S. Polidoro, “On a class of hypoelliptic evolution operators,” Rendiconti del Seminario Matematico Università e Politecnico di Torino, vol. 52, pp. 29–63, 1994.
View at: Google Scholar
A. Pascucci and S. Polidoro, “On the Harnack inequality for a class of hypoelliptic evolution equations,” Transactions of the American Mathematical Society, vol. 356, no. 11, pp. 4383–4394, 2004.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
S. Polidoro and M. A. Ragusa, “Sobolev-Morrey spaces related to an ultraparabolic equation,” Manuscripta Mathematica, vol. 96, no. 3, pp. 371–392, 1998.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
Y. Komori and S. Shirai, “Weighted Morrey spaces and a singular integral operator,” Mathematische Nachrichten, vol. 282, no. 2, pp. 219–231, 2009.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
C. B. Morrey, “On the solutions of quasi-linear elliptic partial differential equations,” Transactions of the American Mathematical Society, vol. 43, no. 1, pp. 126–166, 1938.
View at: Publisher Site | Google Scholar | MathSciNet
Z. Wang and Z. Si, “Commutator theorems for fractional integral operators on weighted Morrey spaces,” Abstract and Applied Analysis, vol. 2014, Article ID 413716, 8 pages, 2014.
View at: Publisher Site | Google Scholar | MathSciNet
Y. X. Hou and P. C. Niu, “Weighted Sobolev-Morrey estimates for hypoelliptic operators with drift on homogeneous groups,” Journal of Mathematical Analysis and Applications, vol. 428, no. 2, pp. 1319–1338, 2015.
View at: Publisher Site | Google Scholar | MathSciNet
A. Bonfiglioli, E. Lanconelli, and F. Uguzzoni, Stratified Lie Groups and Potential Theory for Their Sub-Laplacians, Springer Monographs in Mathematics, Springer, Berlin, Germany, 2007.
View at: Publisher Site
L. Capogna, D. Danielli, S. D. Pauls, and J. T. Tyson, An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem, vol. 259 of Progress in Mathematics, Birkhäuser, Basel, Switzerland, 2007.
Y. X. Hou, X. W. Cui, and X. J. Feng, “Global Hölder estimates for hypoelliptic operators with drift on homogeneous groups,” Miskolc Mathematical Notes, vol. 13, pp. 337–347, 2012.
View at: Google Scholar

Copyright

Copyright © 2017 Yuexia Hou and Pengcheng Niu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies

Views

538

Downloads

698

Citations