We derive the explicit fundamental solutions for a class of degenerate(or singular)one- parameter subelliptic differential operators on groups of Heisenberg(H)type.This extends the result of Kaplan for the sub-Laplaci...We derive the explicit fundamental solutions for a class of degenerate(or singular)one- parameter subelliptic differential operators on groups of Heisenberg(H)type.This extends the result of Kaplan for the sub-Laplacian on H-type groups,which in turn generalizes Folland's result on the Heisenberg group.As an application,we obtain a one-parameter representation formula for Sobolev functions of compact support on H-type groups.By choosing the parameter equal to the homogeneous dimension Q and using the Mose-Trudinger inequality for the convolutional type operator on stratified groups obtained in[18].we get the following theorem which gives the best constant for the Moser- Trudiuger inequality for Sobolev functions on H-type groups. Let G be any group of Heisenberg type whose Lie algebra is generated by m left invariant vector fields and with a q-dimensional center.Let Q=m+2q.Q'=Q-1/Q and Then. with A_Q as the sharp constant,where ▽G denotes the subelliptic gradient on G. This continues the research originated in our earlier study of the best constants in Moser-Trudinger inequalities and fundamental solutions for one-parameter subelliptic operators on the Heisenberg group [18].展开更多
In this note, we compute the fundamental solution for the Hermite operator with singularity at an arbitrary point y∈R^n. We also apply this result to obtain the fundamental solutions for the Grushin operator in R^2 a...In this note, we compute the fundamental solution for the Hermite operator with singularity at an arbitrary point y∈R^n. We also apply this result to obtain the fundamental solutions for the Grushin operator in R^2 and the sub-Laplacian in the Heisenberg group Hn.展开更多
We consider the g-function related to a class of radial functions which gives a characterization of the L^p-norm of a function on the Heisenberg group.
In this article, we deal with weak solutions to non-degenerate sub-elliptic equations in the Heisenberg group, and study the regularities of solutions. We establish horizontal Calderón-Zygmund type estimate in Be...In this article, we deal with weak solutions to non-degenerate sub-elliptic equations in the Heisenberg group, and study the regularities of solutions. We establish horizontal Calderón-Zygmund type estimate in Besov spaces with more general assumptions on coefficients for both homogeneous equations and non-homogeneous equations. This study of regularity estimates expands the Calderón-Zygmund theory in the Heisenberg group.展开更多
In the setting of the Heisenberg group, based on the rotation method, we obtain the sharp （p,p） estimate for the Hardy operator. It will be shown that the norm of the Hardy operator on LP（Hn） is still p/（p- 1）. ...In the setting of the Heisenberg group, based on the rotation method, we obtain the sharp （p,p） estimate for the Hardy operator. It will be shown that the norm of the Hardy operator on LP（Hn） is still p/（p- 1）. This goes some way to imply that the Lp norms of the Hardy operator are the same despite the domains are intervals on R, balls in Rn, or ‘ellipsoids＇ on the Heisenberg group Hn. By constructing a special function, we find the best constant in the weak type （1, 1） inequality for the Hardy operator. Using the translation approach, we establish the boundedness for the Hardy operator from H1 to L1. Moreover, we describe the difference between Mp weights and Ap weights and obtain the characterizations of such weights using the weighted Hardy inequalities.展开更多
In this paper,we compute sub-Riemannian limits of Gaussian curvature associated to two kinds of Schouten-Van Kampen affine connections and the adapted connection for a Euclidean C2-smooth surface in the Heisenberg gro...In this paper,we compute sub-Riemannian limits of Gaussian curvature associated to two kinds of Schouten-Van Kampen affine connections and the adapted connection for a Euclidean C2-smooth surface in the Heisenberg group away from characteristic points and signed geodesic curvature associated to two kinds of Schouten-Van Kampen affine connections and the adapted connection for Euclidean C2-smooth curves on surfaces.We get Gauss-Bonnet theorems associated to two kinds of Schouten-Van Kampen affine connections in the Heisenberg group.展开更多
This paper is devoted to the high-dimensional and multilinear Hausdorff operators on the Heisenberg group Hn. The sharp bounds for the strong type （p,p） （1 〈 p 〈 ∞） estimates of n- dimensional Hausdorff operato...This paper is devoted to the high-dimensional and multilinear Hausdorff operators on the Heisenberg group Hn. The sharp bounds for the strong type （p,p） （1 〈 p 〈 ∞） estimates of n- dimensional Hausdorff operators on Hn are obtained. The sharp bounds for strong （p,p） estimates are further extended to multilinear cases. As an application, we derive the sharp constant for the multilinear Hardy operator on Hn. The weak type （p,p） （1 〈 p 〈 ∞） estimates are also obtained.展开更多
Let L = -△Hn ＋ V be a SchrSdinger operator on Heisenberg group Hn, where AHn is the sublaplacian and the nonnegative potential V belongs to the reverse HSlder class BQ/2 where Q is the homogeneous dimension of Hn. L...Let L = -△Hn ＋ V be a SchrSdinger operator on Heisenberg group Hn, where AHn is the sublaplacian and the nonnegative potential V belongs to the reverse HSlder class BQ/2 where Q is the homogeneous dimension of Hn. Let T1 = （--△Hn ＋V）-1V, T2 = （-△Hn ＋V）-1/2V1/2, and T3 = （--AHn ＋V）-I/2△Hn, then we verify that [b, Ti], i = 1, 2, 3 are bounded on some LP（Hn）, where b ∈ BMO（Hn）. Note that the kernel of Ti, i = 1, 2, 3 has no smoothness.展开更多
Recently, many new features of Sobolev spaces W k,p ？RN ？ were studied in [4-6, 32]. This paper is devoted to giving a brief review of some known characterizations of Sobolev spaces in Euclidean spaces and describin...Recently, many new features of Sobolev spaces W k,p ？RN ？ were studied in [4-6, 32]. This paper is devoted to giving a brief review of some known characterizations of Sobolev spaces in Euclidean spaces and describing our recent study of new characterizations of Sobolev spaces on both Heisenberg groups and Euclidean spaces obtained in [12] and [13] and outlining their proofs. Our results extend those characterizations of first order Sobolev spaces in [32] to the Heisenberg group setting. Moreover, our theorems also provide diff erent characterizations for the second order Sobolev spaces in Euclidean spaces from those in [4, 5].展开更多
Laguerre calculus is a powerful tool for harmonic analysis on the Heisenberg group. Many sub-elliptic partial differential operators can be inverted by Laguerre calculus. In this article, we use Laguerre calculus to f...Laguerre calculus is a powerful tool for harmonic analysis on the Heisenberg group. Many sub-elliptic partial differential operators can be inverted by Laguerre calculus. In this article, we use Laguerre calculus to find explicit kernels of the fundamental solution for the Paneitz operator and its heat equation. The Paneitz operator which plays an important role in CR geometry can be written as follows: $$ {\mathcal{P}_\alpha} = {\mathcal{L}_\alpha} \bar {\mathcal{L}_\alpha} = \frac{1} {4}\left[ {\sum\limits_{j = 1}^n {\left( {Z_j \bar Z_j + \bar Z_j Z_j } \right)} } \right]^2 + \alpha ^2 T^2 $$ Here “Z j ” j=1 n is an orthonormal basis for the subbundle T (1,0) of the complex tangent bundle T ?(H n ) and T is the “missing direction”. The operator $ \mathcal{L}_\alpha $ is the sub-Laplacian on the Heisenberg group which is sub-elliptic if α does not belong to an exceptional set Λ α . We also construct projection operators and relative fundamental solution for the operator $ \mathcal{L}_\alpha $ while α ∈ Λ α .展开更多
Let IHn be the (2n+1)-dimensional Heisenberg group. In this paper, we shall give among other things, the properties of some homogeneous norms relative to dilations on the IHn and prove the equivalence of these norms.
文摘We derive the explicit fundamental solutions for a class of degenerate(or singular)one- parameter subelliptic differential operators on groups of Heisenberg(H)type.This extends the result of Kaplan for the sub-Laplacian on H-type groups,which in turn generalizes Folland's result on the Heisenberg group.As an application,we obtain a one-parameter representation formula for Sobolev functions of compact support on H-type groups.By choosing the parameter equal to the homogeneous dimension Q and using the Mose-Trudinger inequality for the convolutional type operator on stratified groups obtained in[18].we get the following theorem which gives the best constant for the Moser- Trudiuger inequality for Sobolev functions on H-type groups. Let G be any group of Heisenberg type whose Lie algebra is generated by m left invariant vector fields and with a q-dimensional center.Let Q=m+2q.Q'=Q-1/Q and Then. with A_Q as the sharp constant,where ▽G denotes the subelliptic gradient on G. This continues the research originated in our earlier study of the best constants in Moser-Trudinger inequalities and fundamental solutions for one-parameter subelliptic operators on the Heisenberg group [18].
基金partially supported by a William Fulbright Research Grant and a Competitive Research Grant at Georgetown University
文摘In this note, we compute the fundamental solution for the Hermite operator with singularity at an arbitrary point y∈R^n. We also apply this result to obtain the fundamental solutions for the Grushin operator in R^2 and the sub-Laplacian in the Heisenberg group Hn.
基金Supported by the National Natural Science Foundation of China （No. 10371004） and the Specialized Research Fund for the Doctoral Program Higher Education of China （No. 20030001107）
文摘We consider the g-function related to a class of radial functions which gives a characterization of the L^p-norm of a function on the Heisenberg group.
文摘In this article, we deal with weak solutions to non-degenerate sub-elliptic equations in the Heisenberg group, and study the regularities of solutions. We establish horizontal Calderón-Zygmund type estimate in Besov spaces with more general assumptions on coefficients for both homogeneous equations and non-homogeneous equations. This study of regularity estimates expands the Calderón-Zygmund theory in the Heisenberg group.
基金This work was supported in part by the National Natural Science Foundation of China （Grant Nos. 11271175, 11301248, 11401287）, the Natural Science Foundation of Shandong Province （Grant No. ZR2012AQ026）, and the AMEP of Linyi University.
文摘In the setting of the Heisenberg group, based on the rotation method, we obtain the sharp （p,p） estimate for the Hardy operator. It will be shown that the norm of the Hardy operator on LP（Hn） is still p/（p- 1）. This goes some way to imply that the Lp norms of the Hardy operator are the same despite the domains are intervals on R, balls in Rn, or ‘ellipsoids＇ on the Heisenberg group Hn. By constructing a special function, we find the best constant in the weak type （1, 1） inequality for the Hardy operator. Using the translation approach, we establish the boundedness for the Hardy operator from H1 to L1. Moreover, we describe the difference between Mp weights and Ap weights and obtain the characterizations of such weights using the weighted Hardy inequalities.
文摘In this paper,we compute sub-Riemannian limits of Gaussian curvature associated to two kinds of Schouten-Van Kampen affine connections and the adapted connection for a Euclidean C2-smooth surface in the Heisenberg group away from characteristic points and signed geodesic curvature associated to two kinds of Schouten-Van Kampen affine connections and the adapted connection for Euclidean C2-smooth curves on surfaces.We get Gauss-Bonnet theorems associated to two kinds of Schouten-Van Kampen affine connections in the Heisenberg group.
基金Supported by National Natural Science Foundation of China(Grant No.11201287)China Scholarship Council(Grant No.201406895019)
文摘This paper is devoted to the high-dimensional and multilinear Hausdorff operators on the Heisenberg group Hn. The sharp bounds for the strong type （p,p） （1 〈 p 〈 ∞） estimates of n- dimensional Hausdorff operators on Hn are obtained. The sharp bounds for strong （p,p） estimates are further extended to multilinear cases. As an application, we derive the sharp constant for the multilinear Hardy operator on Hn. The weak type （p,p） （1 〈 p 〈 ∞） estimates are also obtained.
基金supported by NSFC 11171203, S2011040004131STU Scientific Research Foundation for Talents TNF 10026+1 种基金supported by NSFC No.10990012,10926179RFDP of China No.200800010009
文摘Let L = -△Hn ＋ V be a SchrSdinger operator on Heisenberg group Hn, where AHn is the sublaplacian and the nonnegative potential V belongs to the reverse HSlder class BQ/2 where Q is the homogeneous dimension of Hn. Let T1 = （--△Hn ＋V）-1V, T2 = （-△Hn ＋V）-1/2V1/2, and T3 = （--AHn ＋V）-I/2△Hn, then we verify that [b, Ti], i = 1, 2, 3 are bounded on some LP（Hn）, where b ∈ BMO（Hn）. Note that the kernel of Ti, i = 1, 2, 3 has no smoothness.
文摘Recently, many new features of Sobolev spaces W k,p ？RN ？ were studied in [4-6, 32]. This paper is devoted to giving a brief review of some known characterizations of Sobolev spaces in Euclidean spaces and describing our recent study of new characterizations of Sobolev spaces on both Heisenberg groups and Euclidean spaces obtained in [12] and [13] and outlining their proofs. Our results extend those characterizations of first order Sobolev spaces in [32] to the Heisenberg group setting. Moreover, our theorems also provide diff erent characterizations for the second order Sobolev spaces in Euclidean spaces from those in [4, 5].
基金supported by a research grant from the United States Air Force Office of Scientific Research(AFOSR) SBIR Phase I (Grant No. FA9550-09-C-0045)a Hong Kong RGC competitive earmarked research(Grant No. 600607)+1 种基金a competitive research grant at Georgetown University (Grant No. GD2236000)supported by Natural Science Foundation of Taiwan,China (Grant No.97-2115-M-002-015)
文摘Laguerre calculus is a powerful tool for harmonic analysis on the Heisenberg group. Many sub-elliptic partial differential operators can be inverted by Laguerre calculus. In this article, we use Laguerre calculus to find explicit kernels of the fundamental solution for the Paneitz operator and its heat equation. The Paneitz operator which plays an important role in CR geometry can be written as follows: $$ {\mathcal{P}_\alpha} = {\mathcal{L}_\alpha} \bar {\mathcal{L}_\alpha} = \frac{1} {4}\left[ {\sum\limits_{j = 1}^n {\left( {Z_j \bar Z_j + \bar Z_j Z_j } \right)} } \right]^2 + \alpha ^2 T^2 $$ Here “Z j ” j=1 n is an orthonormal basis for the subbundle T (1,0) of the complex tangent bundle T ?(H n ) and T is the “missing direction”. The operator $ \mathcal{L}_\alpha $ is the sub-Laplacian on the Heisenberg group which is sub-elliptic if α does not belong to an exceptional set Λ α . We also construct projection operators and relative fundamental solution for the operator $ \mathcal{L}_\alpha $ while α ∈ Λ α .
文摘Let IHn be the (2n+1)-dimensional Heisenberg group. In this paper, we shall give among other things, the properties of some homogeneous norms relative to dilations on the IHn and prove the equivalence of these norms.
