Endpoint estimates for the commutator of pseudo-differential operators

It is well known that the commutator Tb of the Calderón-Zygmund singular integral operator is bounded on L^p(Rⁿ) for 1 < p < +∞ if and only if b ∈ BMO [1]. On the other hand, the commutator T_b is bounded from H¹(Rⁿ) into L¹(Rⁿ) only if the function b is a constant [2]. In this article, we will discuss the boundedness of commutator of certain pseudo-differential operators on Hardy spaces H¹. Let T^σ be the operators that its symbol is S⁰_1,δ with 0 ≤ δ < 1, if b ∈ LMO_∞, then, the commutator [b, T^σ] is bounded from H¹(Rⁿ) into L¹(Rⁿ) and from L¹(Rⁿ) into BMO(Rⁿ); If [b, T^σ] is bounded from H¹(Rⁿ) into L¹(Rⁿ) or L¹(Rⁿ) into BMO(Rⁿ), then, b ∈ LMO_loc.     

Riesz means for the sublaplacian on the Heisenberg group

The uniform boundedness of the Riesz means for the sublaplacian on the Heisenberg groupH ⁿ is considered. It is proved thatS _R ^α are uniformly bounded onL ^p(Hⁿ) for 1≤p≤2 provided α>α(p)=(2n+1)[(1/p)−(1/2)].     

The heat kernel transform for the Heisenberg group

The heat kernel transform H_t is studied for the Heisenberg group in detail. The main result shows that the image of H_t is a direct sum of two weighted Bergman spaces, in contrast to the classical case of Rⁿ and compact symmetric spaces, and the weight functions are found to be (surprisingly) not non-negative.     

Fourier multipliers for Sobolev spaces on the Heisenberg group

In this paper, it is shown that the class of right Fourier multipliers for the Sobolev space W ^k,p (H ⁿ ) coincides with the class of right Fourier multipliers for L ^p (H ⁿ ) for k ∈ ?, 1 < p < ∞. Towards this end, it is shown that the operators R _j $ \bar R $ _j ?^?1 and $ \bar R $ _j R _j ?^?1 are bounded on L ^p (H ⁿ ), 1 < p < ∞, where $$ R_j = \frac{\partial } {{\partial z_j }} - \frac{i} {4}\bar z_j \frac{\partial } {{\partial t}}, \bar R_j = \frac{\partial } {{\partial \bar z_j }} + \frac{i} {4}z_j \frac{\partial } {{\partial t}} $$ and ? is the sublaplacian on H ⁿ . This proof is based on the Calderon-Zygmund theory on the Heisenberg group. It is also shown that when p = 1, the class of right multipliers for the Sobolev space W ^k,1(H ⁿ ) coincides with the dual space of the projective tensor product of two function spaces.     

Laguerre expansion on the Heisenberg group and Fourier-Bessel transform on ℂ n

Given a principal value convolution on the Heisenberg group H _n = ? ⁿ × ?, we study the relation between its Laguerre expansion and the Fourier-Bessel expansion of its limit on ? ⁿ . We also calculate the Dirichlet kernel for the Laguerre expansion on the group H _n .     

The weighted Hardy spaces associated to self-adjoint operators and their duality on product spaces

Let L be a non-negative self-adjoint operator acting on L²(R ⁿ ) satisfying a pointwise Gaussian estimate for its heat kernel. Let w be an A _r weight on R ⁿ × R ⁿ , 1 < r < ∞. In this article we obtain a weighted atomic decomposition for the weighted Hardy space H _L,w ^p (R ⁿ ×R ⁿ ), 0 < p ≤ 1 associated to L. Based on the atomic decomposition, we show the dual relationship between H _L,w ¹ (R ⁿ × R ⁿ ) and BMO_L,w(R ⁿ × R ⁿ ).     

LOWER SEMICONTINUITY OF A CLASS OF FUNCTIONALS IN SBVH

In this paper lower semicontinuity of the functional I(u)=∫ _Ω f(x,u,Δ _Hu)dx is investigated for f being a Carathéodory function defined on H ⁿ × R × R²ⁿ and for u∈SBV _H (Ω), where H ⁿ is the Heisenberg group with dimension 2n+1, Ω∩H ⁿ is an open set and ∇ Hu denotes the approximate derivative of the absolute continuous part D ^a _Hu with respect to D _Hu. In addition, a Lusin type approximation theorem for a SBV _H function is proved.     

A local smoothing estimate for the Schrödinger equation

We show that the Schrödinger operator eitΔ is bounded from Wα,q(Rn) to Lq(Rn×[0,1]) for all α>2n(1/2−1/q)−2/q and q?2+4/(n+1). This is almost sharp with respect to the Sobolev index. We also show that the Schrödinger maximal operator sup0<t<1|eitΔf| is bounded from Hs(Rn) to when s>s₀ if and only if it is bounded from Hs(Rn) to L²(Rn) when s>2s₀. A corollary is that sup0<t<1|eitΔf| is bounded from Hs(R²) to L²(R²) when s>3/4.     

海森堡群上薛定谔算子的黎斯位势的某些性质

设L=-△_(H~n)+V是Heisenberg群H~n上的Schr(o|¨)dinger算子,其中△_(H~n)为H~n上的次Laplacian,V≠0为满足逆H(o|¨)lder不等式的非负函数.本文研究H~n上Riesz位势I_α~L=L~(-α/2)在Campanato型空间A_L~β和Hardy型空间H_L~p上的某些性质.     

Two versions of the Nikodym maximal function on the Heisenberg group

The classical Nikodym maximal function on the Euclidean plane R² is defined as the supremum over averages over rectangles of eccentricity N; its operator norm in L²(R²) is known to be O(logN). We consider two variants, one on the standard Heisenberg group H¹ and the other on the polarized Heisenberg group . The latter has logarithmic L² operator norm, while the former has the L² operator norm which grows essentially of order O(N1/4). We shall imbed these two maximal operators in the family of operators associated to the hypersurfaces {(x₁,x₂,αx₁x₂)} in the Heisenberg group H¹ where the exceptional blow up in N occurs when α=0.     

Intrinsic regular hypersurfaces in Heisenberg groups

We study the ℍ-regular surfaces, a class of intrinsic regular hypersurfaces in the setting of the Heisenberg group ℍ ⁿ = ℂ ⁿ × ℝ = ℝ²ⁿ⁺¹ endowed with a leftinvariant metric d_∞ equivalent to its Carnot-Carathéodory (CC) metric. Here hypersurface simply means topological codimension 1 surface and by the words “intrinsic” and “regular” we mean, respectively notions involving the group structure of ℍ ⁿ and its differential structure as CC manifold. In particular, we characterize these surfaces as intrinsic regular graphs inside ℍ ⁿ by studying the intrinsic regularity of the parameterizations and giving an areatype formula for their intrinsic surface measure.     

The geometry of cc-balls and the constants in the Ball-Box theorem on Heisenberg group algebras

We study the geometry of balls in the Carnot-Carathéodory metric and find the optimal equivalence constants in the Ball-Box Theorem on the Heisenberg group algebras ? _α ⁿ .     

Weak limit and blowup of approximate solutions to H-systems

Let be a continuous function such that H(p)→H₀∈R as |p|→+∞. Fixing a domain Ω in R² we study the behaviour of a sequence (un) of approximate solutions to the H-system Δu=2H(u)ux∧uy in Ω. Assuming that supp∈R³|(H(p)−H₀)p|<1, we show that the weak limit of the sequence (un) solves the H-system and un→u strongly in H¹ apart from a countable set S made by isolated points. Moreover, if in addition H(p)=H₀+o(1/|p|) as |p|→+∞, then in correspondence of each point of S we prove that the sequence (un) blows either an H-bubble or an H₀-sphere.     

Orlicz-Hardy spaces and their duals

We establish the theory of Orlicz-Hardy spaces generated by a wide class of functions.The class will be wider than the class of all the N-functions.In particular,we consider the non-smooth atomic decomposition.The relation between Orlicz-Hardy spaces and their duals is also studied.As an application,duality of Hardy spaces with variable exponents is revisited.This work is different from earlier works about Orlicz-Hardy spaces H(Rn)in that the class of admissible functions is largely widened.We can deal with,for example,(r)≡(rp1(log(e+1/r))q1,0r 6 1,rp2(log(e+r))q2,r1,with p1,p2∈(0,∞)and q1,q2∈(.∞,∞),where we shall establish the boundedness of the Riesz transforms on H(Rn).In particular,is neither convex nor concave when 0p11p2∞,0p21p1∞or p1=p2=1 and q1,q20.If(r)≡r(log(e+r))q,then H(Rn)=H(logH)q(Rn).We shall also establish the boundedness of the fractional integral operators I of order∈(0,∞).For example,I is shown to be bounded from H(logH)1./n(Rn)to Ln/(n.)(log L)(Rn)for 0n.     

Polynomial images of R

Let R be a real closed field and n?2. We prove that: (1) for every finite subset F of Rⁿ, the semialgebraic set Rⁿ?F is a polynomial image of Rⁿ; and (2) for any independent linear forms l₁,…,l_r of Rⁿ, the semialgebraic set {l₁>0,…,l_r>0}⊂Rⁿ is a polynomial image of Rⁿ.     

On the integrality of nth roots of generating functions

Motivated by the discovery that the eighth root of the theta series of the E₈ lattice and the 24th root of the theta series of the Leech lattice both have integer coefficients, we investigate the question of when an arbitrary element f∈R (where R=1+xZ?x?) can be written as f=gn for g∈R, n?2. Let Pn:={gn|g∈R} and let . We show among other things that (i) for f∈R, f∈Pn⇔f (mod μn)∈Pn, and (ii) if f∈Pn, there is a unique g∈Pn with coefficients mod μn/n such that f≡gn (mod μn). In particular, if f≡1 (mod μn) then f∈Pn. The latter assertion implies that the theta series of any extremal even unimodular lattice in Rn (e.g. E₈ in R⁸) is in Pn if n is of the form i²j³k⁵ (i?3). There do not seem to be any exact analogues for codes, although we show that the weight enumerator of the rth order Reed-Muller code of length m² is in Pr² (and similarly that the theta series of the Barnes-Wall lattice BWm² is in Pm²). We give a number of other results and conjectures, and establish a conjecture of Paul D. Hanna that there is a unique element f∈Pn (n?2) with coefficients restricted to the set {1,2,…,n}.     

Contact quasiconformal immersions

Contact immersions of contact manifolds endowed with the associated Carnot-Carathéodory (CC) metric (for example, immersions of the Heisenberg group H ³ ～ ? _CC ³ in itself) are considered. It is assumed that the manifolds have the same dimension and the immersions are quasiconformal with respect to the CC metric. The main assertion is as follows: A quasiconformal immersion of the Heisenberg group in itself, just as a quasiconformal immersion of any contact manifold of conformally parabolic type in a simply connected contact manifold, is globally injective; i.e., such an immersion is an embedding, which, in addition, is surjective in the case of the Heisenberg group. Thus, the global homeomorphism theorem, which is well known in the space theory of quasiconformal mappings, also holds in the contact case.     

Rectifiability and Lipschitz extensions into the Heisenberg group

Denote by \mathbbHⁿ{\mathbb{H}^n} the 2n + 1 dimensional Heisenberg group. We show that the pairs (\mathbbR^k ,\mathbbHⁿ){(\mathbb{R}^k ,\mathbb{H}^n)} and (\mathbbH^k ,\mathbbHⁿ){(\mathbb{H}^k ,\mathbb{H}^n)} do not have the Lipschitz extension property for k  >  n.     

Laguerre expansion on the Heisenberg group and Fourier-Bessel transform on ?n

Given a principal value convolution on the Heisenberg group H _n = ℂ ⁿ × ℝ, we study the relation between its Laguerre expansion and the Fourier-Bessel expansion of its limit on ℂ ⁿ . We also calculate the Dirichlet kernel for the Laguerre expansion on the group H _n . Dedicated to Professor Sheng GONG on the occasion of his 75th birthday