Higher order commutators for a class of rough operators

In this paper we study the (L ^p (u ^p ),L ^q (v ^q )) boundedness of the higher order commutatorsT _Ω,α,b ^m andM _Ω,α,b ^m formed by the fractional integral operatorT _Ω,α , the fractional maximal operatorM _Ω,α , and a functionb(x) in BMO(v), respectively. Our results imporve and extend the corresponding results obtained by Segovia and Torrea in 1993 [9]. The project was supported by NNSF of China.     

Weighted <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> boundedness for multilinear fractional integral on product spaces

For 0 < α < mn and nonnegative integers n ≥ 2, m ≥ 1, the multilinear fractional integral is defined by

The maximal (C,α,β) operator on Hp(T×T)

The two-dimensional classical Hardy space H_p(T×T) on the bidisc are introduced, and it is shown that the maximal operator of the (C,α,β) means of a distribution is bounded from the space H_p(T×T) to L_p(T²) (1/(α+1), 1/(β+1)<p≤∞), and is of weak type (H ₁ ^# (T×T), L₁(T²)), where the Hardy space H ₁ ^# (T×T) is defined by the hybrid maximal function. As a consequence we obtain that the (C, α, β) means of a function f∈H ₁ ^# (T×T)⊃LlogL(T ²) convergs a. e. to the function in question. Moreover, we prove that the (C, α, β) means are uniformly bounded on the spaces H_p(T×T) whenever 1/(α+1), 1(β+1)<p<∞. Thus, in case f∈H_p(T×T), the (C, α, β) means convergs to f in H_p(T×T) norm whenever (1/(α+1), 1/(β+1)<p<∞). The same results are proved for the conjugate (C, α, β) means, too.     

The maximal (C,α,β) operator on Hp(T×T)

The two-dimensional classical Hardy space H_p(T×T) on the bidisc are introduced, and it is shown that the maximal operator of the (C,α,β) means of a distribution is bounded from the space H_p(T×T) to L_p(T²) (1/(α+1), 1/(β+1)<p≤∞), and is of weak type (H ₁ ^# (T×T), L₁(T²)), where the Hardy space H ₁ ^# (T×T) is defined by the hybrid maximal function. As a consequence we obtain that the (C, α, β) means of a function f∈H ₁ ^# (T×T)⊃LlogL(T ²) convergs a. e. to the function in question. Moreover, we prove that the (C, α, β) means are uniformly bounded on the spaces H_p(T×T) whenever 1/(α+1), 1(β+1)<p<∞. Thus, in case f∈H_p(T×T), the (C, α, β) means convergs to f in H_p(T×T) norm whenever (1/(α+1), 1/(β+1)<p<∞). The same results are proved for the conjugate (C, α, β) means, too. This research was made while the author was visiting the Humboldt University in Berlin supported by the Alexander von Humboldt Foundation.     

Best N Term Approximation Spaces for Tensor Product Wavelet Bases

We consider best N term approximation using anisotropic tensor product wavelet bases ("sparse grids"). We introduce a tensor product structure ⊗_q on certain quasi-Banach spaces. We prove that the approximation spaces A^α_q(L₂) and A^α_q(H¹) equal tensor products of Besov spaces B^α_q(L_q), e.g., A^α_q(L₂([0,1]^d)) = B^α_q(L_q([0,1])) ⊗_q · ⊗_q B^α_{q · ·}(L_q([0,1])). Solutions to elliptic partial differential equations on polygonal/polyhedral domains belong to these new scales of Besov spaces.     

On noncommutative weighted local ergodic theorems on <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-spaces

In the present paper we consider a von Neumann algebra M with a faithful normal semi-finite trace τ, and {α _t }, a strongly continuous extension to L ^p (M, τ) of a semigroup of absolute contractions on L ¹(M, τ). By means of a non-commutative Banach Principle we prove for a Besicovitch function b and x ∊ L ^p (M, τ), that the averages 1/T ∫₀ ^T b(t)α _t (x)dt converge bilateral almost uniformly in L ^p (M, τ) as T → 0. Communicated by Dénes Petz     

Discrete spectrum of the perturbed Dirac operator

In this paper we study the asymptotics of the discrete spectrum in the gap (−1, 1) of the perturbed Dirac operatorD(α)=D ₀−αV1 acting inL ₂(R ³;C ⁴) with large coupling constant α. In particular some “non-standard” asymptotic formulae are obtained.     

Boundedness of maximal,potential type,and singular integral operators in the generalized variable exponent Morrey type spaces

We consider generalized Morrey type spaces M^{p( ·),q( ·),w( ·)}( W) {\mathcal{M}^{p\left( \cdot \right),\theta \left( \cdot \right),\omega \left( \cdot \right)}}\left( \Omega \right) with variable exponents p(x), θ(r) and a general function ω(x, r) defining a Morrey type norm. In the case of bounded sets W ì \mathbbRⁿ \Omega \subset {\mathbb{R}^n} , we prove the boundedness of the Hardy–Littlewood maximal operator and Calderón–Zygmund singular integral operators with standard kernel. We prove a Sobolev–Adams type embedding theorem M^{p( ·),q₁( ·),w₁( ·)}( W) ? M^{q( ·),q₂( ·),w₂( ·)}( W) {\mathcal{M}^{p\left( \cdot \right),{\theta_1}\left( \cdot \right),{\omega_1}\left( \cdot \right)}}\left( \Omega \right) \to {\mathcal{M}^{q\left( \cdot \right),{\theta_2}\left( \cdot \right),{\omega_2}\left( \cdot \right)}}\left( \Omega \right) for the potential type operator I ^α(·) of variable order. In all the cases, we do not impose any monotonicity type conditions on ω(x, r) with respect to r. Bibliography: 40 titles.     

Higher order Poincaré inequalities associated with linear operators on stratified groups and applications

 This paper considers the dual of anisotropic Sobolev spaces on any stratified groups 𝔾. For 0≤k<m and every linear bounded functional T on anisotropic Sobolev space W ^m−k,p (Ω) on Ω⊂𝔾, we derive a projection operator L from W ^m,p (Ω) to the collection 𝒫 _k+1 of polynomials of degree less than k+1 such that T(X ^I (Lu))=T(X ^I u) for all uW ^m,p (Ω) and multi-index I with d(I)≤k. We then prove a general Poincaré inequality involving this operator L and the linear functional T. As applications, we often choose a linear functional T such that the associated L is zero and consequently we can prove Poincaré inequalities of special interests. In particular, we obtain Poincaré inequalities for functions vanishing on tiny sets of positive Bessel capacity on stratified groups. Finally, we derive a Hedberg-Wolff type characterization of measures belonging to the dual of the fractional anisotropic Sobolev spaces W ^α,p 𝔾. Received: 25 March 2002; in final form: 10 September 2002 / Published online: 1 April 2003 Mathematics Subject Classification (1991): 46E35, 41A10, 22E25 The second author was supported partly by U.S NSF grant DMS99-70352 and the third author was supported partly by NNSF grant of China.     

Fractional Poisson equations and ergodic theorems for fractional coboundaries

For a given contractionT in a Banach spaceX and 0<α<1, we define the contractionT _α=Σ _j=1 ^∞ a _j T ^j , where {a _j } are the coefficients in the power series expansion (1-t)^α=1-Σ _j=1 ^∞ a _j t ^j in the open unit disk, which satisfya _j >0 anda _j >0 and Σ _j=1 ^∞ a _j =1. The operator calculus justifies the notation(I−T) ^α :=I−T _α (e.g., (I−T _1/2)²=I−T). A vectory∈X is called an, α-fractional coboundary for T if there is anx∈X such that(I−T) ^α x=y, i.e.,y is a coboundary forT _α . The fractional Poisson equation forT is the Poisson equation forT _α . We show that if(I−T)X is not closed, then(I−T) ^α X strictly contains(I−T)X (but has the same closure). ForT mean ergodic, we obtain a series solution (converging in norm) to the fractional Poisson equation. We prove thaty∈X is an α-fractional coboundary if and only if Σ _k=1 ^∞ T ^k y/k ^1-α converges in norm, and conclude that lim _n ‖(1/n ^1-α)Σ _k=1 ⁿ T ^k y‖=0 for suchy. For a Dunford-Schwartz operatorT onL ₁ of a probability space, we consider also a.e. convergence. We prove that iff∈(I−T) ^α L ₁ for some 0<α<1, then the one-sided Hilbert transform Σ _k=1 ^∞ T ^k f/k converges a.e. For 1<p<∞, we prove that iff∈(I−T) ^α L _p with α>1−1/p=1/q, then Σ _k=1 ^∞ T ^k f/k ^1/p converges a.e., and thus (1/n ^1/p ) Σ _k=1 ⁿ T ^k f converges a.e. to zero. Whenf∈(I−T) ^1/q L _p (the case α=1/q), we prove that (1/n ^1/p (logn)^1/q )Σ _k=1 ⁿ T ^k f converges a.e. to zero.     

Classification of framed links in 3-manifolds

We present a short and complete proof of the following Pontryagin theorem, whose original proof was complicated and has never been published in detail. Let M be a connected oriented closed smooth 3-manifold, L ₁(M) be the set of framed links in M up to a framed cobordism, and deg: L ₁(M) → H ₁(M; ℤ) be the map taking a framed link to its homology class. Then for each α ∈ H ₁(M; ℤ) there is a one-to-one correspondence between the set deg⁻¹ α and the group ℤ_2d(α), where d(α) is the divisibility of the projection of α to the free part of H ₁(M; ℤ).     

Boundedness of Multilinear Operators in Herz-type Hardy Space

Let κ∈ℕ. We prove that the multilinear operators of finite sums of products of singular integrals on ℝⁿ are bounded from HK ^α1,p1 _q1 (ℝⁿ) ×···×HK ^αk,pk _qk (ℝⁿ) into HK ^α,p _q (ℝⁿ) if they have vanishing moments up to a certain order dictated by the target spaces. These conditions on vanishing moments satisfied by the multilinear operators are also necessary when α_j≥ 0 and the singular integrals considered here include the Calderón-Zygmund singular integrals and the fractional integrals of any orders. Received September 6, 1999, Revised November 17, 1999, Accepted December 9, 1999     

Asymptotiques d’un systeme dynamique conservatif associe a une equation elliptique non lineaire

LetM be a compact riemannian manifold,h an odd function such thath(r)/r is non-decreasing with limit 0 at 0. Letf(r)=h(r)-γr and assume there exist non-negative constantsA andB and a realp>1 such thatf(r)>Ar ^P-B. We prove that any non-negative solutionu ofu _tt+Δ_gu=f(u) onM x ℝ⁺ satisfying Dirichlet or Neumann boundary conditions on ϖM converges to a (stationary) solution of Δ _g Φ=f(Φ) onM with exponential decay of ‖u-Φ‖C ²(M). For solutions with non-constant sign, we prove an homogenisation result for sufficiently small λ; further, we show that for every λ the map (u(0,·),u _t(0,·))→(u(t,·), u _t(t,·)) defines a dynamical system onW ^1/2(M)⊂C(M)×L ²(M) which possesses a compact maximal attractor.      

Estimates of the weak distance between finite-dimensional Banach spaces

We prove a variant of a theorem of N. Alon and V. D. Milman. Using it we construct for everyn-dimensional Banach spacesX andY a measure space Ω and two operator-valued functionsT: Ω→L(X, Y),S: Ω→L(Y, X) so that ∫_Ω S(ω)oT(ω)dω is the identity operator inX and ∫_Ω||S(ω)||·||T(ω)||dω=O(n ^α ) for some absolute constantα<1. We prove also that any subset of the unitn-cube which is convex, symmetric with respect to the origin and has a sufficiently large volume possesses a section of big dimension isomorphic to ak-cube. Research supported in part by a grant of the Israel Academy of Sciences.     

Compact operators on a Banach space into the space of almost periodic functions

LetS be a topological semigroup andAP(S) the space of continous complex almost periodic functions onS. We obtain characterizations of compact and weakly compact operators from a Banach spaceX into AP(S). For this we use the almost periodic compactification ofS obtained through uniform spaces. For a bounded linear operatorT fromX into AP(S), letT ₅, be the translate ofT bys inS defined byT ₅(x)=(Tx) ₅ . We define topologies on the space of bounded linear operators fromX into AP(S) and obtain the necessary and sufficient conditions for an operatorT to be compact or weakly compact in terms of the uniform continuity of the maps→T ₅. IfS is a Hausdorff topological semigroup, we also obtain characterizations of compact and weakly compact multipliers on AP(S) in terms of the uniform continuity of the map S→μ_s, where μ_s denotes the unique vector measure corresponding to the operatorT ₅.     

On the pontryagin-steenrod-wu theorem

We present a short and direct proof (based on the Pontryagin-Thom construction) of the following Pontryagin-Steenrod-Wu theorem: (a) LetM be a connected orientable closed smooth (n + 1)-manifold,n≥3. Define the degree map deg: π ⁿ (M) →H ⁿ (M; ℤ) by the formula degf =f*[S ⁿ ], where [S ⁿ ] εH ⁿ (M; ℤ) is the fundamental class. The degree map is bijective, if there existsβ εH ₂(M, ℤ/2ℤ) such thatβ ·w ₂(M) ε 0. If suchβ does not exist, then deg is a 2-1 map; and (b) LetM be an orientable closed smooth (n+2)-manifold,n≥3. An elementα lies in the image of the degree map if and only ifρ ₂ α ·w ₂(M)=0, whereρ ₂: ℤ → ℤ/2ℤ is reduction modulo 2.     

Maximal sets in α-recursion theory

Let α be an admissible ordinal, and leta ^* be the Σ₁-projectum ofa. Call an α-r.e. setM maximal if α→M is unbounded and for every α→r.e. setA, eitherA∩(α-M) or (α-A)∩(α-M) is bounded. Call and α-r.e. setM amaximal subset of α^* if α^*−M is undounded and for any α-r.e. setA, eitherA∩(α^*-M) or (⇌^*-A)∩(α^*-M) is unbounded in α^*. Sufficient conditions are given both for the existence of maximal sets, and for the existence of maximal subset of α^*. Necessary conditions for the existence of maximal sets are also given. In particular, if α ≧ ℵ ^L then it is shown that maximal sets do not exist. Research partially supported by NSF Grant GP-34088 X. Some of the results in this paper have been taken from the second author’s Ph. D. Thesis, written under the supervision of Gerald Sacks.     

Additive Maps Preserving Similarity of Operators on Banach Spaces

Let X be an infinite-dimensional complex Banach space and denote by B（X） the algebra of all bounded linear operators acting on X. It is shown that a surjective additive map Φ from B（X） onto itself preserves similarity in both directions if and only if there exist a scalar c, a bounded invertible linear or conjugate linear operator A and a similarity invariant additive functional ψ on B（X） such that either Φ（T） = cATA^-1 ＋ ψ（T）I for all T, or Φ（T） = cAT＊A^-1 ＋ ψ（T）I for all T. In the case where X has infinite multiplicity, in particular, when X is an infinite-dimensional Hilbert space, the above similarity invariant additive functional ψ is always zero.     

An algebraic invariant for Jordan automorphisms on β(<Emphasis Type="Italic">H</Emphasis>): The set of idempotents

Let H be an infinite dimensional complex Hilbert space. Denote by B(H) the algebra of all bounded linear operators on H, and by I(H) the set of all idempotents in B(H). Suppose that Φ is a surjective map from B(H) onto itself. If for every λ ∈ -1,1,2,3, and A, B ∈ B(H),A-λB ∈I(H) ⇔ Φ(A) -λΦ(B) ∈I(H, then Φ is a Jordan ring automorphism, i.e. there exists a continuous invertible linear or conjugate linear operator T on H such that Φ(A) = TAT ^-1 for all A ∈ B(H), or Φ(A) = TA*T ^-1 for all A ∈ B(H); if, in addition, A-iB ∈I(H)⇔ Φ(A)-iΦ(B) ∈I(H), here i is the imaginary unit, then Φ is either an automorphism or an anti-automorphism.     

Ordering Trees with Nearly Perfect Matchings by Algebraic Connectivity

Let T2k＋1 be the set of trees on 2k＋1 vertices with nearly perfect matchings and α（T） be the algebraic connectivity of a tree T. The authors determine the largest twelve values of the algebraic connectivity of the trees in T2k＋1. Specifically, 10 trees T2,T3,... ,T11 and two classes of trees T（1） and T（12） in T2k＋1 are introduced. It is shown in this paper that for each tree T^′1,T^″1∈T（1）and T^′12,T^″12∈T（12） and each i,j with 2≤i〈j≤11,α（T^′1）=α（T^″1）〉α（Tj）〉α（T^′12）=α（T^″12）.It is also shown that for each tree T with T∈T2k＋1／（T（1）∪{T2,T3,…,T11}∪T（12））,α（T^′12）〉α（T）.