Operator martingale decompositions and the Radon-Nikodým property in Banach spaces

We consider submartingales and uniform amarts of maps acting between a Banach lattice and a Banach lattice or a Banach space. In this measure-free setting of martingale theory, it is known that a Banach space Y has the Radon-Nikodým property if and only if every uniformly norm bounded martingale defined on the Chaney-Schaefer l-tensor product , where E is a suitable Banach lattice, is norm convergent. We present applications of this result. Firstly, an analogues characterization for Banach lattices Y with the Radon-Nikodým property is given in terms of a suitable set of submartingales (supermartingales) on . Secondly, we derive a Riesz decomposition for uniform amarts of maps acting between a Banach lattice and a Banach space. This result is used to characterize Banach spaces with the Radon-Nikodým property in terms of uniformly norm bounded uniform amarts of maps that are norm convergent. In the case 1<p<∞, our results yield Lp(μ,Y)-space analogues of some of the well-known results on uniform amarts in L¹(μ,Y)-spaces.     

Uniform Homeomorphisms of Unit Spheres and Property H of Lebesgue–Bochner Function Spaces

Assume that the unit spheres of Banach spaces X and Y are uniformly homeomorphic.Then we prove that all unit spheres of the Lebesgue–Bochner function spaces L_p(μ, X) and L_q(μ, Y)are mutually uniformly homeomorphic where 1 ≤ p, q ∞. As its application, we show that if a Banach space X has Property H introduced by Kasparov and Yu, then the space L_p(μ, X), 1 ≤ p ∞,also has Property H.     

Distance regularity of Kerdock codes

A code is called distance regular, if for every two codewords x, y and integers i, j the number of codewords z such that d(x, z) = i and d(y, z) = j, with d the Hamming distance, does not depend on the choice of x, y and depends only on d(x, y) and i, j. Using some properties of the discrete Fourier transform we give a new combinatorial proof of the distance regularity of an arbitrary Kerdock code. We also calculate the parameters of the distance regularity of a Kerdock code.     

Multiresolution Approximations and Wavelet Bases of WeightedLpSpaces

We study boundedness and convergence on L ^p (R ⁿ ,d) of the projection operators P _j given by MRA structures with non-necessarily compactly supported scaling function. As a consequence, we prove that if w is a locally integrable function such that w ^-(1/p–1)(x) (1+|x|)^-N is integrable for some N > 0, then the Muckenhoupt A _p condition is necessary and sufficient for the associated wavelet system to be an unconditional basis for the weighted space L ^p (R ⁿ ,w(x) dx), 1 < p < .     

A Probabilistic Characterization of the Dominance Order on Partitions

We present a probabilistic characterization of the dominance order on partitions. Let ν be a partition and Y _ν its Ferrers diagram, i.e. a stack of rows of cells with row i containing ν _i cells. Let the cells of Y _ν be filled with independent and identically distributed draws from the random variable X = B i n(r, p) with r ≥ 1 and p ∈ (0, 1). Given j, t ≥ 0, let P(ν, j, t) be the probability that the sum of all the entries in Y _ν is j while the sum of the entries in each row of Y _ν is no more than t. It is shown that if ν and μ are two partitions of n, ν dominates μ if and only if P(ν, j, t) ≤ P(μ, j, t) for all j, t ≥ 0. It is shown that the same result holds if X is any log-concave integer valued random variable with {i : P(X = i) > 0} = {0, 1,…,r} for some r ≥ 1.     

On the variation of Tate-Shafarevich groups of elliptic curves over hyperelliptic curves

Let E be an elliptic curve over F=F_q(t) having conductor (p)·∞, where (p) is a prime ideal in F_q[t]. Let d∈F_q[t] be an irreducible polynomial of odd degree, and let . Assume (p) remains prime in K. We prove the analogue of the formula of Gross for the special value L(E⊗_FK,1). As a consequence, we obtain a formula for the order of the Tate-Shafarevich group Ш(E/K) when L(E⊗_FK,1)≠0.     

Endpoints of set-valued contractions in metric spaces

Suppose (X,d) be a complete metric space, and suppose F:X→CB(X) be a set-valued map satisfies H(Fx,Fy)≤ψ(d(x,y)), , where ψ:[0,∞)→[0,∞) is upper semicontinuous, ψ(t)<t for each t>0 and satisfies lim inf_t→∞(t−ψ(t))>0. Then F has a unique endpoint if and only if F has the approximate endpoint property.     

Banach spaces in various positions

We formulate a general theory of positions for subspaces of a Banach space: we define equivalent and isomorphic positions, study the automorphy index a(Y,X) that measures how many non-equivalent positions Y admits in X, and obtain estimates of a(Y,X) for X a classical Banach space such as ?p,Lp,L₁,C(ωω) or C[0,1]. Then, we study different aspects of the automorphic space problem posed by Lindenstrauss and Rosenthal; namely, does there exist a separable automorphic space different from c₀ or ?₂? Recall that a Banach space X is said to be automorphic if every subspace Y admits only one position in X; i.e., a(Y,X)=1 for every subspace Y of X. We study the notion of extensible space and uniformly finitely extensible space (UFO), which are relevant since every automorphic space is extensible and every extensible space is UFO. We obtain a dichotomy theorem: Every UFO must be either an L_∞-space or a weak type 2 near-Hilbert space with the Maurey projection property. We show that a Banach space all of whose subspaces are UFO (called hereditarily UFO spaces) must be asymptotically Hilbertian; while a Banach space for which both X and X^∗ are UFO must be weak Hilbert. We then refine the dichotomy theorem for Banach spaces with some additional structure. In particular, we show that an UFO with unconditional basis must be either c₀ or a superreflexive weak type 2 space; that a hereditarily UFO Köthe function space must be Hilbert; and that a rearrangement invariant space UFO must be either L_∞ or a superreflexive type 2 Banach lattice.     

Extreme points of Banach lattices related to conditional expectations

Let (X,F,μ) be a complete probability space, B a sub-σ-algebra, and Φ the probabilistic conditional expectation operator determined by B. Let K be the Banach lattice {f∈L¹(X,F,μ):‖Φ(|f|)_‖∞<∞} with the norm ‖f‖=‖Φ(|f|)_‖∞. We prove the following theorems:

(1)

The closed unit ball of K contains an extreme point if and only if there is a localizing set E for B such that supp(Φ(χE))=X.

(2)

Suppose that there is n∈N such that f?nΦ(f) for all positive f in L^∞(X,F,μ). Then K has the uniformly λ-property and every element f in the complex K with is a convex combination of at most 2n extreme points in the closed unit ball of K.

     

Riesz transforms for Dunkl Hermite expansions

In the present paper, we establish that Riesz transforms for Dunkl Hermite expansions introduced by Nowak and Stempak are singular integral operators with Hörmander's type condition. We prove that they are bounded on L^p(R^d,d_μκ)$L^p({\mathbb{R}}^d,d{\mu }_{\kappa })$ for 1<p<∞$1<p<\infty$ and from L¹(R^d,d_μκ)$L^1({\mathbb{R}}^d,d{\mu }_{\kappa })$ into L^1,∞(R^d,d_μκ)$L^{1,\infty }({\mathbb{R}}^d,d{\mu }_{\kappa })$.     

Lévy processes and Fourier multipliers

We study Fourier multipliers which result from modulating jumps of Lévy processes. Using the theory of martingale transforms we prove that these operators are bounded in Lp(Rd) for 1<p<∞ and we obtain the same explicit bound for their norm as the one known for the second order Riesz transforms.     

Cores for Feller semigroups with an invariant measure

Let be an elliptic differential operator with unbounded coefficients on RN and assume that the associated Feller semigroup (T(t))_t?0 has an invariant measure μ. Then (T(t))_t?0 extends to a strongly continuous semigroup (Tp(t))_t?0 on Lp(μ)=Lp(RN,μ) for every 1?p<∞. We prove that, under mild conditions on the coefficients of A, the space of test functions is a core for the generator (Ap,Dp) of (Tp(t))_t?0 in Lp(μ) for 1?p<∞.     

A dominated ergodic theorem for some bilinear averages

Let T be a positive invertible linear operator with positive inverse on some Lp(μ), 1?p<∞, where μ is a σ-finite measure. We study the convergence in the Lp(μ)-norm and the almost everywhere convergence of the bilinear operators

     

Necessary and sufficient conditions for boundedness of some commutators with weighted Lipschitz functions

In this paper we show that b∈Lipβ,μ if and only if the commutator [b,T] of the multiplication operator by b and the singular integral operator T is bounded from Lp(μ) to Lq(μ1−q), where 1<p<q<∞, 0<β<1 and 1/q=1/p−β/n. Also we will obtain that b∈Lipβ,μ if and only if the commutator [b,Iα] of the multiplication operator by b and the fractional integral operator Iα is bounded from Lp(μ) to Lr(μ1−(1−α/n)r), where 1<p<∞, 0<β<1 and 1/r=1/p−(β+α)/n with 1/p>(β+α)/n.     

On fractional maximal function and fractional integrals associated with the Dunkl operator on the real line

In this paper we obtain necessary and sufficient conditions on the parameters for the boundedness of the Dunkl-type fractional maximal operator Mβ, and the Dunkl-type fractional integral operator Iβ from the spaces Lp,α(R) to the spaces Lq,α(R), 1<p<q<∞, and from the spaces L1,α(R) to the weak spaces WLq,α(R), 1<q<∞. In the case , we prove that the operator Mβ is bounded from the space Lp,α(R) to the space L∞,α(R), and the Dunkl-type modified fractional integral operator is bounded from the space Lp,α(R) to the Dunkl-type BMO space BMOα(R). By this results we get boundedness of the operators Mβ and Iβ from the Dunkl-type Besov spaces to the spaces , 1<p<q<∞, 1/p−1/q=β/(2α+2), 1?θ?∞ and 0<s<1.     

Mapping properties for oscillatory integrals in d-dimensions

For aj,bj?1, j=1,2,…,d, we prove that the operator maps into itself for , where , and k(x,y)=φ(x,y)eig(x,y), φ(x,y) satisfies (1.2) (e.g. φ(x,y)=|x−y|^iτ,τ real) and the phase g(x,y)=xa⋅yb. We study operators with more general phases and for these operators we require that aj,bj>1, j=1,2,…,d, or al=bl?1 for some l∈{1,2,…,d}.     

An operator-valued Tb theorem

We establish conditions similar to the Tb theorem of David, Journé and Semmes which guarantee the boundedness of an integral transformation T with L(X)-valued kernel on , where 1<p<∞ and X is a Banach space with the unconditionality property of martingale differences (UMD).     

Dixmier traces as singular symmetric functionals and applications to measurable operators

We unify various constructions and contribute to the theory of singular symmetric functionals on Marcinkiewicz function/operator spaces. This affords a new approach to the non-normal Dixmier and Connes-Dixmier traces (introduced by Dixmier and adapted to non-commutative geometry by Connes) living on a general Marcinkiewicz space associated with an arbitrary semifinite von Neumann algebra. The corollaries to our approach, stated in terms of the operator ideal L^(1,∞) (which is a special example of an operator Marcinkiewicz space), are: (i) a new characterization of the set of all positive measurable operators from L^(1,∞), i.e. those on which an arbitrary Connes-Dixmier trace yields the same value. In the special case, when the operator ideal L^(1,∞) is considered on a type I infinite factor, a bounded operator x belongs to L^(1,∞) if and only if the sequence of singular numbers {s_n(x)}_n?1 (in the descending order and counting the multiplicities) satisfies . In this case, our characterization amounts to saying that a positive element x∈L^(1,∞) is measurable if and only if exists; (ii) the set of Dixmier traces and the set of Connes-Dixmier traces are norming sets (up to equivalence) for the space , where the space is the closure of all finite rank operators in L^(1,∞) in the norm ∥.∥_(1,∞).     

Multilinear Calderón-Zygmund operators on the product of Lebesgue spaces with non-doubling measures

Under the assumption that μ is a non-doubling measure on Rd, the author proves that for the multilinear Calderón-Zygmund operator, its boundedness from the product of Hardy space H¹(μ)×H¹(μ) into L1/2(μ) implies its boundedness from the product of Lebesgue spaces Lp₁(μ)×Lp₂(μ) into Lp(μ) with 1<p₁,p₂<∞ and p satisfying 1/p=1/p₁+1/p₂.