Distributional Chébli-Trimèche transforms

In this paper we investigate the distributional Chébli-Trimèche transforms. We use the so-called kernel method and we are inspired by the papers of Dube and Pandey [L.S. Dube, J.N. Pandey, On the Hankel transform of distributions, Tôhoku Math. J. 27 (1975) 337-354] and Koh and Zemanian [E.L. Koh, A.H. Zemanian, The complex Hankel and I-transformations of generalized functions, SIAM J. Appl. Math. 16 (1968) 945-957] about distributional Hankel transforms. We note that our procedure, supported in a representation of the elements in the corresponding dual spaces, is simpler than the methods described in the above mentioned papers. Some applications of our distributional theory are presented.     

A local two radii theorem on the Chébli-Trimèche hypergroup

Let 0<r₁<r₂<r₁+r₂<R. It is given a necessary and sufficient condition so that the null function is the unique solution f∈C^∞(]−R,+R[) of the system

(1)     

Lambert's W, infinite divisibility and Poisson mixtures

The Lambert W function is shown to be the Laplace exponent of a positive infinitely divisible law (i.e. W is a Bernstein function) called the standard Lambert law. This law is a generalized gamma convolution. At least three Poisson mixture families are defined in terms of W. One of these is the generalized Poisson laws which are shown to be generalized negative-binomial convolutions. Mixing with positive stable laws yields further generalizations.     

A W-transform-based criterion for the existence of bounded extensions of E-operators

Let Γ(H) be the symmetric Fock space over a Hilbert space H and ε:H→Γ(H) the exponential mapping. By an E-operator we mean an operator defined on ε(H). For an E-operator A, the composition mapping Φ=A°ε is called its W-transform. In this paper, we obtain a criterion based on the W-transform for checking whether or not an E-operator becomes a bounded linear operator on the Fock space.     

Integral representation of the W-weighted Drazin inverse for Hilbert space operators

This paper studies the integral representation of the W-weighted Drazin inverse for bounded linear operators between Hilbert spaces. By using operator matrix blocks, some integral representations of the W-weighted Drazin inverse for Hilbert space operators are established.     

Generalized composition operators from F(p, q, s) spaces to Bloch-type spaces

Let H(B) denote the space of all holomorphic functions on the unit ball B of Cⁿ. Let φ be a holomorphic self-map of B and g ∈ H(B) such that g(0) = 0. In this paper, we investigate the boundedness and compactness of the generalized composition operator

     

Self-improving properties of inequalities of Poincaré type on measure spaces and applications

We show that the self-improving nature of Poincaré estimates persists for domains in rather general measure spaces. We consider both weak type and strong type inequalities, extending techniques of B. Franchi, C. Pérez and R. Wheeden. As an application in spaces of homogeneous type, we derive global Poincaré estimates for a class of domains with rough boundaries that we call ?-John domains, and we show that such domains have the requisite properties. This class includes John (or Boman) domains as well as s-John domains. Further applications appear in a companion paper.     

Polynomial sequential continuity on C(K,E) spaces

We show that, for bounded sequences in C(K,E), the polynomial sequential convergence is not equivalent to the pointwise polynomial sequential convergence. We introduce several conditions on E under which different versions of the result are true when K is a scattered compact space. These conditions are related with some others appeared in the literature and they seem to be of independent interest.     

Generalized difference sequence spaces and their dual spaces

The definition of the pα-, pβ- and pγ-duals of a sequence space was defined by Et [Internat. J. Math. Math. Sci. 24 (2000) 785-791]. In this paper we compute pα- and N-duals of the sequence spaces Δ^m_v(X) for X=?_∞, c and c₀, and compute β- and γ-duals of the sequence spaces Δ^m_v(X) for X=?_∞, c and c₀.     

Weak barrelledness for C(X) spaces

Buchwalter and Schmets reconciled C_c(X) and C_p(X) spaces with most of the weak barrelledness conditions of 1973, but could not determine if -barrelled ⇔ ?^∞-barrelled for C_c(X). The areas grew apart. Full reconciliation with the fourteen conditions adopted by Saxon and Sánchez Ruiz needs their 1997 characterization of Ruess' property (L), which allows us to reduce the C_c(X) problem to its 1973 status and solve it by carefully translating the topology of Kunen (1980) and van Mill (1982) to find the example that eluded Buchwalter and Schmets. The more tractable C_p(X) readily partitions the conditions into just two equivalence classes, the same as for metrizable locally convex spaces, instead of the five required for C_c(X) spaces. Our paper elicits others, soon to appear, that analytically characterize when the Tychonov space X is pseudocompact, or Warner bounded, or when C_c(X) is a df-space (Jarchow's 1981 question).     

A characterization of finite symplectic polar spaces of odd prime order

A sufficient condition for the representation group for a nonabelian representation (Definition 1.1) of a finite partial linear space to be a finite p-group is given (Theorem 2.9). We characterize finite symplectic polar spaces of rank r at least two and of odd prime order p as the only finite polar spaces of rank at least two and of prime order admitting nonabelian representations. The representation group of such a polar space is an extraspecial p-group of order p1+2r and of exponent p (Theorems 1.5 and 1.6).     

Sums of commutators in non-commutative Banach function spaces

Let M be a II_∞-factor and denote by τ its normal faithful semi-finite trace. For any rearrangement invariant Köthe function space X on [0,+∞[, let X(M,τ) be the associated non-commutative Banach function space. This paper is concerned with ideals in M of the form I_X(M,τ)=M∩X(M,τ) that are contained in L^p(M,τ) for some p>0. It is proved that an element T in I_X(M,τ) is a finite sum of commutators of the form [A,B] with A∈I_X(M,τ) and B∈M if and only if the function belongs to X, where ν_T is the Brown spectral measure of T and t→λ_t(T) is the non-increasing rearrangement of the function λ→|λ| with respect to ν_T. This extends to general Banach function spaces a result obtained by Kalton for quasi-Banach ideals of compact operators and implies that the Dixmier's trace of a quasi-nilpotent element in L^1,∞(M,τ) is always zero.     

On p-approximation properties for p-operator spaces

This paper has a two-fold purpose. Let 1<p<∞. We first introduce the p-operator space injective tensor product and study various properties related to this tensor product, including the p-operator space approximation property, for p-operator spaces on Lp-spaces. We then apply these properties to the study of the pseudofunction algebra PFp(G), the pseudomeasure algebra PMp(G), and the Figà-Talamanca-Herz algebra Ap(G) of a locally compact group G. We show that if G is a discrete group, then most of approximation properties for the reduced group C^∗-algebra , the group von Neumann algebra VN(G), and the Fourier algebra A(G) (related to amenability, weak amenability, and approximation property of G) have the natural p-analogues for PFp(G), PMp(G), and Ap(G), respectively. The p-completely bounded multiplier algebra McbAp(G) plays an important role in this work.     

Some embedding theorems for Hörmander-Beurling spaces

In this paper we prove a number of results on sequence space representations and embedding theorems of Hörmander-Beurling spaces. As a consequence and using sharp results of Meise, Taylor and Vogt, a result of Kaballo on short sequences and hypoelliptic operators is extended to ω-hypoelliptic differential operators and to the vector-valued setting.     

Explicit characterizations of orthogonality in lS(C)

In a previous paper, the author used a notion of orthogonality introduced in another article to establish characterizations for orthogonality in the spaces l_S^p(C), 1?p<∞, thus obtaining generalizations of the usual characterization of orthogonality in the Hilbert spaces l_S²(C), via inner products. In this paper we make explicit these characterizations for some of the spaces l_S^p(C). We finish by presenting some remarks and open problems.     

Irreducibility and p-adic monodromies on the Siegel moduli spaces

We generalize the surjectivity result of the p-adic monodromy for the ordinary locus of a Siegel moduli space by Faltings and Chai (independently by Ekedahl) to that for any p-rank stratum. We discuss irreducibility and connectedness of some p-rank strata of the moduli spaces with parahoric level structure. Finer results are obtained on the Siegel 3-fold with Iwahori level structure.     

Herz spaces and restricted summability of Fourier transforms and Fourier series

A general summability method, the so-called θ-summability is considered for multi-dimensional Fourier transforms and Fourier series. A new inequality for the Hardy-Littlewood maximal function is verified. It is proved that if the Fourier transform of θ is in a Herz space, then the restricted maximal operator of the θ-means of a distribution is of weak type (1,1), provided that the supremum in the maximal operator is taken over a cone-like set. From this it follows that over a cone-like set a.e. for all f∈L₁(Rd). Moreover, converges to f(x) over a cone-like set at each Lebesgue point of f∈L₁(Rd) if and only if the Fourier transform of θ is in a suitable Herz space. These theorems are extended to Wiener amalgam spaces as well. The Riesz and Weierstrass summations are investigated as special cases of the θ-summation.     

Weighted differentiation composition operators from H and Bloch spaces to nth weighted-type spaces on the unit disk

We characterize the boundedness and compactness of weighted differentiation composition operators from the space of bounded analytic functions, the Bloch space and the little Bloch space to nth weighted-type spaces on the unit disk.