The nonnegative weight function pairs u, v for which the operator maps the nonnegative nonincreasing functions in LP(v) boundedly into weak Lq(u) are characterized. This result is used, in particular, both to generalize and to provide an alternate proof of certain strong type inequalities recently obtained by Ariño and Muckenhouptfor the Hardy averaging operator restricted to nonnegative nonincreasing functions.

Generalized (distribution-valued) Ornstein-Uhlenbeck processes, which by definition are solutions of generalized Langevin equations, arise in many investigations on fluctuation limits of particle systems (eg. Bojdecki and Gorostiza [1], Dawson, Fleischmann and Gorostiza [5], Fernández [7], Gorostiza [8,9], Holley and Stroock [10], Itô [12], Kallianpur and Pérez-Abreu [16], Kallianpur and Wolpert [14], Kotelenez [17], Martin-Löf [19], Mitoma [22], Uchiyama [25]). The state space for such a process is the strong dual Φ′ of a nuclear space Φ. A generalized Langevin equation for a Φ′-valued process X ≡ {Xt} is a stochastic evolution equation of the form

where {At} is a family of linear operators on Φ and Z ≡ {Zt} is a Φ'-valued semimartingale (in some sense) with independent increments. Equations of the type (1.1) where Z does not have independent increments also arise in applications (eg. [9,14,20]) but here we are interested precisely in the case when Z has independent increments (we restrict the term generalized Langevin equation to this case in accordance with the classical Langevin equation).

Let F be a skew field with a valuation (also called total) subring B, i.e. x in F\ B implies x-1 in B. Such rings are useful not only in the investigation and construction of division algebras (see for example [5],[6],[12]) but also in geometry ([15]).

Associated with B is an invariant subring R of F and a value group G. We investigate the relationship between properties like the distributivity of R and properties like being lattice ordered of G.

For a vector solution u(x, t) with finite energy of the Navier Stokes equations with body forces and boundary values on a region Ω ⊆ R3 for t > 0, conditions are established on the L6/5(Ω) and L2(Ω) norms of derivatives of the data that ensure the estimates and max , up to any given integer value of the weighted order 2r+s, where r or s = s1 + s2 + s3 > 0 and 0 < T < ∞.

Root systems and Coxeter groups are important tools in multivariable analysis. This paper is concerned with differential-difference and integral operators, and orthogonality structures for polynomials associated to Coxeter groups. For each such group, the structures allow as many parameters as the number of conjugacy classes of reflections. The classical orthogonal polynomials of Gegenbauer and Jacobi type appear in this theory as two-dimensional cases. For each Coxeter group and admissible choice of parameters there is a structure analogous to spherical harmonics which relies on the connection between a Laplacian operator and orthogonality on the unit sphere with respect to a group-invariant measure. The theory has been developed in several papers of the author [4,5,6,7]. In this paper, the emphasis is on the study of an intertwining operator which allows the transfer of certain results about ordinary harmonic polynomials to those associated to Coxeter groups. In particular, a formula and a bound are obtained for the Poisson kernel.

Necessary multiplier conditions for Laguerre expansions are derived and discussed within the framework of weighted Lebesgue spaces.

For any group K and finite-dimensional (right) K-module V let be the right regular representation of K on the algebra of polynomial functions on V. An Isotypic Component of is the sum of all k-submodules of on which π restricts to an irreducible representation can then be written as f = ΣƬ ƒƬ with ƒƬ in .

Explicit solutions to the recurrence relation for associated continuous Hahn polynomials are derived using 3F2 contiguous relations. These solutions are used to obtain a new continued fraction and the associated absolutely continuous measure. An exceptional case is shown to yield entry 33 in Chapter 12 of Ramanujan's second notebook.

Finite extensions of complex commutative Banach algebras are naturally related to corresponding finite covering maps between the carrier spaces for the algebras. In the case of function rings, the finite extensions are induced by the corresponding finite covering maps, and the topological properties of the coverings are strongly reflected in the algebraic properties of the extensions and conversely. Of particular interest to us is the class of finite covering maps for which the induced extensions of function rings admit primitive generators. This is exactly the class of polynomial covering maps and the extensions are algebraic extensions defined by the underlying Weierstrass polynomials.

The purpose of this paper is to develop a suitable Galois theory for finite extensions of function rings induced by finite covering maps and to apply it in the case of Weierstrass polynomials and polynomial covering maps.

We give complex weight functions with respect to which the Jacobi, Laguerre, little q-Jacobi and Askey-Wilson polynomials are orthogonal. The complex functions obtained are weight functions in a wider range of parameters than the real weight functions. They also provide an alternative to the recent distributional weight functions of Morton and Krall, and the more recent hyperfunction weight functions of Kim.

We consider the positive zeros j″vk, k = 1, 2,…, of the second derivative of the Bessel function Jν(x). We are interested first in how many zeros there are on the interval (0,jν1), where jν1 is the smallest positive zero of Jν(x). We show that there exists a number ƛ = —0.19937078… such that and . Moreover, j″v1 decreases to 0 and j″ν2 increases to j″01 as ν increases from ƛ to 0. Further, j″vk increases in —1 < ν< ∞, for k = 3,4,… Monotonicity properties are established also for ordinates, and the slopes at the ordinates, of the points of inflection when — 1 < ν < 0.

We continue to develop the theory of previous papers concerning transforms corresponding to Mellin multipliers which involve products and/or quotients of Γ-functions. We show that, by working with certain subspaces of Lp,μ consisting of smooth functions, we can simplify considerably the restrictions on the parameters which were necessary in the Lp,μ setting. As a result, operators in our class become homeomorphisms on these subspaces under conditions of great generality.