Local functionals on C∞(G) and probability

A characterization of local functionals on C_∞(G), the space of real continuous functions with compact supports on a locally compact space G, is given. Such functionals were defined by Gel'fand and Vilenkin, as they occur in the analysis of generalized random processes with independent values. The preceding characterization is then used in a representation of the characteristic functionals of the above processes on C_∞(G), and results analogous to those of Gel'fand and Vilenkin on a Schwartz space are obtained. Since C_∞(G) is not nuclear, this study presents new problems and it largely complements the earlier work.     

Multipliers from L1(G) to a Lipschitz space

Let G be a metric locally compact Abelian group. We prove that the spaces (L₁, Lip(α, p)), (L₁, lip(α, p)), Lip(α, p) and lip(α, p)^~ are isometrically isomorphic, where Lip(α, p) and lip(α, p) denote the Lipschitz spaces defined on G, (L₁, A) is the space of multipliers from L₁ to A, and lip(α, p)^~ denotes the relative completion of lip(α, p). We also show that $\text{L}{}_1\text{1}\text{Lip}\text{(α, p) =}\text{lip}\text{(α, p) = L}{}_1\text{1}\text{lip}\text{(α, p)}$.     

Invariant means and invariant ideals in L∞(G) for a locally compact group G

For a nondiscrete σ-compact locally compact Hausdorff group G, L_∞(G) is a commutative Banach algebra under pointwise multiplication which has many nonzero proper closed invariant ideals; there is at least a continuum of maximal invariant ideals {N_α} such that N_α1 + N_α2 = L_∞(G) whenever α₁ ≠ α₂. It follows from the construction of these ideals that when G is also amenable as a discrete group, then LIM?TLIM contains at least a continuum of mutually singular elements each of which is singular to any element of TLIM. The supports of left-invariant means are in the maximal ideal space of L_∞(G); the structure of these supports leads to the notion of stationary and transitive maximal ideals. To prove that both these types of maximal ideals are dense among all maximal ideals, one shows that the intersection of all nonzero closed invariant ideals is zero. This is the case even though the intersection of any sequence of closed invariant ideals is not zero and the intersection of all the maximal invariant ideals is not zero.     

Projections onto translation—Invariant subspaces of L1(G)

Let G be a locally compact abelian group. A translation-invariant subspace in L₁(G) may or may not be complemented depending on the structure of its hull in ?. Techniques for deciding this complementation problem in a variety of situations are developed and illustrated with examples. A complete characterization is obtained for those ideals with a discrete hull.     

Uniqueness of norm on L(G) and C(G) when G is a compact group

Let G be a compact group. If the trivial representation of G is not weakly contained in the left regular representation of G on L₀²(G) and X is either L^p(G) for 1<p?∞ or C(G), then we show that every complete norm |·| on X that makes translations from (X,|·|) into itself continuous is equivalent to ||·||_p or ||·||_∞ respectively. If 1<p?∞ and every left invariant linear functional on L^p(G) is a constant multiple of the Haar integral, then we show that every complete norm |·| on L^p(G) that makes translations from (L^p(G),|·|) into itself continuous and that makes the map t?L_t from G into bounded is equivalent to ||·||_p.     

Weakly compact operators and multipliers on C0(S, A)

The problem of defining a convolution of operator-valued measures defined on a locally compact semitopological semigroup which represent weakly compact operators is considered in this paper. Convolution of the corresponding weakly compact operators is then defined. A characterization of multipliers on function spaces is given in terms of their representing measures.     

Regularizing effects for ut + Aϑ(u) = 0 in L1

Various initial-boundary value problems and Cauchy problems can be written in the form $\text{du}\text{dt}\text{+ A?(u) = 0}$, where ?:R→$\text{R}$ is nondecreasing and A is the linear generator of strongly continuous nonexpansive semigroup e^?tA in an L¹ space. For example, if A = ?Δ (subject, perhaps, to suitable boundary conditions) we obtain equations arising in flow in a porous medium or plasma physics (depending on the choice of ?) while if $\text{A =}\text{?}\text{?x}$ acting in L¹($\text{R}$) we have a scalar conservation law. In this paper we show that if M, m > 0 and m?′² ? ν??′' ? M?′², where ν ? {1,?1}, then (roughly speaking), the norm of $\text{t}\text{du}\text{dt}$ may be estimated in terms of the initial data u₀ in L¹. Such estimates give information about the regularity of solutions, asymptotic behaviour, etc., in applications. Side issues, such as the introduction of sufficiently regular approximate problems on which estimates can be made and the assignment of a precise meaning to the operator A?, are also dealt with. These considerations are of independent interest.     

Espaces Hp au dessus de l'espace hermitien hyperbolique de Cn (n > 1), II

Growth of harmonic functions and boundary values in a Sobolev space; comparison between the H^p space associated to Calderón Lusin's area integral and the H^p space associated to the maximal admissible function; study of Littlewood-Paley functions for the harmonic functions for the invariant laplacian of the hermitian hyperbolic space; relation between a Littlewood-Paley function and the bounded mean oscillation functions; study of H¹ and H¹ B.M.O. duality.     

Lp estimates for strongly singular integrals on spaces of homogeneous type

A theory for strongly singular integrals on spaces of homogeneous type is developed and used to obtain sharp multiplier theorems for various groups.     

A note on Banach spaces containing c0 or C∞

It is shown that every separable Banach space X containing a subspace isomorphic to c₀ has a subspace Y with basis such that $\text{X}\text{Y}\text{~}\text{c}{}_0\text{C}{}_{\mathrm{\infty }}$ and the latter space has a shrinking basis and an unconditional FDD. Moreover, it is shown that X ⊕ C_∞ has a basis if X has the bounded approximation property.     

On the Lp-Theory of C0-Semigroups Associated with Second-Order Elliptic Operators, II

We study positive C₀-semigroups on L_p associated with second-order uniformly elliptic divergence-type operators with singular lower-order terms, subject to a wide class of boundary conditions. We obtain an interval (p_min, p_max) in the L_p-scale where these semigroups can be defined, including the case 2∉(p_min, p_max). We present an example showing that the result is optimal. We also show that the semigroups are analytic with angles of analyticity and spectra of the generators independent of p, for the whole range of p where the semigroups are defined.     

Kernel density estimation for spatial processes: the L1 theory

The purpose of this paper is to investigate kernel density estimators for spatial processes with linear or nonlinear structures. Sufficient conditions for such estimators to converge in L₁ are obtained under extremely general, verifiable conditions. The results hold for mixing as well as for nonmixing processes. Potential applications include testing for spatial interaction, the spatial analysis of causality structures, the definition of leading/lagging sites, the construction of clusters of comoving sites, etc.     

On the Lp-Theory of C0-Semigroups Associated with Second-Order Elliptic Operators, I

We study L_p-theory of second-order elliptic divergence-type operators with measurable coefficients. To this end, we introduce a new method of constructing positive C₀-semigroups on L_p associated with sesquilinear (not necessarily sectorial) forms in L₂. A precise condition ensuring that the elliptic operator is associated with a quasi-contractive C₀-semigroup on L_p is established.     

Subdirect sums of P(P0)-matrices and totally nonnegative matrices

In this paper, the problem of when the sub-direct sum of two strictly diagonally dominant P-matrices is a strictly diagonally dominant P-matrix is studied. In particular, it is shown that the subdirect sum of overlapping principal submatrices of strictly diagonally dominant P-matrices is a strictly diagonally dominant P-matrix. It is also established that the 2-subdirect sum of two totally nonnegative matrices is a totally nonnegative matrix under some conditions. It is obtained that a partial totally nonnegative matrix, whose graph of the specified entries is a monotonically labeled 2-chordal graph, has a totally nonnegative completion. Finally, a positive answer to the question (IV) in Fallat and Johnson [Shaun M. Fallat, C.R. Johnson, J.R. Torregrosa, A.M. Urbano, P-matrix completions under weak symmetry assumptions, Linear Algebra Appl. 312 (2000) 73-91] is given for P₀-matrices.