We consider the Riesz transforms ∇L−1/2, where L ≡ −divA(x)∇, and A is an accretive, n× n matrix with bounded measurable complex entries, defined on Rn. We establish boundedness of these operators on Lp(Rn), for the range pn < p ≤ 2, where pn = 2n/(n + 2), n ≥ 2, and we obtain a weak-type estimate at the endpoint pn. The case p = 2 was already known: it is equivalent to the solution of the squa...

Example 1 (Spectrum of Multiplication Operators) Let • (X, M, µ) be a σ–finite (actually semifinite will do) measure space, • 1 ≤ p ≤ ∞ and • a : X → C be a bounded measurable function on X. (Aϕ)(x) = a(x)ϕ(x) Point spectrum: Let λ ∈ C. Then λ1l − A is injective ⇐⇒ ϕ ∈ L p (X, M, µ), (λ − a(x))ϕ(x) = 0 a.e. =⇒ ϕ(x) = 0 a.e. ⇐⇒ λ − a(x) = 0 a.e.

Córdoba–Fefferman collections are defined and used to characterize functions whose corresponding maximal functions are locally integrable. Córdoba–Fefferman collections are also used to show that, if Mx and My respectively denote the one-dimensional Hardy–Littlewood maximal operators in the horizontal and vertical directions in R, MHL denotes the standard Hardy–Littlewood maximal operator in R,...

In this paper, we provide a notion of structure preserving maps (i.e. knowledge-belief morphisms) between knowledge-belief spaces. Then we show that under the condition that the knowledge operators of the players in a knowledge-belief space operate just on measurable subsets of the space there is a unique (up to isomorphism) universal knowledge-belief space to which every knowledge-belief space...

This paper settles the existence question for a rather general class of convex optimal design problems with a volume constraint. In low dimensions, we prove existence of an optimal configuration for general convex minimization problems ruled by bounded measurable degenerate elliptic operators. Under a mild continuity assumption on the medium, the free boundary is proven to have an appropriate w...

In this article, we will consider second order uniformly elliptic operators of divergence form defined on Rn with measurable coefficients. Mainly, we will give estimates on the dimension of space of solutions that grow at most polynomially of degree d. More precisely, in terms of a rectangular coordinate system {x1, . . . , xn}, a second order uniformly elliptic operator of divergence form, L, ...

This paper is a study of derivations on unbounded operator algebras in connection with those in operator algebras. In particular we study spatiality of derivations in several situations. We give the characterization of derivations on general «-algebras by using positive linear functionals. We also show that a derivation with some range-property on a left ■EW*-algebra induced by an unbounded Hub...

We find a Gaussian off-diagonal heat kernel estimate for uniformly elliptic operators with measurable coefficients acting on regions Ω ⊆ R , where the order 2m of the operator satisfies N < 2m. The estimate is expressed using certain Riemannian-type metrics, and a geometrical result is established allowing conversion of the estimate into terms of the usual Riemannian metric on Ω. Work of Barbat...

We introduce and analyze expected uncertain utility theory (EUU). A prior and an interval utility characterize an EUU decision maker. The decision maker uses her subjective prior to transform each uncertain prospect f into an interval-valued prospect f which assigns an interval [x, y] of prizes to each state. The decision maker ranks prospects according to their expected interval utilities E(u(...

Regular operators on L p-spaces are characterised in terms of an operator bound which is associated with certain generalisations of the Feynman-Kac formula. of multiplication by the characteristic functions of elements of the-algebra S, so that if f is a bounded S-measurable function, Q(f) = R f dQ is the operator of multiplication by f. When is it true that we can nd a positive number C such t...