The new class of weights called A p weights is introduced. We prove that a characterization and an unconditional basis of the weighted Lp space Lp(Rn,w(x)dx) with w ∈ A p (1 < p < ∞) are given by the Haar wavelets and the Haar scaling function. As an application of these results, we establish a greedy basis by using the Haar wavelets and the Haar scaling function again.

This paper presents a personal review of the use of Linear Programming (LP) techniques in Military Operational Analysis studies over the period 1967 – 2008. Applications described include Logistics Modelling, Reinforcement and Deployment modelling, Air Defence Balance of Investment (BoI) Analysis, and Cross-capability BoI Studies. In particular, it emphasises the use of LP to explore the soluti...

We give a complete characterization of integral functionals which are Lipschitzian on a Lebesgue space Lp with p finite. When the measure is atomless, we characterize the integral functionals which are locally Lipschitzian on such Lebesgue spaces Lp. In every cases, the Lipchitzian properties of the integral functional are described by Lipschitzian properties of the integrand. Mathematics Subje...

Increasing sequences of contractive projections on a reflexive LP space share an unconditionality property similar to that exhibited sequences of self-adjoint projections on a Hilbert space. A slight variation of this property is shown to be precisely the correct condition on a reflexive Banach space to ensure that every operator with a contractive AC-functional calculus is scalar-type spectraL

In this paper, we consider the motion of incompressible magnetohydrodynamics (MHD) with resistivity in a domain bounded by free surface. An electromagnetic field generated some currents an external keeps MHD flow domain. On surface, boundary conditions for and transmission fields are imposed. We proved local well-posedness general setting domains from mathematical point view. The solutions obta...

A metric space X has Markov type 2, if for any reversible finite-state Markov chain {Zt} (with Z0 chosen according to the stationary distribution) and any map f from the state space to X, the distance Dt from f(Z0) to f(Zt) satisfies E(D t ) ≤ K tE(D 1) for some K = K(X) < ∞. This notion is due to K. Ball (1992), who showed its importance for the Lipschitz extension problem. However until now, ...

Let (Ω,Σ) be a measurable space and let X be a Banach space. Throughout this paper G will be a countably additive vector measure G : Σ → X. Consider the space L1(G) of (classes of real) G-integrable functions, following the definition of Bartle, Dunford and Schwartz [1] and Lewis [9]. The properties of this space have been studied by Kluvánek and Knowles [7], Okada [10] and Curbera [2]. In the ...

We investigate the structure of shift-invariant spaces generated by a finite number of compactly supported functions in Lp(R) (1 ≤ p ≤ ∞). Based on a study of linear independence of the shifts of the generators, we characterize such shift-invariant spaces in terms of the semi-convolutions of the generators with sequences on Z. Moreover, we show that such a shiftinvariant space provides Lp-appro...

We consider an abstract non-negative self-adjoint operator H on an L-space. We derive a characterization for the restriction estimate ‖dEH(λ)‖Lp→Lp′ ≤ Cλ d 2 ( 1 p − 1 p )−1 in terms of higher order derivatives of the semigroup e . We provide an alternative proof of a result in [1] which asserts that dispersive estimates imply restriction estimates. We also prove L − L ′ estimates for the deriv...

A Banach space X has the average distance property (ADP) if there exists a unique real number r such that for each positive integer n and all x1, . . . , xn in the unit sphere of X there is some x in the unit sphere of X such that 1 n n ∑ k=1 ‖xk − x‖ = r. We show that lp does not have the average distance property if p > 2. This completes the study of the ADP for lp spaces.