On order structure and operators in L ∞(μ)

It is known that there is a continuous linear functional on L _∞ which is not narrow. On the other hand, every order-to-norm continuous AM-compact operator from L _∞(μ) to a Banach space is narrow. We study order-to-norm continuous operators acting from L _∞(μ) with a finite atomless measure μ to a Banach space. One of our main results asserts that every order-to-norm continuous operator from L _∞(μ) to c ₀(Γ) is narrow while not every such an operator is AM-compact.     

(L p –L q )-boundedness of pseudodifferential operators on the n-dimensional torus

Mathematical Notes -     

Bellman function for the estimates of Littlewood–Paley type and asymptotic estimates in the p−1 problem

We utilize the method of Bellman functions to derive new $L^p$-estimates of Littlewood–Paley type involving $p?1$. Among the applications to singular integrals we improve the $2(p?1)$ bounds for the Ahlfors–Beurling operator on $L^p(\mathbb{C})$ when $p\to \infty$. In addition, dimensionless estimates of Riesz transforms in the classical as well as in the Ornstein–Uhlenbeck setting are attained. To cite this article: O. Dragi?evi?, A. Volberg, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

On the asymptotic optimality of the cμ-rule in queueing networks

Although the cμ-rule is proven to be exactly or asymptotically optimal in various parallel queueing networks, its performance in non-parallel queueing networks is relatively unexplored. We study the performance of the cμ-rule (that is, the performance of implementing the cμ-rule at all servers) in non-parallel queueing networks. We numerically show that the cμ-rule can perform poorly even in a simple tandem queue with two job types. We derive conditions under which the cμ-rule is asymptotically optimal in diffusion scale.     

On lower and on sharp asymptotic estimates of solutions of Emden–Fowler-type equations

It is shown that, for the Lennard-Jones potential, there exist far-range van-der-Waals forces. It is claimed that such potentials occur rarely, just as potentials for the Schrödinger equation whose semiclassical solutions coincide with exact solutions.     

On basicity of exponentials inL P (dμ) and general prediction problems

Let be a nonnegative measure on the unit circle in the complex plane and 1<p<. It is of interest to find conditions on so that the set of exponentialse ⁱⁿ form a strongM-basis forL ^p (d). Some partial results are proved which can shed some light on this important open question. These results are of fundamental importance in the prediction theory of stochastic processes and other fields of applications. These results is then used to obtain a theorem which reduces some prediction problems to easier ones.To 80^th birthday of Paul ErdsThis research is supported by Office of Naval Research Grant No N00014-89-J-1824.     

L p -error estimates for positive linear operators using the second-order τ-modulus

ПустьМ[а, b] — множество вещественных функци йf, измеримых и ограниче нных на отрезке [а, b], иL: M[a, b] → M[a, b] — лин ейный положительный оператор. В статье пол учены оценки погрешностей аппроксимации вL _p -нор ме, выраженные в термина х так называемого усредненного локаль ного модуля гладкост и, илиτ-модуля. ПустьL облад ает свойствами: (e _i ,i(t):=t ⁱ дляi=0,1,2)Le ₀ =e ₀ ,Le ₁ =e ₁ ,Le ₂ (х)=х ² +β(х), x∈[a,b] и i)A ≦ min (1, (b-a)²/4) для 1 <p<∞, ii)A ≦ min (e ^?1 , (b — a)²) дляр=1 приA:=∥β∥_t8. Тогда выполнены оцен ки: $$\begin{gathered} \left\| {f - Lf} \right\|_p \leqq C\frac{p}{{p - 1}}\tau _2 (f;\sqrt A )_p 1< p< \infty \hfill \\ \left\| {f - Lf} \right\|_1 \leqq C\tau _2 \left( {f;\sqrt {A\ln \frac{1}{A}} } \right)_1 , \hfill \\ \end{gathered} $$ где положительные по стоянныеС не завися т от оператораL, функцииfи параметрар.     

On the asymptotic stability ofθ-methods for delay differential equations

Stability regions of -methods for the linear delay differential test equations

On the finiteness of the Morse index for Schr?dinger operators

We show that if M is a complete Riemannian manifold and H = Δ + V is a Schr?dinger operator, then the existence of a positive solution of Hu?=?0 outside a compact set is equivalent to the finiteness of the Morse index of H.     

On spectral estimates for Schrödinger-type operators: The case of small local dimension

The behavior of the discrete spectrum of the Schrödinger operator - Δ -V is determined to a large extent by the behavior of the corresponding heat kernel P(t; x,y) as t → 0 and t→ ∞. If this behavior is power-like, i.e.,

$\left\| {P(t; \cdot , \cdot )} \right\|_{L^\infty } = O(t^{ - \delta /2} ),t \to 0,\left\| {P(t; \cdot , \cdot )} \right\|_{L^\infty } = O(t^{ - D/2} ),t \to \infty ,$

then it is natural to call the exponents δ and D the local dimension and the dimension at infinity, respectively. The character of spectral estimates depends on a relation between these dimensions. The case where δ < D, which has been insufficiently studied, is analyzed. Applications to operators on combinatorial and metric graphs are considered.

     

On the martingale problem and Feller and strong Feller properties for weakly coupled Lévy type operators

This paper considers the martingale problem for a class of weakly coupled Lévy type operators. It is shown that under some mild conditions, the martingale problem is well-posed and uniquely determines a strong Markov process $(X,\Lambda )$. The process $(X,\Lambda )$, called a regime-switching jump diffusion with Lévy type jumps, is further shown to possess Feller and strong Feller properties under non-Lipschitz conditions via the coupling method.     

On complemented subspaces of the spaces (l p ) N ∩l q (l q )

Summary In this paper we study the complemented subspaces of the spaces (l^p)^N∩l^q(l^q), with 1≤p<q≤∞ or q=0, thereby showing that if (l^p)^N∩l^q(l^q)=F⊕G then either F or G contains a complemented copy of the whole space. Entrata in Redazione il 19 aprile 1999. Ricevuta versione finale il 4 febbraio 2000. Research supported by the Italian MURST.     

Lp-gradient estimates for the commutators of the Kato square roots of second-order elliptic operators on ℝn

Let L=-div(A▽) be a second-order divergent-form elliptic operator,where A is an accretive n×n matrix with bounded and measurable complex coefficients on Herein,we prove that the commutator [b,L1/2]of the Kato square root L1/2 and b with ▽b∈Ln(Rn)(n> 2),is bounded from the homogenous Sobolev space L1p(Rn) to Lp(Rn)(p-(L)