On the Asymptotically Optimal Rate of Approximation of Multiplier Operators from L p into L q

The asymptotic behavior of the n -widths of multiplier operators from L _p [0,1] into L _q [0,1] is studied. General upper and lower bounds for the n -widths in terms of the multipliers are established. Moreover, it is shown that these upper and lower bounds coincide for some important concrete examples. August 3, 1994. Date revised: November 15, 1996.     

双线性Fourier乘子在Triebel-Lizorkin和Besov空间的有界性

本文利用Littlewood-Paley分解,Fourier变换和逆变换等方法,研究了双线性Fourier乘子在非齐次正光滑性Triebel-Lizorkin空间和Besov空间的有界性.     

Some Estimates for Radial Fourier Multiplier Operators with Slowly Decaying Kernels

We consider the restriction to radial functions of a class of radial Fourier multiplier operators containing the Bochner-Riesz multiplier operator. The convolution kernel K(x) of an operator in this class decays too slowly at infinity to be integrable, but has enough oscillation to achieve L^p -boundedness for p inside a suitable interval (a, b). We prove boundedness results for the maximal operator Kf(x) = sup_{r>0 r}ⁿ∣K(r) * f(x)∣ associated with such a kernel. The maximal operator is shown to be weak type bounded at the lower critical index a, restricted weak type bounded at the upper critical index b, and strong type bounded between. This together with our assumptions on K(x) leads to the pointwise convergence result lim_γ→ γⁿ K(γ·) * f(x) = cf(x) a. e. for radial f ? L^P(?ⁿ), a ≥ p > b.     

COMMUTATORS OF MULTIPLIER OPERATORS

Denote M~l={ω∈C~∞(R~K\{0}:|ω~((β))(ξ)|≤C_β|ξ|~(l-|β|)},l is an integer.R_((-α))~((m))is the n-foldcomposition of Taylor series remainder operator,m=(m_1,…,m_n)∈Z~n.Z is the set ofnon-negative integers,α∈(R~K)n.DenoteThe main results are as follows:(i) If γ_1,γ_2∈Z~K and l is an integer such that |γ_1|+|γ_2|+l=|m|=m_1+…+m_n,0≤|γ_1|≤{m_4},and ω∈M~l,then we havewhereis a conseant.(ii)In the same sense of notation as in (i),but now|m|=1,we havewhereThese results extend the corresponding ones given by coifman-Meyer in [4] andCohen,J.in [2],and,in a sense,extend those given by Calderón,A.P.in [1].     

Derived space of L p (G, H)

Let G denote a locally compact abelian group and H a separable Hilbert space. Let L _p (G, H), 1 ≤ p < ∞, be the space of H-valued measurable functions which are in the usual L _p space. Motivated by the work of Helgason [1], Figa-Talamanca [11] and Bachelis [2, 3], we have defined the derived space of the Banach space L _p (G, H) and have studied its properties. Similar to the scalar case, we prove that if G is a noncompact, locally compact abelian group, then L _p ⁰ (G, H) = {0} holds for 1 ≤ p < 2. Let G be a compact abelian group and Γ be its dual group. Let S _p (G, H) be the L ₁(G) Banach module of functions in L _p (G, H) having unconditionally convergent Fourier series in L _p -norm. We show that S _p (G, H) coincides with the derived space L _p ⁰ (G, H), as in the scalar valued case. We also show that if G is compact and abelian, then L _p ⁰ (G, H) = L ₂(G, H) holds for 1 ≤ p ≤ 2. Thus, if F ∈ L _p (G, H), 1 ≤ p < 2 and F has an unconditionally convergent Fourier series in L _p -norm, then F ∈ L ₂(G, H). Let Ω be the set of all functions on Γ taking only the values 1, ?1 and Ω* be the set of all complex-valued functions on Γ having absolute value 1. As an application of the derived space L _p ⁰ (G, H), we prove the following main result of this paper. Let G be a compact abelian group and F be an H-valued function on the dual group Γ such that $$ \sum \omega (\gamma )F(\gamma )\gamma $$ is a Fourier-Stieltjes series of some measure µ ∈ M(G, H) for every scalar function ω such that |ω(γ)| = 1. Then F ∈ l ²(Γ, H).     

On Measuring the Efficiency of Kernel Operators in L p (R d )

It is well known that it is possible to enhance the approximation properties of a kernel operator by increasing its support size. There is an obvious tradeoff between higher approximation order of a kernel and the complexity of algorithms that employ it. A question is then asked: how do we compare the efficiency of kernels with comparable support size? We follow Blu and Unser and choose as a measure of the efficiency of the kernels the first leading constant in a certain error expansion. We use time domain methods to treat the case of globally supported kernels in L _p (R ^d), 1≤p≤∞.

     

Price Operators Analysis in L p -Spaces

An integral type representation and various extension theorems for monotone linear operators in L _p -spaces are considered in relation to market price modelling. As application, a characterization of the existence of a risk-neutral probability measure equivalent to the applied underlying one is provided in terms of the given prices. These results are in the line of the fundamental theorem of asset pricing. Here, in particular, the risk-neutral probability measure considered has the advantage of having its density laying in pre-considered upper and lower bounds.     

Asymptotic Behaviour of Iterates of Volterra Operators on L p (0, 1)

Given k ∈ L¹ (0,1) satisfying certain smoothness and growth conditions at 0, we consider the Volterra convolution operator V_k defined on L^p (0,1) by

关于(s,p)-w-亚正常算子类

本文在W-亚正常算子类的基础上，引入（s，p）-w-亚正常算子类，进而讨论了该类算子的特征，包含关系，对一类特殊的该类算子还考虑了其平方性质．     

Optimal Weak Type Estimates for p -Adic Hardy Operators

The present article investigates the weak boundedness of fractional p-adic Hardy operators and their adjoints on Lebesgue spaces. It also furnishes the corresponding operator norms for such operators on these spaces.     

Weighted Composition Lambert-Type Operators on L p Spaces

In this paper boundedness of a weighted composition Lambert-type operator T = M _w EM _u C _φ acting between two different L ^p (Σ) spaces is characterized using some properties of conditional expectation operator. Moreover, we establish criteria for hyponormality for these types of operators on L ²(Σ).     

Global L 2-Boundedness Theorems for a Class of Fourier Integral Operators

We prove sharp interpolatory estimates between directional Haar projections and Riesz Transforms. We apply those to prove a conjecture of L. Tartar that arose within the theory of compensated compactness. To provide a frame of reference for the analytical estimates addressed in this paper we briefly review the core ideas of compensated compactness as developed by Murat and Tartar.     

M p(G)≠M q(G) (p −1+q −=1)

There exists a compact groupG havingM _4/3(G)≠M ₄(G). This answers in the negative (the dual reformulation of) a question of Eymard (Séminaire Bourbaki, 1969/70).     

The (L) summability of double Fourier series

Summary In this note the author applies the method of (L) summability to the double Fourier series to obtain a summability criterion for it.     

L p Spectral Independence of Elliptic Operators via Commutator Estimates

Let {T _p:q ₁ p q ₂} be a family of consistent C ₀ semigroups on L ^p(), with q ₁,q ₂ [1,) and open. We show that certain commutator conditions on T _p and on the resolvent of its generator A _p ensure the p independence of the spectrum of A _p for p [q ₁,q ₂.Applications include the case of Petrovskij correct systems with Hölder continuous coefficients, Schrödinger operators, and certain elliptic operators in divergence form with real, but not necessarily symmetric, or complex coefficients.     

Weighted Compact Commutator of Bilinear Fourier Multiplier Operator

Let T_σ be the bilinear Fourier multiplier operator with associated multiplier σ satisfying the Sobolev regularity that sup κ∈Z∥σ_κ∥W~s(R~(2n)) ∞ for some s ∈ (n, 2n]. In this paper, it is proved that the commutator generated by T_σ and CMO(R~n) functions is a compact operator from L~(p1)(R~n, w_1) × L~(p2)(R~n, w_2) to L~p(R~n, ν_w) for appropriate indices p_1, p_2, p ∈ (1, ∞) with1 p=1/ p_1 +1/ p_2 and weights w_1, w_2 such that w = (w_1, w_2) ∈ A_(p/t)(R~(2n)).     

Groups G2(p) as quotients of (2,3,7;2p)

It is proved that for any prime the group is a quotient of      

Extension of Finite Rank Operators and Operator Ideals with the Property (I)

We present criteria and related techniques which help to decide whether the adjoint operator ideal (𝔄*, A *) of an injective and totally accessible maximal Banach ideal (𝔄, A ) is itself also totally accessible. This approach (which involves the transfer of the principle of local reflexivity to operator ideals) is based on the extension of finite rank operators, viewed as elements of the adjoint ideal (𝔄*, A *). Using the local properties (I) and (S) of the corresponding product ideal 𝔄* ∘ 𝔏_∞, these methods even enable us to show that 𝔏_∞ and 𝔏₁ cannot be totally accessible – answering an open question of Defant and Floret .     

Некоторые вопросы теории приближения в пространствах L p(x) (E)

Some aspects of approximation theory are studied in the paper for the spaces of integrable functions with variable exponent. In particular, necessary and suffcient conditions on the variable exponent are established that guarantee the basis property of the trigonometric system in the corresponding normed spaces. Namely, these conditions require that the periodic variable exponent p(x) > 1 satisfy the Dini-Lipschitz condition.