A limit theorem for order preserving nonexpansive operators inL 1

We prove the following theorem:Let T be an order preserving nonexpansive operator on L ₁ (μ) (or L ₁ ⁺ ) of a σ-finite measure, which also decreases theL _∞-norm, and let S=tI+(1?t)T for 0<t<1. Then for everyf ∈ L_p (1<p<∞),the sequence S ⁿf converges weakly in L_p. (The assumptions do not imply thatT is nonexpansive inL _p for anyp>1, even ifμ is finite.) For the proof we show that ∥S ⁿ⁺¹ f?S ⁿf∥ _p → 0 for everyf ∈L _p, 1<p<∞, and apply toS the following theorem:Let T be order preserving and nonexpansive in L ₁ ⁺ , and assume that T decreases theL _∞-norm. Then forg ∈L _p (1<p<∞) Tⁿg is weakly almost convergent. If forf ∈ L_p we have T ⁿ⁺¹ f?T ⁿ f → 0weakly, then T ⁿf converges weakly in L_p (1<p<∞).     

Zero density of L-functions related to Maass forms

Let f(z) be a Hecke-Maass cusp form for SL ₂(?), and let L(s, f) be the corresponding automorphic L-function associated to f. For sufficiently large T, let N(σ, T) be the number of zeros ρ = β +iγ of L(s, f) with |γ| ? T, β ? σ, the zeros being counted according to multiplicity. In this paper, we get that for 3/4 ? σ ? 1 ? ?, there exists a constant C = C(?) such that N(σ,T) ? T ^2(1?σ)/σ(logT) ^C , which improves the previous results.     

A pointwise abelian Ergodic theorem for Lp semigroups, 1 ⩽ p < ∞

Let (X, ∑, μ) be a σ-finite measure space and L_p(μ) = L_p(X, ∑, μ), 1 ? p ? ∞, the usual Banach spaces of complex-valued functions. Let {T_t: t ? 0} be a strongly continuous semigroup of positive L_p(μ) operators for some 1 ? p < ∞. Denote by R_λ the resolvent of {T_t}. We show that f?L_p(μ) implies λR_λf(x) → f(x) a.e. as λ → ∞.     

Sequences of contractions on L1-spaces

Let T_n, n = 1,2,… be a sequence of linear contractions on the space where is a finite measure space. Let M be the subspace of L¹ for which T_ng → g weakly in L¹ for g?M. If T_n1 → 1 strongly, then T_nf → f strongly for all f in the closed vector sublattice in L¹ generated by M.This result can be applied to the determination of Korovkin sets and shadows in L¹. Given a set G ? L¹, its shadow S(G) is the set of all f?L¹ with the property that T_nf → f strongly for any sequence of contractions T_n, n = 1, 2,… which converges strongly to the identity on G; and G is said to be a Korovkin set if S(G) = L¹. For instance, if 1 ?G, then, where M is the linear hull of G and $\text{B}$_M is the sub-σ-algebra of $\text{B}$ generated by {x?X: g(x) > 0} for g?M. If the measure algebra is separable, has Korovkin sets consisting of two elements.     

L p-approximation by Kantorovič operators

Оператор Канторович а дляf∈L _p(I), I=[0,1], определяе тся соотношением $$P_n (f,x) = (n + 1)\sum\limits_{k = 0}^n {\left( {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right)} x^k (1 - x)^{n - 1} \int\limits_{I_k } {f(t)dt,} $$ гдеI _k=[k/(n}+1),(k+1)/(n+ 1)],n∈N. Доказывается, что есл ир>1 иf∈W _p ² (I), т.е.f абсол ютно непрерывна наI иf″∈L _p(I), то $$\left\| {P_n f - f} \right\|_p = O(n^{ - 1} ).$$ Далее, установлено, чт о еслиf∈L _p(I),p>1 и ∥P _n f-f∥_р=О(n ^?1), тоf∈S, гдеS={f∶f аб-солютно непрерывна наI, x(1?x)f′(x)=∝ ₀ ^x h(t)dt, гдеh∈L _p(I) и ∝ ₀ ¹ h(t)dt=0}. Если жеf∈L_p(I),p>1, то из условия ∥P _n(f)?f∥_pL=o(n^?1) вытекает, чтоf постоянна почти всюду.     

Caratheodory inequality for analytic operator function

Abstract. Suppose H is a complex Hilbert space, AH (△) denotes the set of all analytic operator functions on     

Regularity of hyperbolic equations underL 2(0,T; L 2 (Γ))-Dirichlet boundary terms

With Ω an open bounded domain inR ⁿ with boundary Γ, letf(t; f ₀,f ₁;u) be the solution to a second order linear hyperbolic equation defined on Ω, under the action of the forcing termu(t) applied in the Dirichlet B.C., and with initial dataf ₀ ∈L ₂ (Ω) andf ₁ ∈H ^?1 (Ω). In a previous paper [6], we proved (among other things) that the mapu → f ? f _t , from the Dirichlet input into the solution is continuous fromL ₂(0,T; L ₂ (Γ)) intoL ₂(0,T; L ₂(Ω))?L₂ (0, T; H ^?1 (Ω)). Here, we make crucial use of this result to present the following marked improvement: the mapu → f ?f _t is continuous fromL ₂ (0, T; L ₂ (Γ)) intoC([0, T]; L ₂ (Ω))?C([0, T]; H ^?1 (Ω)). Our approach uses the cosine operator model introduced in [6], along with crucial relevant fact from cosine operator theory. A new trace theory result, on which we base our proof here, plays also a decisive role in the corresponding quadratic optimal control problem [7]. Whenu, instead, acts in the Neumann B. C. and Ω is either a sphere or a parallelepiped, the same approach leads to the same improvement over results obtained in [6] to the regularity int of the solution (i.e., fromL ₂ (0, T) toC[0, T]).     

On the Behaviour of the spectral characteristic of Feigenbaum’s map

Let T _g : [?1, 1] ?? [?1, 1] be the Feigenbaum map. It is well known that T _g has a Cantor-type attractor F and a unique invariant measure ??₀ supported on F. The corresponding unitary operator (U _g ??)(x) = ??(g(x)) has pure point spectrum consisting of eigenvalues ?? _n,r , n ?? 1, 0 ?? r ?? 2 ^n?1 ? 1 with eigenfunctions e _r ⁽ⁿ⁾ (x). Suppose that f ?? C ¹([?1, 1]), f?? is absolutely continuous on [?1, 1] and f?? ?? L ^p ([?1, 1], d??₀), p > 1. Consider the sum of the amplitudes of the spectral measure of f: $$ Sn(f): = \sum\limits_{r = 0}^{2^n - 1} {|\rho _r^{(n)} |^2 ,\rho _r^{(n)} = \int\limits_{ - 1}^1 {f(x)\overline {e_r^{(n)} (x)} d\mu _o } } (x). $$ Using the thermodynamic formalism for T _g we prove that S _n (f) ?? 2^?n q ⁿ , as n ?? ??, where the constant q ?? (0, 1) does not depend on f.     

The (Lp, Lq) bilinear Hardy-Littlewood function for the tail

Let (X, B, μ, T) be a measure preserving dynamical system on a finite measure space. Consider the maximal function

On approximation of convex functions by rational ones in integral metrics

Пустьf - действительн означная конечная фу нкция на конечном отрезке Δ=[а, b] вещественной оси, |Δ|=b?a, M(f) = sup {|f(x)|: x∈Δ}, R_n(f,p Δ) = inf∥f?r∥_Lp(Δ) (0 < p < ∞), где нижняя грань бере тся по всем рациональ ным функциямr порядка не вышеп, K(М, Δ) класс всех выпуклых на отре зке Δ функцийf, для кот орыхM(f)≦M. Теорема.При любом вещ ественном р, 0<р<∞ и вс ехп=1, 2, ... sup {R_n(f, p, Δ):f∈K(M, Δ)} ≦ C(p)M|Δ|^1/pn^?2,где С(р) - величина, зави сящая лишь от р.     

Convolution functions on compact Abelian groups

ПустьG — бесконечная компактная абелева г руппа с группой характеров?, и дляr>0A _r (G) обозначает множест во всехf∈L ₁ (G), преобразование Фурь е которыхf принадлеж итl _r (?). Пусть, далее, дляr>0 иs>0A(r, s)(G) обозначает множество всех такихf∈L ₁ (G), чтоf принадлежит про странству ЛоренцаL(r,s)(?). Теорема 1. Пусть 1<р≦2, 1<q≦2и 1/r=1/р+ 1/q?1. ТогдаL _p (G)*L _q (G)?A _r ,(G), 1/r+1/r′=1, причем равенство имеет место в том и тол ько том случае, когда p=q=2. Теорема 2. Пусть p, q, r удовл етворяют условиям те оремы 1 и 1/s=1/p+1/q. Тогда

1. существуютf∈L _p (G) и h∈L_q(G) т акие, чтоf*h ? A(β,γ)(G) ни для какихβ0;
2. если 0<s ₀ , то существую тf ∈L _p (G) и h ∈L _q ,(G) такие, что f*h∈A(r′, s ₀)(G).

Из теоремы 2 следует, чт о неравенство Юнга в определенном смысле неулучшаемо. Непосредственными с ледствиями теоремы 2 я вляются также один результат Р. Л. Липсмана и теорема У. Б. Тевари—А. К. Гупта.     

A new proof of certain littlewood-paley inequalities

The aim of this article is to give a new proof of the L^p-inequalities for the Littlewood-Paley g_*-function. Our main tool is a pointwise equality, relating a function f, and the associated functional g_*(f), which has the form f²=h(f)+g _* ² (f), where h(f) is an explicit function. We obtain this equality as a particular case of a more general one, which is reminiscent of a well-known identity in the stochastic calculus setting, namely the Itô formula. Once the above equality is proved, L^p-estimates for g_*(f) are obviously equivalent to L^p/2-estimates for h(f). We obtain these last estimates (more precisely, H^p/2-estimates for h(f) by using a slight extension of the Coifman-Meyer-Stein theorem relating the so-called tent-spaces and the Hardy spaces. We observe that our methods clearly show that the restriction p>2n/n+1 is closely related to cancellation and size properties of the gradient of the Poisson kernel.     

Analogues of the “zero-two” law for positive linear contractions inL p andC(X)

LetT be a positive linear contraction inL ^p (1≦p<∞), then we show that lim ‖T ^pf −T ⁿ⁺¹ f‖ _p ≦(1 − ε)2^1/p (f∈L _p ⁺ , ε>0 independent off) implies already lim_n n→∞ ‖T ⁿf −T ⁿ⁺¹ n+1f ‖_p p=0. Several other related results as well as uniform variants of these are also given. Finally some similar results inLsu/t8 andC(X) are shown.     

Eigenvalue results for pseudomonotone perturbations of maximal monotone operators

Let X be an infinite-dimensional real reflexive Banach space such that X and its dual X* are locally uniformly convex. Suppose that T: X?D(T) → 2 ^X * is a maximal monotone multi-valued operator and C: X?D(C) → X* is a generalized pseudomonotone quasibounded operator with L ? D(C), where L is a dense subspace of X. Applying a recent degree theory of Kartsatos and Skrypnik, we establish the existence of an eigensolution to the nonlinear inclusion 0 ∈ T _x + λ C _x , with a regularization method by means of the duality operator. Moreover, possible branches of eigensolutions to the above inclusion are discussed. Furthermore, we give a surjectivity result about the operator λT + C when λ is not an eigenvalue for the pair (T, C), T being single-valued and densely defined.     

关于Orlicz空间的(?)-乘积问题

Let T_M1(G). T_M2(F) be the unite sphere of K_M1^*(G) and L_M2^*(F). respectively. In this paper we obtain the following theorem.     

Fourier Series with the Continuous Primitive Integral

Fourier series are considered on the one-dimensional torus for the space of periodic distributions that are the distributional derivative of a continuous function. This space of distributions is denoted A_c(\mathbbT){\mathcal{A}}_{c}(\mathbb{T}) and is a Banach space under the Alexiewicz norm, ||f||_\mathbbT=sup_{|I| ￡ 2p}|ò_I f|\|f\|_{\mathbb{T}}=\sup_{|I|\leq2\pi}|\int_{I} f|, the supremum being taken over intervals of length not exceeding 2π. It contains the periodic functions integrable in the sense of Lebesgue and Henstock–Kurzweil. Many of the properties of L ¹ Fourier series continue to hold for this larger space, with the L ¹ norm replaced by the Alexiewicz norm. The Riemann–Lebesgue lemma takes the form [^(f)](n)=o(n)\hat{f}(n)=o(n) as |n|→∞. The convolution is defined for f ? A_c(\mathbbT)f\in{\mathcal{A}}_{c}(\mathbb{T}) and g a periodic function of bounded variation. The convolution commutes with translations and is commutative and associative. There is the estimate ||f*g||_￥ ￡ ||f||_\mathbbT ||g||_BV\|f\ast g\|_{\infty}\leq\|f\|_{\mathbb{T}} \|g\|_{\mathcal{BV}}. For g ? L¹(\mathbbT)g\in L^{1}(\mathbb{T}), ||f*g||_\mathbbT ￡ ||f||_{\mathbb T} ||g||₁\|f\ast g\|_{\mathbb{T}}\leq\|f\|_{\mathbb {T}} \|g\|_{1}. As well, [^(f*g)](n)=[^(f)](n) [^(g)](n)\widehat{f\ast g}(n)=\hat{f}(n) \hat{g}(n). There are versions of the Salem–Zygmund–Rudin–Cohen factorization theorem, Fejér’s lemma and the Parseval equality. The trigonometric polynomials are dense in A_c(\mathbbT){\mathcal{A}}_{c}(\mathbb{T}). The convolution of f with a sequence of summability kernels converges to f in the Alexiewicz norm. Let D _n be the Dirichlet kernel and let f ? L¹(\mathbbT)f\in L^{1}(\mathbb{T}). Then ||D_n*f-f||_\mathbbT?0\|D_{n}\ast f-f\|_{\mathbb{T}}\to0 as n→∞. Fourier coefficients of functions of bounded variation are characterized. The Appendix contains a type of Fubini theorem.     

WEIGHTED Lp-APPROXIMATION OF DERIVATIVES BY THE METHOD OF BETA OPERATORS

The author consides Beta operators β_nf on suitable Sobolev type subspace of L_p[0, ∞) and characterizes the global rate of approximation of derivatives f^(τ) through corresponding derivatives (β_nf)^(τ) in an appropriate weighted L_p-metric by the rate of Ditzian and Totik's τ-th order weighted modulus of Smoothness.     

Vanishing theorem for sheaves of microfunctions at the boundary on cr–manifolds

Let X be a complex analytic manifold. Consider S?M?Xreal analytic submonifolds with codium ^R _MS=1,and let ω be a connected component of M\S. Let p∈S X_MT_M ^*X where T^* _Xdenotes the conormal bundle to M in X, and denote by ν(p) the complex radial Euler field at p. Denote by μ*(Ox) (for * = M, ω) the microlocalization of the sheaf of holomorphic functions along *.

Under the assumption dim^R(T_pT_M ^*X? ν(p)) = 1, a theorem of vanishing for the cohomology groups H^jμM(Ox)p is proved in [K-S 1, Prop. 11.3.1], j being related to the number of positive and negative eigenvalue for the Levi form of M.

Under the hypothesis dim^R(T_pT_S ^*X∩ν(p))=1, a similar result is proved here for the cohomology groups of the complex of microfunctions at the boundary μω(Ox).Stating this result in terms of regularity at the boundary for CR–hyperfunctions a local Bochner–type theorem is then obtained.     

The natural operators lifting connections to higher order cotangent bundles

We prove that the problem of finding all Mf _m -natural operators C: Q ? QT ^r * lifting classical linear connections ? on m-manifolds M into classical linear connections C _M (?) on the r-th order cotangent bundle T ^r *M = J ^r (M, ?)₀ of M can be reduced to the well known one of describing all M f _m -natural operators D: Q ? ? ^p T ? ? ^q T* sending classical linear connections ? on m-manifolds M into tensor fields D _M (?) of type (p, q) on M.     

Functional equations and linear transformations,IIIB: Factorization

Let (X, m) and (X ^～ ,m ^～ ) be topologically isomorphic locally compact abelian groups. Let the isomorphism \(A:X^ \sim \xrightarrow{{onto}}X\) be such that (i) $$A:u - v \to A_{u - v} = A_u A_{ - v} = A_u A_v^{ - 1} = xy^{ - 1;} $$ (ii) $$m^ \sim (E) = m(AE)$$ for each Borel subsetE ofX. Define convolution, whenever it exists, onX andX ^～ by \((f \circ g)(x) = \int\limits_x {f(xy^{ - 1} )g(y)dm} (y)\) and \((f^ \sim *g^ \sim )(x) = \int\limits_{X^ \sim } {f^ \sim (x - y)g^ \sim (y)dm^ \sim } (y)\) , respectively. Let σ be a relatively invariant positive measure onX, i.e., for eachf ∈ K X, $$\int\limits_X {f(vy^{ - 1} )\sigma d(v) = \alpha (y)\int {f(v)d\sigma (v); \alpha (y)} > 0} $$ for ally∈X. Define a mappingR _μ byR _μ f(x)=α(x) ^?μ f(x ^?1 ), and setf(x ^?1 )=f ^⊥ (x). LetB _1p denote the space of equivalence classes of functionsf for which \((\int\limits_X {|f} (y)|\alpha (y)^{1/P} d\sigma (y))< \) <∞. LetL _p (m), L _p (m ^～ ) andL _p =L _p (σ), 1≤p<∞, denote, respectively, the spaces of equivalence classes of functions for which \(\parallel f\parallel _{p(m)}^p = (\int\limits_X {|f|^p dm} )< \infty , \parallel f\parallel _{p(m - )}^p = \) \( = (\int\limits_X {|f|} ^p dm^ \sim )< \infty \) and \(\parallel f\parallel _p^p = (\int {|f|} ^p |d\sigma )< \infty \) . Let be the mappingSf=f ^～ , wheref ^～ (x)=α(A _x ) ^1/p f(A _x );x∈X ^～ . Supposef, g∈L _p . Set α(x)^1/pf(x)=F(x), α(x)^1/pg(x)=H(x). A linear bounded mappingT: L _p →L _p , 1≤p<∞, is said to belong to classI (i.e.,T∈I∩C(L _p ,L _p )) if there exist functionsV andK such thatg=Tf means(V?H)=(K?F ^⊥ ); similarly, the bounded linear mappingT: L _p →L _p ,1≤ p<∞, is said to belong to classG (i.e.,T∈G∩C(L _p ,L _p )) if there exist functionsV andK such thatg=Tf means(V?H)=(K?F). The following theorem is proved, and its applications to Watson transforms are considered.