The maximal Fejér operator on the spacesH p (R×···×R)

It is shown that the maximal operator of the Fejér means of a tempered distribution is bounded from thed-dimensional Hardy spaceH _p (R×···×R) toL _p (R ^d ) (1/2<p<∞) and is of weak type (H ₁ ^?i ,L ₁) (i=1,…,d), where the Hardy spaceH ₁ ^?i is defined by a hybrid maximal function. As a consequence, we obtain that the Fejér means of a functionf ∈H ₁ ^?i ?L(logL) ^d?1 converge a.e. to the function in question. Moreover, we prove that the Fejér means are uniformly bounded onH _p (R×···×R) whenever 1/2<p<∞. Thus, in casef ∈H _p (R×···×R) the Fejér means converge tof inH _p (R×···×R) norm. The same results are proved for the conjugate Fejér means, too.     

On characterization of multivariate stable distributions via random linear statistics

LetX ₁,X ₂, ...,X _n be independent and identically distributed random vectors inR ^d , and letY=(Y ₁,Y ₂, ...,Y _n )′ be a random coefficient vector inR ⁿ , independent ofX _j ^/′ . We characterize the multivariate stable distributions by considering the independence of the random linear statistic $$U = Y_1 X_1 + Y_2 X_2 + \cdot \cdot \cdot + Y_n X_n $$ and the random coefficient vectorY.     

On Hardy-Bessel potential spaces over the ring of integers in a local field

Штрихарц [3] дал характе ристику пространствL ^p (R ⁿ ) бесселевых потенциа лов порядка α функций из п ространстваL ^p (R ⁿ ) с пом ощьюL ^p -норм функционалов $$D_\alpha f(x) = \left( {\smallint _0^\infty \left( {\smallint _{\rm B} |f(x + rt) - f(x)|dt} \right)^2 r^{ - 2\alpha - 1} dr} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}$$ для 0<α<1, гдеB обозначает ед иничный шар. Целью нас тоящей статьи является изучение пр остранств потенциал а Харди-БесселяF _α ^p (P ⁰) (0<p<1, α>1/р?1) в терминах функц ионаловS _r ^α f(x) (1τ≤2), которые в случаеR ⁿ } соответствуютD _α f (x), гдеР ⁰ - кольцо целых в локальном поле. Получ ено приложение, относяще еся к регулярности бе сселевых потенциалов.     

Følner sequences and Hilbert’s Irreducibility Theorem over Q

Letf(X; T ₁, ...,T _n) be an irreducible polynomial overQ. LetB be the set ofb teZ ⁿ such thatf(X;b) is of lesser degree or reducible overQ. Let ?={F _j}{F _j } _j?1 ^∞ be a Følner sequence inZ ⁿ — that is, a sequence of finite nonempty subsetsF _j ?Z ⁿ such that for eachvteZ ⁿ , $\mathop {lim}\limits_{j \to \infty } \frac{{\left| {F_j \cap (F_j + \upsilon )} \right|}}{{\left| {F_j } \right|}} = 1$ Suppose ? satisfies the extra condition that forW a properQ-subvariety ofP ⁿ ?A ⁿ and ?>0, there is a neighborhoodU ofW(R) in the real topology such that $\mathop {lim sup}\limits_{j \to \infty } \frac{{\left| {F_j \cap U} \right|}}{{\left| {F_j } \right|}}< \varepsilon $ whereZ ⁿ is identified withA ⁿ (Z). We prove $\mathop {lim}\limits_{j \to \infty } \frac{{\left| {F_j \cap B} \right|}}{{\left| {F_j } \right|}} = 0$ .     

Weighted norm inequalities for the vectorvalued Hardy-Littlewood maximal functions of one parameter rectangles

Let N denote the Hardy-Littlewood maximal operator for the familyR of one parameter rectangles. In this paper, we obtain that for 1 _w ^p (l^r) to L _W ^P (l^r) if and only if w ∈ A_P(R); for 1≤p<∞, N is bounded from L _W ^P (l^r) to weak L _W ^P (l^r) if and only if W ∈ A_P(R). Here we say W∈A_p (1), if $$\begin{gathered} \mathop {sup}\limits_{R \in R} \left( {\tfrac{1}{{|R|}}\smallint _r wdx} \right)\left( {\tfrac{1}{{|R|}}\smallint _R w^{ - 1/(p - 1)} dx} \right)^{p - 1}< \infty ,1< p< \infty , \hfill \\ (Nw)(x) \leqslant Cw(x)a.e.,p = 1 \hfill \\ \end{gathered} $$ ,     

Polynomial approximation inL p(S) forp>0

For a simple polytopeS inR ^d andp>0 we show that the best polynomial approximationE _n(f)_p≡E_n(f)L_p(S) satisfies $$E_n \left( f \right)_p \leqslant C\omega _S^r \left( {f,\frac{1}{n}} \right)p,$$ where ω _S ^r is a measure of smoothness off. This result is the best possible in the sense that a weak-type converse inequality is shown and a realization of ω _S ^r (f,t)_p via polynomial approximation is proved.     

Homogenization of foliated annuli

Let \(\Omega = \Omega _0 \backslash \bar \Omega _1\) be a regular annulus inR ^N and \(\phi :\bar \Omega \to R\) be a regular function such that φ=0 on ?Ω₀, φ=1 on ?Ω₁ and ▽φ ≠ 0. Let K_n be the subset of functions v ε W^1,p (Ω) such that v=0 on ?Ω₀, v=1 on ?Ω₁, v=(unprescribed) constant on n given level surfaces of φ. We study the convergence of sequences of minimization problems of the type $$Inf\left\{ {\int\limits_\Omega {\frac{1}{{a_n \circ \phi }}G(x,(a_n \circ \phi )\nabla v)dx;v \in K_n } } \right\},$$ where a_n ε L^∞ (0,1) and G: (x, ζ) ε Ω × R^N → G(x, ζ εR is convex with respect to ξ and verifies some standard growth conditions.     

Inequalities for entire functions of exponential type satisfying f(z) \equiv e^{i\gamma } e^{i\tau z} \overline {f(\bar z)}

Let f be an entire function of exponential type satisfying the condition $ f(z) \equiv e^{i\gamma } e^{i\tau z} \overline {f(\bar z)} $ for some real γ. Lower and upper estimates for ∫ _?∞ ^∞ |f′(x)| ^p dx in terms of ∫ _?∞ ^∞ |f(x)| ^p dx, for such a function f belonging to L ^p (R), have been known in the case where p ? [1, ∞) and γ = 0. In this paper, these estimates are shown to hold for any p ? (0, ∞) and any real γ.     

Interpolation and frames in certain Banach spaces of entire functions

The important class of generalized bases known as frames was first introduced by Duffin and Schaeffer in their study of nonharmonic Fourier series in L ² (?π, π) [4]. Here we consider more generally the classical Banach spacesE ^p(1 ≤ p ≤ ∞) consisting of all entire functions of exponential type at most π that belong to L^p (?∞, ∞) on the real axis. By virtue of the Paley-Wiener theorem, the Fourier transform establishes an isometric isomorphism between L ² (?π, π) andE ² . When p is finite, a sequence {λ _n} of complex numbers will be called aframe forE ^p provided the inequalities $$A\left\| f \right\|^p \leqslant \sum {\left| {f\left( {\lambda _\pi } \right)} \right|^p } \leqslant B\left\| f \right\|^p $$ hold for some positive constants A and B and all functions f inE ^p. We say that {λ _n} is aninterpolating sequence forE ^p if the set of all scalar sequences {f (λ _n)}, with f εE ^p, coincides with ?^p. If in addition {λ _n} is a set of uniqueness forE ^p, that is, if the relations f(λ _n)=0(?∞<n<∞), with f εE ^p, imply that f ≡0, then we call {λ _n} acomplete interpolating sequence. Plancherel and Pólya [7] showed that the integers form a complete interpolating sequence forE ^p whenever1<p<∞. In Section 2 we show that every complete interpolating sequence forE ^p(1<p<∞) remains stable under a very general set of displacements of its elements. In Section 3 we use this result to prove a far-reaching generalization of another classical interpolation theorem due to Ingham [6].     

A moment theorem for completely hyperexpansive operators

Given a family $ \{ A_m^x \} _{\mathop {m \in \mathbb{Z}_ + ^d }\limits_{x \in X} } $ (X is a non-empty set) of bounded linear operators between the complex inner product space $ \mathcal{D} $ and the complex Hilbert space ? we characterize the existence of completely hyperexpansive d-tuples T = (T ₁, … , T _d ) on ? such that A _m ^x = T ^m A ₀ ^x for all m ? ? ₊ ^d and x ? X.     

Global existence for the Cauchy problem of the parabolic–parabolic Keller–Segel system on the plane

This paper is concerned with the Cauchy problem for the Keller–Segel system $$\left\{\begin{array}{l@{\quad}l}u_t = \nabla \cdot (\nabla u - u \nabla v) & \hbox{in } {\bf R}^{2} \times(0,\infty),\\v_t = \Delta v - \lambda v + u & \hbox{ in } {\bf R}^2 \times(0,\infty),\\u(x,0) = u_0 (x) \geq 0, \; v(x,0) = v_0 (x) \geq 0 & \hbox{ in} {\bf R}^2\end{array}\right.$$ with a constant λ ≥ 0, where ${(u_0, v_0) \in (L^1 ({\bf R}^2) \cap L^\infty ({\bf R}^2) ) \times (L^1 ({\bf R}^2) \cap H^1 ({\bf R}^2))}$ . Let $$m (u_0;{\bf R}^2) = \int\limits_{{\bf R}^2} u_0 (x) dx$$ . The same method as in [9] yields the existence of a blowup solution with m (u ₀; R ²) > 8π. On the other hand, it was recently shown in [7] that under additional hypotheses ${u_0 \log (1 + |x|^2) \in L^1 ({\bf R}^2)}$ and ${u_0 \log u_0 \in L^1 ({\bf R}^2)}$ , any solution with m(u ₀; R ²) < 8π exists globally in time. In[18], the extra assumptions were taken off, but the condition on mass was restricted to m (u ₀; R ²) < 4π. In this paper, we prove that any solution with m (u ₀; R ²) < 8π exists globally in time under no extra conditions. Furthermore the global existence of solutions is obtained under some condition on u ₀ also in the critical case m (u ₀; R ²) = 8π.     

A contribution to stability theory for nonlinear Neumann problem

In this paper we study the stability of the solutions of some nonlinear Neumann problems, under perturbations of the domains in the Hausdorff complementary topology. We consider the problem $${{\left\{\begin{array}{c}-\text{ div}\;\left(a\left( x,\nabla u_{\Omega}\right)\right)=0 \;\text{in}\; \Omega \\ {a\left( x, \nabla u_{\Omega}\right) \cdot \nu=0\; \text{on}\; \partial\Omega}\end{array}\right.}}$$ where ${{\mathbf{R}^n \times \mathbf{R}^n \rightarrow \mathbf{R}^n}}$ is a Caratheodory function satisfying the standard monotonicity and growth conditions of order p, 1?<?p?<???. If ?? _h is a uniformly bounded sequence of connected open sets in R ⁿ , n ??? 2, we prove that if ${{\Omega_{h}^{c} \rightarrow \Omega^{c}}}$ in the Hausdorff metric, ${|\Omega_{h}| \rightarrow |\Omega|}$ and the geodetic distances satisfy the inequality ${d_{\Omega}\left( x,y\right) \leq \liminf_{h} d_{\Omega_{h}} \left( x,y\right)}$ for every ${x, y \in \Omega,}$ then ${\nabla u_{\Omega_h} \rightarrow\nabla u_{\Omega}}$ strongly in L ^p , provided that W ^{1, ??}(??) is dense in the space L ^{1, p} (??) of all functions whose gradient belongs to L ^p (??, R ⁿ ).     

Exactn-widths of hardy-sobolev classes

Let $\tilde h^r _{\infty ,\beta } $ and $\tilde H^r _{\infty ,\beta } $ denote those 2π-periodic, real-valued functions onR that are analytic in the strip |Imz|<β and satisfy the restrictions |Ref ^(r)(z)| ≤ 1 and |f ^(r)(z)| ≤ 1, respectively. We determine the Kolmogorov, linear, and Gel’fand widths of $\tilde h^r _{\infty ,\beta } $ inL _q[0, 2π], 1 ≤q ≤ ∞, and $\tilde H^r _{\infty ,\beta } $ inL _∞[0, 2π].     

Time-Varying Fractionally Integrated Processes with Finite or Infinite Variance and Nonstationary Long Memory

Recently, Philippe et al. (C.R. Acad. Sci. Paris. Ser. I 342, 269–274, 2006; Theory Probab. Appl., 2007, to appear) introduced a new class of time-varying fractionally integrated filters A(d)x _t =∑ _j=0 ^∞ a _j (t)x _t?j , B(d)x _t =∑ _j=0 ^∞ b _j (t)x _t?j depending on arbitrary given sequence d=(d _t ,t∈?) of real numbers, such that A(d)^?1=B(?d), B(d)^?1=A(?d) and such that when d _t ≡d is a constant, A(d)=B(d)=(1?L) ^d is the usual fractional differencing operator. Philippe et al. studied partial sums limits of (nonstationary) filtered white noise processes X _t =B(d)ε _t and Y _t =A(d)ε _t in the case when (1) d is almost periodic having a mean value $\bar{d}\in (0,1/2)$ , or (2) d admits limits d _±=lim? _t→±∞ d _t ∈(0,1/2) at t=±∞. The present paper extends the above mentioned results of Philippe et al. into two directions. Firstly, we consider the class of time-varying processes with infinite variance, by assuming that ε _t ,t∈? are iid rv’s in the domain of attraction of α-stable law (1<α≤2). Secondly, we combine the classes (1) and (2) of sequences d=(d _t ,t∈?) into a single class of sequences d=(d _t ,t∈?) admitting possibly different Cesaro limits $\bar{d}_{\pm}\in(0,1-(1/\alpha))$ at ±∞. We show that partial sums of X _t and Y _t converge to some α-stable self-similar processes depending on the asymptotic parameters $\bar{d}_{\pm}$ and having asymptotically stationary or asymptotically vanishing increments.     

Multi-point boundary value problems for one-dimensionalp-Laplacian at resonance

In this paper, we consider the multi-point boundary value problems for one-dimensional p-Laplacian at resonance: $(\phi _p (x'(t)))' = f(t,x(t),x'(t))$ subject to the boundary value conditions: $(\phi _p (x'(0)) = \sum\limits_{i = 1}^{n - 2} {\alpha _i \phi _p (x'(\xi _i ))} $ , $(\phi _p (x'(1)) = \sum\limits_{j = 1}^{m - 2} {\beta _j \phi _p (x'(\eta _i ))} $ where ? _p (s)=|s|^p-2 s, p>1,α_i(1≤i≤n-2)∈R,β_{jit}(1≤j≤m-2)∈R, 0<ξ₁<ξ₂<...<ξ_n-2<1, 0<η₁<η₂<...<η_m-2<1, By applying the extension of Mawhin’s continuation theorem, we prove the existence of at least one solution. Our result is new.     

Sharp estimates for operator\bar \partial _M on aq-concave CR manifoldon aq-concave CR manifold

We construct integral operatorsR _r andH _r on the spaces of differential forms of the type (o, r) withr <q on a regularq-concave CR manifoldM such that $$f(z) = \bar \partial _M R_r (f)(z) + R_{r + 1} (\bar \partial _M f)(z) + H_r (f)(z),$$ for a differential formf ∈ L _(0,r) ^s (M) and forz ∈ M′ ?M, whereH _r is compact andR _r admits sharp estimates.     

Local linear independence of the translates of a box spline

Let Ξ=(ξ _i ) _l ⁿ be a sequence of vectors inR ^m . The box splineM _Ξ is defined as the distribution given by $$M_\Xi :\varphi \to \int_{[0,1]^n } \varphi \left( {\sum\limits_{i = 1}^n {\lambda (i)\xi _i } } \right)d\lambda ,\varphi \in C_c^\infty (R^m ).$$ . Suppose that Ξ contains a basis forR ^m . ThenM _Ξ∈L _∞(R ^m ). Assume $$\Xi \subset V: = z^m .$$ . Consider the translatesM _v :=M _Ξ(·?v),v∈V. It is known that (M _v ) _V is linearly dependent unless (*) $$|\det Z| = 1forallbasesZ \subset \Xi$$ . This paper demonstrates that under condition (*), (M _v ) _V is locally linearly independent, i.e., $$\{ M_v ;\sup p M_v \cap A \ne \not 0\}$$ is linearly independent over any open setA.     

On the order of magnitude of double fourier transforms. II

Denote by $\hat f$ the (complex) Fourier transform of a functionf which belongs toL ¹(R ²). We shall assume thatf is odd inx andy, orf is even inx and odd iny, orf is odd inx and even iny. Among others, we prove that iff ∈L ¹(R ²) and (x, y)=(0,0) is a strong Lebesgue point off, then $\left| t \right|\left| v \right|\hat f(t,v)$ tends to 0 as |t|, |v|→∞ in the sense (C;α,β) for allα,β>1.     

Certain additive maps on m-power closed Lie ideals

Let R be a prime ring with extended centroid C and m a fixed positive integer >?1. A Lie ideal L of R is called m-power closed if ${u^m \in L}$ for all ${u \in L}$ . We prove that if char R = 0 or a prime p?>?m, then every non-central, m-power closed Lie ideal L of R contains a nonzero ideal of R except when dim _C RC?=?4, m is odd, and ${u^{m-1} \in C}$ for all ${u \in L}$ . Moreover, the additive maps d : L ?? R satisfying d(u ^m )?=?mu ^m-1 d(u) (resp. d(u ^m )?=?u ^m-1 d(u)) for all ${u \in L}$ are completely characterized if char R = 0 or a prime p?>?2(m ? 1).     

Wolff’s inequality in multi-parameter Morrey spaces

We prove Wolff inequalities for multi-parameter Riesz potentials and Wolff potentials in Lebesque spaces L ^p (R ^d ) and multi-parameter Morrey spaces ${L^p_\lambda (R^d)}$ , where ${R^d=R^{n_1} \times R^{n_2} \times \cdots \times R^{n_k},\, \lambda = (\lambda _1,\ldots ,\lambda _k})$ and 0?<?λ _i ≤ n _i , 1?≤ i ≤ k, in the dyadic case as well as in the non-dyadic (continuous) case.