On perfectly homogeneous bases inF-spaces

It is proved that in anF-space with a basis (χ _n ) _n ^{= 1/∞} , (χ _n ) _n ^{= 1/∞} is equivalent to the unit-vectors basis ofc ₀,l _p (p>0), or (s) if and only if (χ _n ) _n ^{= 1/∞} is equivalent to each of itsɛ-normalized block basic sequences for eachɛ>0. This result is an extension of a theorem of M. Zippin.     

Zero sets and multiplier theorems for star-invariant sub spaces

Given an inner function θ, let {Kskθ/p}:= H^p ∩θ {Hsk0/p} be the corresponding star-invariant subspace of the Hardy spaceH ^p. We show that, unless θ is a finite Blaschke product, the zero sets for K _θ ^p -spaces are different for different p’s. We also investigate the (non)stability of zero sets when passing from {Kskθ/p} to {Ksku/q}, whereq > p and u is an inner function divisible by θ. This problem is motivated by the Beurling-Malliavin multiplier theorem for entire functions, and we solve it (at least in a natural special case) by proving an appropriate multiplier theorem for K _θ ^p .     

On the number of elements of given order in a finitep-group

A. Kulakoff [9] proved that forp>2 the numberN _k =N _k (G) of solutions of the equationx ^{p k} =e in a non-cyclicp-groupG is divisible byp ^k+1. This result is a generalization of the well-known theorem of G. A. Miller asserting that the numberC _k =C _k (G) of cyclic subgroups of orderp ^k >p>2 is divisible byp. In this note we show that, as a rule: (1) ifk>1, thenN _k ≡0(modp ^k+p ); (2) ifk>2, thenC _k ≡0(modp ^p ). These facts are generalizations of many results from [1–5,8,9].     

Riesz means for the sublaplacian on the Heisenberg group

The uniform boundedness of the Riesz means for the sublaplacian on the Heisenberg groupH ⁿ is considered. It is proved thatS _R ^α are uniformly bounded onL ^p(Hⁿ) for 1≤p≤2 provided α>α(p)=(2n+1)[(1/p)−(1/2)].     

A new inductive approach to the lace expansion for self-avoiding walks

Summary. We introduce a new inductive approach to the lace expansion, and apply it to prove Gaussian behaviour for the weakly self-avoiding walk on ℤ ^d where loops of length m are penalised by a factor e ^{−β/m p} (0<β≪1) when: (1) d>4, p≥0; (2) d≤4, . In particular, we derive results first obtained by Brydges and Spencer (and revisited by other authors) for the case d>4, p=0. In addition, we prove a local central limit theorem, with the exception of the case d>4, p=0. Received: 29 October 1997 / In revised form: 15 January 1998     

On the maximal ergodic theorem for certain subsets of the integers

It is shown that the set of squares {n ²|n=1, 2,…} or, more generally, sets {n ^t|n=1, 2,…},t a positive integer, satisfies the pointwise ergodic theorem forL ²-functions. This gives an affirmative answer to a problem considered by A. Bellow [Be] and H. Furstenberg [Fu]. The previous result extends to polynomial sets {p(n)|n=1, 2,…} and systems of commuting transformations. We also state density conditions for random sets of integers in order to be “good sequences” forL ^p-functions,p>1.     

The free interpolation problem and basis property of a system of rational functions in weighted Hardy spaces

The paper considers the free interpolation problem in the Hardy weighted space H p(ρ). It is assumed that the weight ρ has unique singularity of the order αonly at the point 1. Particularly, for α − p[(α + 1)p ⁻¹] > 0 the corresponding free interpolation problem is stated and its solvability is proved.     

Extinction and Positivity for a Doubly Nonlinear Degenerate Parabolic Equation

The aims of this paper are to discuss the extinction and positivity for the solution of the initial boundary value problem and Cauchy problem of ut = div（[↓△u^m｜p-2↓△u^m）. It is proved that the weak solution will be extinct for 1 〈 p ≤ 1 ＋ 1/m and will be positive for p 〉 1 ＋ 1/m for large t, where m 〉 0.     

A Fatou theorem for α-harmonic functions in Lipschitz domains

We study α-harmonic functions in Lipschitz domains. We prove a Fatou theorem when the boundary function is bounded and L^p-H?lder continuous of order β with βp > 1. Mathematics Subject Classifications (2000): Primary 31B25; Secondary 60J50 *Research supported by NSF Grant DMS0244737     

Almost-all results on the p λ problem. II

Summary Suppose that 1/2 ≦ λ < 1. Balog and Harman proved that for any real θ there exist infinitely many primes p satisfying p^λ-θ < p^{-(1-λ)/2+ ε} (with an asymptotic result). In the present paper we establish that for almost all θ in the interval 0 ≦ θ < 1 there exist infinitely many primes p such that {p^λ-θ} < p^{-min{(2-λ)/6,(14-9λ)/32}+ε}. Thus we obtain a better result for almost all θ than for a single θ if λ>1/2.     

On the pointwise ergodic theorem onL p for arithmetic sets

The purpose of this note is to show how the results of [B] on the pointwise ergodic theorem forL ²-functions may be extended toL ^p for certainp<2. More precisely, we give a proof of the almost sure convergence of the means (t≧1) given a dynamical system (Ω,B, μ, T) andf of classL ^p(Ω,μ),p>(√5+1)/2.     

Espaces de Banach stables

We define the notion of “stable Banach space” by a simple condition on the norm. We prove that ifE is a stable Banach space, then every subspace ofL ^p(E) (1≦p<∞) is stable. Our main result asserts that every infinite dimensional stable Banach space containsl ^p, for somep, 1≦p<∞. This is a generalization of a theorem due to D. Aldous: every infinite dimensional subspace ofL ¹ containsl ^p, for somep in the interval [1, 2].     

On the multiplier conjecture

The Multiplier Theorem is a celebrated theorem in the Design theory. The conditionp>λ is crucial to all known proofs of the multiplier theorem. However in all known examples of difference sets μ _p . is a multiplier for every primep with (p, v)=1 andp‖n. Thus there is the multiplier conjecture: “The multiplier theorem holds without the assumption thatp>λ”. The general form of the multiplier theorem may be viewed as an attempt to partially resolve the multiplier conjecture, where the assumption “p>λ” is replaced by “n ₁>λ”. Since then Newman (1963), Turyn (1964), and McFarland (1970) attempted to partially resolve the multiplier conjecture (see [7], [8], [9]). This paper will prove the following result using the representation theory of finite groups and the algebraic number theory: LetG be an abelian group of orderv,v ₀ be the exponent ofG, andD be a (v, k, λ)-difference set inG. Ifn=2_{n 1}, then the general form of the multiplier theorem holds without the assumption thatn ₁>λ in any of the following cases:

New a Priori Estimates for q-Nonlinear Elliptic Systems with Strong Nonlinearities in the Gradient, 1<q<2

We consider q-nonlinear nondiagonal elliptic systems, where 1 < q < 2, with strong nonlinear terms in the gradient. Under a smallness condition on the gradient of a solution in the Morrey space L^q,n−q, we estimate the L^p-norm of the gradient for p > q and the Holder norm of the solution for the case n = 2. An abstract theorem on “quasireverse Holder inequalities” proved by the author earlier is used essentially. Bibliography: 24 titles. Dedicated to N. N. Uraltseva on the occasion of her 70th birthday __________ Published in Zapiski Nauchnykh Seminarov POMI, Vol. 310, 2004, pp. 19–48.     

Gap series constructions for the <Emphasis Type="Italic">p</Emphasis>-Laplacian

Highly oscillatory bounded solutions of div(∇u|∇u| ^p−2) = 0 are constructed when p > 2. Fatou’s theorem is shown to fail for this equation. Tom Wolff wrote this paper in 1984, but he never published it. With his family’s permission, we have edited it for publication here. Except for the shorter proof of Lemma 2.1 and the citations of [1] and [12], our alterations to the paper have mostly been typographical. We thank Juan Manfredi for help on Section 3.     

On complete submanifolds with parallel mean curvature in negative pinched manifolds

A rigidity theorem for oriented complete submanifolds with parallel mean curvature in a complete and simply connected Riemannian (n p)-dimensional manifold Nn p with negative sectional curvature is proved. For given positive integers n(≥ 2), p and for a constant H satisfying H ＞ 1 there exists a negative number τ(n,p, H) ∈ (-1, 0) with the property that if the sectional curvature of N is pinched in [-1, τ(n,p, H)], and if the squared length of the second fundamental form is in a certain interval, then Nn p is isometric to the hyperbolic space Hn p(-1). As a consequence, this submanifold M is congruent to Sn(1/ H2-1) or theVeronese surface in S4(1/√H2-1).     

Onl p-complemented copies in Orlicz spaces II

For anyp > 1, the existence is shown of Orlicz spacesL ^F andl ^F with indicesp containingsingular l ^p-complemented copies, extending a result of N. Kalton ([6]). Also the following is proved:Let 1 <α ≦β < ∞and H be an arbitrary closed subset of the interval [α, β].There exist Orlicz sequence spaces l ^F (resp. Orlicz function spaces L^F)with indices α and β containing only singular l ^p-complemented copies and such that the set of values p > 1for which l ^p is complementably embedded into l^F (resp. L ^F)is exactly the set H (resp. H ∪ {2&#x007D;). An explicitly defined class of minimal Orlicz spaces is given. Supported in part by CAICYT grant 0338-84.     

Positive undecidable numberings in the Ershov hierarchy

A sufficient condition is given under which an infinite computable family of Σ^-1 _a -sets has computable positive but undecidable numberings, where a is a notation for a nonzero computable ordinal. This extends a theorem proved for finite levels of the Ershov hierarchy in [1]. As a consequence, it is stated that the family of all Σ^-1 _a -sets has a computable positive undecidable numbering. In addition, for every ordinal notation a > 1, an infinite family of Σ^-1 _a -sets is constructed which possesses a computable positive numbering but has no computable Friedberg numberings. This answers the question of whether such families exist at any—finite or infinite—level of the Ershov hierarchy, which was originally raised by Badaev and Goncharov only for the finite levels bigger than 1.     

On the maximal operator of (<Emphasis Type="Italic">C</Emphasis>, α)-means of Walsh–Kaczmarz–Fourier series

Simon [J. Approxim. Theory, 127, 39–60 (2004)] proved that the maximal operator σ^α,κ,* of the (C, α)-means of the Walsh–Kaczmarz–Fourier series is bounded from the martingale Hardy space H _p to the space L _p for p > 1 / (1 + α), 0 < α ≤ 1. Recently, Gát and Goginava have proved that this boundedness result does not hold if p ≤ 1 / (1 + α). However, in the endpoint case p = 1 / (1 + α ), the maximal operator σ^α,κ,* is bounded from the martingale Hardy space H _1/(1+α) to the space weak- L _1/(1+α). The main aim of this paper is to prove a stronger result, namely, that, for any 0 < p ≤ 1 / (1 + α), there exists a martingale f ∈ H _p such that the maximal operator σ^α,κ,* f does not belong to the space L _p .