Singular measures with smalH(p, q)-projections

We construct a singular probability measure μ on the complex sphere such that the Poisson integral of μ is a pluriharmonic function in the ball and the Fourier transform of μ is asp→∞. Supported by the Centre de Recerca Matemàtica (Institut d’Estudis Catalans, Barcelona) under a grant from DGICYT (Spain); partially supported by the RFFI grant 96-01-00693.     

Multiplication properties of the spacesB

Summary Let A be either B _p, ^qs or F _p, ^qs , where - ∞<s <∞; 0<p, q≦∞ (spaces of Besov-Hardy-Sobolev type, defined on R_n). (i) If g ∈C _ϱ (H?lder-Zygmund spaces), then f → gf is a bounded operator from A into A, provided that ϱ=ϱ(s, p, q, n) is large enough. (ii) There are given sufficient conditions for s, p, and q ensuring that A is a subalgebra of C (space of uniformly continuous bounded functions on R_n). Entrata in Redazione il 17 marzo 1976.     

Minimax goodness-of-fit testing in multivariate nonparametric regression

We consider an unknown response function f defined on Δ = [0, 1] ^d , 1 ≤ d ≤ ∞, taken at n random uniform design points and observed with Gaussian noise of known variance. Given a positive sequence r _n → 0 as n → ∞ and a known function f ₀ ∈ L ₂(Δ), we propose, under general conditions, a unified framework for goodness-of-fit testing the null hypothesis H ₀: f = f ₀ against the alternative H ₁: f ∈ $ \mathcal{F} $ \mathcal{F} , ∥f − f ₀∥ ≥ r _n , where $ \mathcal{F} $ \mathcal{F} is an ellipsoid in the Hilbert space L ₂(Δ) with respect to the tensor product Fourier basis and ∥ · ∥ is the norm in L ₂(Δ). We obtain both rate and sharp asymptotics for the error probabilities in the minimax setup. The derived tests are inherently non-adaptive. Several illustrative examples are presented. In particular, we consider functions belonging to ellipsoids arising from the well-known multidimensional Sobolev and tensor product Sobolev norms as well as from the less-known Sloan-Woźniakowski norm and a norm constructed from multivariable analytic functions on the complex strip.     

Weak generators of the algebra of measures and unicellularity of convolution operators

A general scheme which enables us to consider convolution operators with measures acting in a wide class of spaces of distributions on the interval [0, a), 0<a<∞, is represented. It is proved that if a measure μ is a weak generator of the algebra of measures on [0, a), then C_μ (the convolution operator with μ) is unicellular. We give a condition for a measure μ under which the unicellularity of C_μ implies that μ is a weak generator of the algebra of measures. The following statement is also proved. Let , K_θ=H²⊖θH², and let P_θ be the orthogonal projector from H² onto K_θ; in addition, let μ be a weak generator of the algebra of measures on [0, a) and , z ∈ (here is the unit disk and F^-1 is the inverse Fourier transformation). Let ψ∈H^∞ and let p be a polynomial such that p o(ψ−φ)∈θH^∞. Then the operator x→P_θψx, acting in K_θ, is unicellular. Bibliography: 13 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 217, 1994, pp. 36–53.     

The Gleason’s problem for some polyharmonic and hyperbolic harmonic function spaces

Let Ω ⊆ ℝⁿ be a bounded convex domain with C ² boundary. For 0 < p, q ⩽ ∞ and a normal weight φ, the mixed norm space H _k ^p,q,φ (Ω) consists of all polyharmonic functions f of order k for which the mixed norm ∥ · ∥_p,q,φ < ∞. In this paper, we prove that the Gleason’s problem (Ω, a, H _k ^p,q,φ ) is always solvable for any reference point a ∈ Ω. Also, the Gleason’s problem for the polyharmonic φ-Bloch (little φ-Bloch) space is solvable. The parallel results for the hyperbolic harmonic mixed norm space are obtained.     

Ising model in half-space: A series of phase transitions in low magnetic fields

For the Ising model in half-space at low temperatures and for the “unstable boundary condition,” we prove that for each value of the external magnetic field μ, there exists a spin layer of thickness q(μ) adjacent to the substrate such that the mean spin is close to −1 inside this layer and close to +1 outside it. As μ decreases, the thickness of the (−1)-spin layer changes jumpwise by unity at the points μ_q, and q(μ) → ∞ as μ → +0. At the discontinuity points μ_q of q(μ), two surface phases coexist. The surface free energy is piecewise analytic in the domain Re μ > 0 and at low temperatures. We consider the Ising model in half-space with an arbitrary external field in the zeroth layer and investigate the corresponding phase diagram. We prove Antonov’s rule and construct the equation of state in lower orders with the precision of x⁷, x = e^−7ɛ. In particular, with this precision, we find the points of coexistence of the phases 0, 1, 2 and the phases 0, 2, 3, where the phase numbers correspond to the height of the layer of unstable spins over the substrate. Dedicated to Roland L’vovich Dobrushin __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 153, No. 2, pp. 220–261, November, 2007.     

Computation of the ε entropy of the space of continuous functions with the Hausdorff metric

The number K_p,q, i.e., the number of (p, q) corridors of closed domains which are convex in the vertical direction, consist of elementary squares of the integral lattice, are situated within a rectangle of the size q × p, and completely cover the side of length p of this rectangle under projection is computed. The asymptotic (K_p,q/q²)^1/p → λ, as p, q → ∞, where λ = 0.3644255… is the maximum root of the equation₁F₁(-1/2 − 1/(16λ), 1/2, 1/(4λ)) = 0,₁F₁ being the confluence hypergeometric function, is established. These results allow us to compute the ε entropy of the space of continuous functions with the Hausdorff metric. Translated from Matematicheskie Zametki, Vol. 21, No. 1, pp. 39–50, January, 1977.     

NON-EXISTENCE OF POSITIVE ENTIRE SOLUTIONS FOR ELLIPTIC INEQUALITIES OF P-LAPLACIAN

In this paper,the nonexistence of positive entire solutions for div(|Du|^1-2Du)≥q(x)f(u),x∈R^N,is establisbed,where p＞1,DU=(D,u……dnu),qsR^N→(o,∞)and f2(0,∞)→(o,∞)are continuous functions.     

An answer to a question by J. Milnor

We consider two commuting automorphismsT ₁,T ₂ of the Lebesque space (M, M, μ) such thath _m,n=h(T ₁ ^m T ₂ ⁿ )<∞ whereh is the measure-theoretic entropy. Under additional assumptions we show the existence of the limits lim (1/m)h _m,n wherem→∞,n→∞,m/n→ω and ω is an irrational number.     

Nash Inequalities for Markov Processes in Dimension One

In this paper, we give characterizations of Nash inequalities for birth-death process and diffusion process on the line. As a by-product, we prove that for these processes, transience implies that the semigroups P(t) decay as ∥P(t)∥_1→∞≥Ct ₋₁. Sufficient conditions for general Markov chains are also obtained. Received November 29, 2000, Revised February 21, 2001, Accepted March 16, 2001     

Extension to R + n+1 of singular integrals and fractional integrals

In this paper, a natural R ₊ ⁿ⁺¹ extension of singular integrals, i.e.,T _κ:f→K*φ _t *t with K a standard C-Z kernel and ϕ usual one, is investigated. One of the main results is: Let (dμ, udx) ∈C₁ and u-Mw, w∈A_∞, then T_k is of type (L^p(udx), L^p(dμ)). As a related topic, a maximal operator is proved to be of type , where , provided (dμ, udx) ∈C₁ and u∈ A_∞. Supported by National Science Foundation of China     

Large derivatives,backward contraction and invariant densities for interval maps

In this paper, we study the dynamics of a smooth multimodal interval map f with non-flat critical points and all periodic points hyperbolic repelling. Assuming that |Dfⁿ(f(c))|→∞ as n→∞ holds for all critical points c, we show that f satisfies the so-called backward contracting property with an arbitrarily large constant, and that f has an invariant probability μ which is absolutely continuous with respect to Lebesgue measure and the density of μ belongs to L^p for all p<ℓ_max/(ℓ_max-1), where ℓ_max denotes the maximal critical order of f. In the appendix, we prove that various growth conditions on the derivatives along the critical orbits imply stronger backward contraction.     

Integral inequalities with weights for maxiamal operators

Let μ be a measure on the upper half-space R ₊ ⁿ⁺¹ , and v a weight onR ⁿ, we give a characterization for the pair (v, μ) such that ∥M(fv)∥_{L Θ (μ}) ⩽ c ∥f∥_{L Θ (μ}), where Φ is an N-function satisfying Δ₂ condition andMf(x,t), is the maximal function onR ₊ ⁿ⁺¹ , which was introduced by Ruiz, F. and Torrea, J.. Supported by NSFC.     

A question of Gol’dberg concerning entire functions with prescribed zeros

Let (z_j) be a sequence of complex numbers satisfying |z_j|→ ∞ asj → ∞ and denote by n(r) the number of z_j satisfying |z_j|≤ r. Suppose that lim inf_{r → ⇈} log n(r)/ logr > 0. Let ϕ be a positive, non-decreasing function satisfying ∫^∞ (ϕ(t)t logt)⁻¹ dt < ∞. It is proved that there exists an entire functionf whose zeros are the z_j such that log log M(r,f) = o((log n(r))²ϕ(log n(r))) asr → ∞ outside some exceptional set of finite logarithmic measure, and that the integral condition on ϕ is best possible here. These results answer a question by A. A. Gol’dberg.     

ON THE DIFFUSION PHENOMENON OF QUASILINEAR HYPERBOLICWAVES

Introduction1.1.ConsiderthefollowingquasilinearhyperbolicCauchyproblemwithlineardamping{:;;!OTt=-:i<:,;>>L06,(11)wherexER",t20,anda(.)isasmoothfunctionsatisfyinga(y)~1 O(lyl")aslyl-0,orEN.(1.2)Thepurposeofthispaperistoshowthat,atleastwhenn53,theasymptoticprofileofthesolutionu(x,t)of(l.1)isgivenbythesolutionv(x,t)ofthecorrespondingparabolicproblem{:;.t>ivj:相似文献   

Sharp forms of nevanlinna’s error terms

Let f(z) be a meromorphic function in the plane. If ψ(t)/t andp(t) are two positive, continuous and non-decreasing functions on [1,∞) with ∫ ₁ ^∞ dt/ψ(t) = ∞ and ∫ ₁ ^∞ dt/p(t) = ∞, then asr → ∞ outside a small exceptional set, provided that the divergence of the integral ∫ ₁ ^r dt/ψ(t) is slow enough. The same forms for the logarithmic derivative and for the ramification term are obtained. It is shown by example that the estimates are best possible. Author supported by Max-Planck-Gesellschaft Z.F.D.W and by NSFC.     

A condition for asymptotic finite-dimensionality of an operator semigroup

Let X be a Banach space and let T: X → X be a power bounded linear operator. Put X ₀ = {x ∈ X ∣ T ⁿ x → 0}. Assume given a compact set K ⊂ X such that lim inf _n→∞ ρ{T ⁿ x, K} ≤ η < 1 for every x ∈ X, ∥x∥ ≤ 1. If $\eta < \tfrac{1} {2} $\eta < \tfrac{1} {2} , then codim X ₀ < ∞. This is true in X reflexive for $\eta \in [\tfrac{1} {2},1) $\eta \in [\tfrac{1} {2},1) , but fails in the general case.     

Rearrangement invariant subspaces of Lorentz function spaces

The main result is that for 2≦q≦p<∞ the only subspaces of the Lorentz function spaceL _pq [0, 1] which are isomorphic to r.i. function spaces on [0, 1] are, up to equivalent renormings,L _pq [0, 1] andL ₂[0, 1].     

The dual method in critical point theory. Multiplicity results for indefinite functional

Sunto Si dimostra l'esistenza di soluzioni multiple per l'equazione Lu=A(u) ove L è un operatore indefinito ed autoaggiunto in uno spazio di Hilbert; A è un operatore variazionale, monotono, dispari che «cresce» in maniera opportuna più che linearmente per u . Dal suddetto risultato si deduce, tra l'altro, l'esistenza di soluzioni periodiche multiple per sistemi hamiltoniani ed equazioni delle onde non lineari sulla sfera n-dimensionale Sⁿ.

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Work partially supported by C.N.R. and M.P.I.     

Vague convergence of sums of independent random variables

A sequence (μ _n) of probability measures on the real line is said to converge vaguely to a measureμ if∫ fdμ _n →∫ fdμ for every continuous functionf withcompact support. In this paper one investigates problems analogous to the classical central limit problem under vague convergence. Let ‖μ‖ denote the total mass ofμ andδ ₀ denote the probability measure concentrated in the origin. For the theory of infinitesimal triangular arrays it is true in the present context, as it is in the classical one, that all obtainable limit laws are limits of sequences of infinitely divisible probability laws. However, unlike the classical situation, the class of infinitely divisible laws is not closed under vague convergence. It is shown that for every probability measureμ there is a closed interval [0,λ], [0,e ⁻¹] ⊂ [0,λ] ⊂ [0, 1], such thatβμ is attainable as a limit of infinitely divisible probability laws iffβ ε [0,λ]. In the independent identically distributed case, it is shown that if (x ₁ + ... +x _n)/a _n, a_n → ∞, converges vaguely toμ with 0<‖μ‖<1, thenμ=‖μ‖δ ₀. If furthermore the ratiosa _n+1/a _n are bounded above and below by positive numbers, thenL(x)=P[|X ₁|>x] is a slowly varying function ofx. Conversely, ifL(x) is slowly varying, then for everyβ ε (0, 1) one can choosea _n → ∞ so that the limit measure=βδ ₀. To the memory of Shlomo Horowitz This research was partially supported by the National Science Foundation.