Wavelet expansions of fractal measures

Let μ be a measure on ℝⁿ that satisfies the estimate μ(B _r(x))≤cr ^α for allx ∈ ℝⁿ and allr ≤ 1 (B _r(x) denotes the ball of radius r centered atx. Let ϕ _j,k ^(ɛ) (x)=2 ^nj2ϕ^(ɛ)(2 ^j x-k) be a wavelet basis forj ∈ ℤ, κ ∈ ℤⁿ, and ∈ ∈E, a finite set, and letP _j (T)=Σ_ɛ,k <T,ϕ _j,k ^(ɛ) >ϕ _j,k ^(ɛ) denote the associated projection operators at levelj (T is a suitable measure or distribution). Iff ∈Ls ^p(dμ) for 1 ≤p ≤ ∞, we show thatP _j(f dμ) ∈ L^p(dx) and ||P _j (fdμ)||L _p(dx)≤c2 ^{j((n-α)/p′))}||f||L _p(dμ) for allj ≥ 0. We also obtain estimates for the limsup and liminf of ||P _j (fdμ)||L _p(dx) under more restrictive hypotheses. Communicated by Guido Weiss     

Finite quasihypermetric spaces

Let (X, d) be a compact metric space and let (X) denote the space of all finite signed Borel measures on X. Define I: (X) → ℝ by I(μ) = ∫ _X ∫ _X d(x, y)dμ(x)dμ(y), and set M(X) = sup I(μ), where μ ranges over the collection of measures in (X) of total mass 1. The space (X, d) is quasihypermetric if I(μ) ≦ 0 for all measures μ in (X) of total mass 0 and is strictly quasihypermetric if in addition the equality I(μ) = 0 holds amongst measures μ of mass 0 only for the zero measure. This paper explores the constant M(X) and other geometric aspects of X in the case when the space X is finite, focusing first on the significance of the maximal strictly quasihypermetric subspaces of a given finite quasihypermetric space and second on the class of finite metric spaces which are L ¹-embeddable. While most of the results are for finite spaces, several apply also in the general compact case. The analysis builds upon earlier more general work of the authors [11] [13].      

Leading coefficients of Kazhdan–Lusztig polynomials for Deodhar elements

We show that the leading coefficient of the Kazhdan–Lusztig polynomial P _x,w (q) known as μ(x,w) is always either 0 or 1 when w is a Deodhar element of a finite Weyl group. The Deodhar elements have previously been characterized using pattern avoidance in Billey and Warrington (J. Algebraic Combin. 13(2):111–136, [2001]) and Billey and Jones (Ann. Comb. [2008], to appear). In type A, these elements are precisely the 321-hexagon avoiding permutations. Using Deodhar’s algorithm (Deodhar in Geom. Dedicata 63(1):95–119, [1990]), we provide some combinatorial criteria to determine when μ(x,w)=1 for such permutations w. The author received support from NSF grants DMS-9983797 and DMS-0636297.     

Sumsets in difference sets

We study some properties of sets of differences of dense sets in ℤ² and ℤ³ and their interplay with Bohr neighbourhoods in ℤ. We obtain, inter alia, the following results.

Higher order Riesz transforms associated with Bessel operators

In this paper we investigate Riesz transforms R _μ ^(k) of order k≥1 related to the Bessel operator Δ_μ f(x)=-f”(x)-((2μ+1)/x)f’(x) and extend the results of Muckenhoupt and Stein for the conjugate Hankel transform (a Riesz transform of order one). We obtain that for every k≥1, R _μ ^(k) is a principal value operator of strong type (p,p), p∈(1,∞), and weak type (1,1) with respect to the measure dλ(x)=x ^2μ+1  dx in (0,∞). We also characterize the class of weights ω on (0,∞) for which R _μ ^(k) maps L ^p (ω) into itself and L ¹(ω) into L ^1,∞(ω) boundedly. This class of weights is wider than the Muckenhoupt class of weights for the doubling measure dλ. These weighted results extend the ones obtained by Andersen and Kerman.     

Balanced Convex Partitions of Measures in ℝ2

 Let n≥2 be an integer and let μ₁ and μ₂ be measures in ℝ² such that each μ _i is absolutely continuous with respect to the Lebesgue measure and μ₁(ℝ²)=μ₂(ℝ²)=n. Let u≠0 be a vector on the plane. We show that if μ₁(B)=μ₂(B)=n for some bounded domain B, then there exist positive integers n ₁,n ₂ with n ₁+n ₂=n and disjoint open half-planes D ₁,D ₂ such that , μ₁(D ₁)=μ₂(D ₁)=n ₁ and μ₁(D ₂)=μ₂(D ₂)=n ₂; or there exist positive integers n ₁,n ₂,n ₃ with n ₁+n ₂+n ₃=n and disjoint open convex domains D ₁,D ₂,D ₃ such that , μ₁(D ₁)=μ₂(D ₁)=n ₁, μ₁(D ₂)= μ₂(D ₂)=n ₂, μ₁(D ₃)=μ₂(D ₃)=n ₃ and such that the ray is parallel to u. We also show a similar result for partitioning of point sets on the plane. Received: November 24, 1999 Final version received: February 9, 2001     

The Quasi-Canonical Solution Operator to $${\bar \partial }$$ Restricted to the Fock-Space

We consider the solution operator S: ℱ_μ,(p,q) → L ²(μ)_{(p, q)} to the -operator restricted to forms with coefficients in ℱ_μ = {f: f is entire and ∫_ℂn |f(z)|² dμ(z) < ∞}. Here ℱ_μ,(p,q) denotes (p,q)-forms with coefficients in ℱ_μ, L ²(μ) is the corresponding L ²-space and μ is a suitable rotation-invariant absolutely continuous finite measure. We will develop a general solution formula S to . This solution operator will have the property Sv ⊥ ℱ_(p,q) ∀v ∈ ℱ_(p,q+1). As an application of the solution formula we will be able to characterize compactness of the solution operator in terms of compactness of commutators of Toeplitz-operators : ℱ_μ → L ²(μ).     

The connection between self‐associated two‐dimensional vector functionals and third degree forms

We deal with the 2‐orthogonal, 2‐symmetric self‐associated sequence (2‐orthogonal Tchebychev polynomials) and its cubic components. We prove that all the forms (linear functionals) arising are third degree forms. Therefore, an introduction to third degree forms is provided. We look for the connection between these components which are 2‐orthogonal with respect to the functional vector ^t(w₀{^μ},w₁ ^μ) and orthogonal sequences with respect to w₀ ^μ, μ=0,1,2. Associated forms w₀ ^μ)¹) and their inverse w₀ ^μ)^-1 are also studied through the symmetrized w₀}₀ ^μ, μ=0,1,2. Further, we give integral representations for some of these forms. This revised version was published online in June 2006 with corrections to the Cover Date.     

Some estimates related to fractal measures and Laplacians on manifolds

Let Δ be the Laplace-Beltrami operator on ann dimensional completeC ^∞ manifoldM In this paper we establish an estimate ofe ^tΔ (dμ) valid for allt>0 wheredμ is a locally uniformly α dimensional measure onM 0≤α≤n The result is used to study the mapping properties of (I-tΔ)^-β considered as an operator fromL ^p (M dμ) toL ^p (M dx) wheredx is the Riemannian measure onM β>(n−α)/2p′ 1/p+1/p′=1 1≤p≤∞     

Radial solutions to the wave equation

We study some boundedness properties of radial solutions to the Cauchy problem associated to the wave equation (∂ _t ²-▵ _x )u(t,x)=0 and meanwhile we give a new proof of the solution formula. Received: July 7, 1998?Published online: March 19, 2002     

Micro-Local Analysis with Fourier Lebesgue Spaces. Part?I

Let ω,ω ₀ be appropriate weight functions and q∈[1,∞]. We introduce the wave-front set, WF_FL^q(w)(f)\mathrm{WF}_{\mathcal{F}L^{q}_{(\omega)}}(f) of f ? S￠f\in \mathcal{S}' with respect to weighted Fourier Lebesgue space FL^q_(w)\mathcal{F}L^{q}_{(\omega )}. We prove that usual mapping properties for pseudo-differential operators Op (a) with symbols a in S^(w₀)_r,0S^{(\omega _{0})}_{\rho ,0} hold for such wave-front sets. Especially we prove that

On some properties of Musielak-Orlicz spaces and their subspaces of order continuous elements

Some criterions in order thatl ¹ embeds complementably inE ^Φ(μ) and inL ^Φ(μ) are given. It is also proved that every idealL inL ^Φ(μ) such thatI _Φ(x/‖x‖Φ)=1 for anyxεL/{0} is contained inE ^Φ(μ).     

Un teorema di Radon-Nikodym per funzioni d’insieme subadditive

Riassunto In questo articolo si danno delle condizioni necessarie e sufficienti affinchè per una fissata coppia di funzioni d’insieme ν, μ crescenti esista una funzionef tale che ν=∫fdμ. Si ottiene cosi una proposizione comprendente il teorema di R-N. classico e dei teoremi di R-N., presentati da altri autori, riguardanti le funzioni d’insieme finitamente additive e le funzioni d’insieme subadditive e continue per successioni crescenti.

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Résumé Soient ν, μ:A→[0,+∞) deux fonctions d’ensemble croissantes sur une σ-algèbre d’ensemblesA⊂T(X), telles que pour chaqueA∈A avec ν(A)=μ(A)=0 on a l’égalité μ(A)=μ(A∪S) ∀S∈A (c’est le cas des fonctions sousadditives!). Dans cet article on démontre qu’il existe une fonctionf A-measurable telle que ν=∫fdμ si et seulement si pour chaquer∈(0, + ∞) il y a un ensembleA _r∈A qui vérifie les trois conditions suivantes: (1) ,B∈A avecB⊂A; (2) A (3) limν(A _r)=0. On déduit ainsi une proposition qui a été donnée parI. Forana: ?Si ν, μ sont simplement additives, il existe une fonctionf telle que ν=∫fdμ si et seulement si ν≪μ et la fonction d’ensemble additive a une decomposition de Hahn pour chaquer∈(0, + ∞), c’est-á-dire il y aA _r∈A tel que ?.

     

Asymptotic behavior of eigenvalues of two-parameter nonlinear Sturm-Liouville problems

The nonlinear two-parameter Sturm-Liouville problemu ^"+μg(u)=λf(u) is studied for μ, λ>0. By using Ljusternik-Schnirelman theory on the general level set developed by Zeidler, we shall show the existence of ann-th variational eigenvalue λ=λ_n(μ). Furthermore, for specialf andg, the asymptotic formula of λ₁(μ)) as μ→∞ is established.     

Life span of solutions of a weakly coupled parabolic system

We consider the Cauchy problem for the weakly coupled parabolic system ∂ _t w _λ−Δ w _λ = F(w _λ) in R ^N , where λ > 0, w _λ = (u _λ, v _λ), F(w _λ) = (v _λ ^p , u _λ ^q ) for some p, q ≥ 1, pq > 1, and w_l(0) = (lj₁, l^\fracq+1p+1j₂)w_{\lambda}(0) = ({\lambda}{\varphi}_1, {\lambda}^{\frac{q+1}{p+1}}{\varphi}_2), for some nonnegative functions φ₁, φ₂ ?\in C ₀(R ^N ). If (p, q) is sub-critical or either φ₁ or φ₂ has slow decay at ∞, w _λ blows up for all λ > 0. Under these conditions, we study the blowup of w _λ for λ small.     

Linear functional differential equations of neutral type asymptotically autonomous

Summary It is studied the relationship between the solutions of the linear functional differential equations(1) (d/dx) D(x_t)=L(x_t) and its perturbed equation(2) [(d/dx) D(x_t)−G(t, x_t)]= =L(x_t)+F(t, x_t) and is proved, under certain hypotheses which will be precised bellow that, if μ is a simple characteristic root of(1), then there exist a σ > 0 and a non zero vector a such that system(2) has a solution satisfying where δ(t)=αd{F(t, ϕ_μ)+μG(t, ϕ_μ)+F(t, X₀G(t, ϕ_μ))}, ϕ_μ(θ)=c·exp (μθ), −r⩾θ⩾0 and α, d, X₀ are given constants. Entrata in Redazione il 5 gennaio 1972.     

Value distribution of the Painlevé Transcendents

We consider the solutions of the First Painlevé Differential Equationω″=z+6w ², commonly known as First Painlevé Transcendents. Our main results are the sharp order estimate λ(w)≤5/2, actually an equality, and sharp estimates for the spherical derivatives ofw andf(z)=z ⁻¹ w(z ²), respectively:w#(z)=O(|z|^3/4) andf#(z)=O(|z|^3/2). We also determine in some detail the local asymptotic distribution of poles, zeros anda-points. The methods also apply to Painlevé’s Equations II and IV.     

Triangle-free distance-regular graphs

Let Γ denote a distance-regular graph with diameter d≥3. By a parallelogram of length 3, we mean a 4-tuple xyzw consisting of vertices of Γ such that ∂(x,y)=∂(z,w)=1, ∂(x,z)=3, and ∂(x,w)=∂(y,w)=∂(y,z)=2, where ∂ denotes the path-length distance function. Assume that Γ has intersection numbers a ₁=0 and a ₂≠0. We prove that the following (i) and (ii) are equivalent. (i) Γ is Q-polynomial and contains no parallelograms of length 3; (ii) Γ has classical parameters (d,b,α,β) with b<−1. Furthermore, suppose that (i) and (ii) hold. We show that each of b(b+1)²(b+2)/c ₂, (b−2)(b−1)b(b+1)/(2+2b−c ₂) is an integer and that c ₂≤b(b+1). This upper bound for c ₂ is optimal, since the Hermitian forms graph Her₂(d) is a triangle-free distance-regular graph that satisfies c ₂=b(b+1). Work partially supported by the National Science Council of Taiwan, R.O.C.     

环域上一类拟线性椭圆型方程的正径向解的存在性

The existence of positive radial solutions of the equation -din( |Du|p-2Du)=f(u) is studied in annular domains in Rn,n≥2. It is proved that if f(0)≥0, f is somewherenegative in (0,∞), limu→0^ f‘ (u)=0 and limu→∞ (f(u)/u^p-1)=∞, then there is alarge positive radial solution on all annuli. If f(0)≤0 and satisfies certain conditions, then the equation has no radial solution if the annuli are too wide.     

On the asymptotic maximal density of a set avoiding solutions to linear equations modulo a prime

Given a finite family F\mathcal{F} of linear forms with integer coefficients, and a compact abelian group G, an F\mathcal{F}-free set in G is a measurable set which does not contain solutions to any equation L(x)=0 for L in F\mathcal{F}. We denote by d_F(G)d_{\mathcal{F}}(G) the supremum of μ(A) over F\mathcal{F}-free sets A⊂G, where μ is the normalized Haar measure on G. Our main result is that, for any such collection F\mathcal{F} of forms in at least three variables, the sequence d_F(\mathbb Z_p)d_{\mathcal{F}}({\mathbb {Z}}_{p}) converges to d_F(\mathbb R/\mathbb Z)d_{\mathcal{F}}({\mathbb {R}}/{\mathbb {Z}}) as p→∞ over primes. This answers an analogue for ℤ _p of a question that Ruzsa raised about sets of integers.