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     

On Projections of Semi-Algebraic Sets Defined by Few Quadratic Inequalities

Let S⊂ℝ ^k+m be a compact semi-algebraic set defined by P ₁≥0,…,P _ℓ ≥0, where P _i ∈ℝ[X ₁,…,X _k ,Y ₁,…,Y _m ], and deg (P _i )≤2, 1≤i≤ℓ. Let π denote the standard projection from ℝ ^k+m onto ℝ ^m . We prove that for any q>0, the sum of the first q Betti numbers of π(S) is bounded by (k+m) ^{O(q ℓ)}. We also present an algorithm for computing the first q Betti numbers of π(S), whose complexity is . For fixed q and ℓ, both the bounds are polynomial in k+m. The author was supported in part by an NSF Career Award 0133597 and a Sloan Foundation Fellowship.     

Solutions in Sobolev spaces of vector refinement equations with a general dilation matrix

In this paper, we present a necessary and sufficient condition for the existence of solutions in a Sobolev space W_p^k(ℝ^s) (1≤p≤∞) to a vector refinement equation with a general dilation matrix. The criterion is constructive and can be implemented. Rate of convergence of vector cascade algorithms in a Sobolev space W_p^k(ℝ^s) will be investigated. When the dilation matrix is isotropic, a characterization will be given for the L_p (1≤p≤∞) critical smoothness exponent of a refinable function vector without the assumption of stability on the refinable function vector. As a consequence, we show that if a compactly supported function vector φ∈L_p(ℝ^s) (φ∈C(ℝ^s) when p=∞) satisfies a refinement equation with a finitely supported matrix mask, then all the components of φ must belong to a Lipschitz space Lip(ν,L_p(ℝ^s)) for some ν>0. This paper generalizes the results in R.Q. Jia, K.S. Lau and D.X. Zhou (J. Fourier Anal. Appl. 7 (2001) 143–167) in the univariate setting to the multivariate setting. Dedicated to Professor Charles A. Micchelli on the occasion of his 60th birthday Mathematics subject classifications (2000) 42C20, 41A25, 39B12. Research was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC Canada) under Grant G121210654.     

On the stability of solutions to quadratic programming problems

We consider the parametric programming problem (Q _p ) of minimizing the quadratic function f(x,p):=x ^T Ax+b ^T x subject to the constraint Cx≤d, where x∈ℝ ⁿ , A∈ℝ ^n×n , b∈ℝ ⁿ , C∈ℝ ^m×n , d∈ℝ ^m , and p:=(A,b,C,d) is the parameter. Here, the matrix A is not assumed to be positive semidefinite. The set of the global minimizers and the set of the local minimizers to (Q _p ) are denoted by M(p) and M ^loc (p), respectively. It is proved that if the point-to-set mapping M ^loc (·) is lower semicontinuous at p then M ^loc (p) is a nonempty set which consists of at most ? _m,n points, where ? _m,n = is the maximal cardinality of the antichains of distinct subsets of {1,2,...,m} which have at most n elements. It is proved also that the lower semicontinuity of M(·) at p implies that M(p) is a singleton. Under some regularity assumption, these necessary conditions become the sufficient ones. Received: November 5, 1997 / Accepted: September 12, 2000?Published online November 17, 2000     

The Frattini Subalgebra of Restricted Lie Superalgebras

In the present paper, we study the Frattini subalgebra of a restricted Lie superalgebra （L, [p]）. We show first that if L = A1 ＋ A2 ＋… ＋An, then Фp（L） = Фp（A1） ＋Фp（A2） ＋…＋Фp（An）, where each Ai is a p-ideal of L. We then obtain two results： F（L） = Ф（L） = J（L） = L if and only if L is nilpotent; Fp（L） and F（L） are nilpotent ideals of L if L is solvable. In addition, necessary and sufficient conditions are found for Фp-free restricted Lie superalgebras. Finally, we discuss the relationships of E-p-restricted Lie superalgebras and E-restricted Lie superalgebras.     

Points surrounding the origin

Suppose d > 2, n > d+1, and we have a set P of n points in d-dimensional Euclidean space. Then P contains a subset Q of d points such that for any p ∈ P, the convex hull of Q∪{p} does not contain the origin in its interior. We also show that for non-empty, finite point sets A ₁, ..., A _d+1 in ℝ ^d , if the origin is contained in the convex hull of A _i ∪ A _j for all 1≤i<j≤d+1, then there is a simplex S containing the origin such that |S∩A _i |=1 for every 1≤i≤d+1. This is a generalization of Bárány’s colored Carathéodory theorem, and in a dual version, it gives a spherical version of Lovász’ colored Helly theorem. Dedicated to Imre Bárány, Gábor Fejes Tóth, László Lovász, and Endre Makai on the occasion of their sixtieth birthdays. Supported by the Norwegian research council project number: 166618, and BK 21 Project, KAIST. Part of the research was conducted while visiting the Courant Institute of Mathematical Sciences. Supported by NSF Grant CCF-05-14079, and by grants from NSA, PSC-CUNY, the Hungarian Research Foundation OTKA, and BSF.     

Dual space,carleson measure and sequence interpolation onA p (ϕ)(O<p<1)

LetD={z∈Σ:|z|<1} and ϕ be a normal function on [0,1). Forp∈(0,1) such a function ϕ is used to define a Bergman spaceA ^p (ϕ) onD with weight ϕ ^p (|·|)/(1-|·|²). In this paper, the dual space ofA ^p (ϕ) is given, four characteristics of Carleson measure onA ^p (ϕ) are obtained. Moreover, as an application, three sequence interpolation theorems inA ^p (ϕ) are derived. Supported by the Doctoral Program Foundation of Institute of Higher Education, P.R. China.     

SOME MULTIPLIER THEOREMS FOR ANISOTROPIC HARDY SPACES ——In Memory of Professor Yongsheng Sun

Let A be a symmetric expansive matrix and Hp(Rn) be the anisotropic Hardy space associated with A. For a function m in L∞(Rn), an appropriately chosen function η in Cc∞(Rn) and j ∈ Z define mj(ξ) = m(Ajξ)η(ξ). The authors show that if 0 ＜ p ＜ 1 and (m)j belongs to the anisotropic nonhomogeneous Herz space K11/p-1,p(Rn), then m is a Fourier multiplier from Hp(Rn) to Lp(Rn). For p = 1, a similar result is obtained if the space K10,1(Rn) is replaced by a slightly smaller space K(w).Moreover, the authors show that if 0 ＜ p ≤ 1 and if the sequence {(mj)V} belongs to a certain mixednorm space, depending on p, then m is also a Fourier multiplier from Hp(Rn) to Lp(Rn).     

The Dichotomy Between Traces on d-sets F in R^n and the Density of D（R^n／Г） in Function Spaces

A space Apq^s （R^n） with A ： B or A = F and s ∈R, 0 〈 p, q 〈 ∞ either has a trace in Lp（Г）, where Г is a compact d-set in R^n with 0 〈 d 〈 n, or D（R^n／Г） is dense in it. Related dichotomy numbers are introduced and calculated.     

Compactness of Commutators of Riesz Potential on Morrey Spaces

In this paper, the authors give a characterization of the (L ^{p, λ} , L ^{q, λ} )-compactness for the Riesz potential commutator [b,I _α ]. More precisely, the authors prove that the commutator [b,I _α ] is a compact operator from the Morrey space L ^{p, λ} (ℝ ⁿ ) to L ^{q, λ} (ℝ ⁿ ) if and only if b ∈ VMO(ℝ ⁿ ), the BMO-closure of . The research was supported by NSF of China (Grant: 10571015, 10826046) and SRFDP of China (Grant: 20050027025).     

Generalizations of spectrally multiplicative surjections between uniform algebras

Let $ A $ A and ℬ be unital semisimple commutative Banach algebras. It is shown that if surjections S,T: $ A $ A → ℬ with S(1)=T(1)= 1 and α ∈ ℂ \ {0} satisfy r(S(a)T(b) − α)= r(ab− α) for all a,b ∈ $ A $ A , then S=T and S is a real algebra isomorphism, where r(a) is the spectral radius of a. Let I be a nonempty set, A and B be uniform algebras. Let ρ, τ: I → A and S,T: I → B be maps satisfying σ _π (S(p)T(q)) ⊂ σ _π (ρ(p) τ(q)) for all p,q ∈ I, where σ _π (f) is the peripheral spectrum of f. Suppose that the ranges ρ(I), τ(I) ⊂ A and S(I),T(I) ⊂ B are closed under multiplication in a sense, and contain peaking functions “enough”. There exists a homeomorphism ϕ: Ch(B)→Ch(A) such that S(p)(y)= ρ(p)(ϕ(y)) and T(p)(y)= τ(p)(ϕ(y)) for every p ∈ I and y ∈ Ch(B), where Ch(A) is the Choquet boundary of A.     

Sudakov-type minoration for gaussian chaos processes

Consider a setA of symmetricn×n matricesa=(a _i,j) _i,j≤n . Consider an independent sequence (g _i) _i≤n of standard normal random variables, and letM=Esup_a∈A|Σ_i,j⪯na_i,jg_ig_j|. Denote byN ₂(A, α) (resp.N _t(A, α)) the smallest number of balls of radiusα for thel ₂ norm ofR ^{n 2} (resp. the operator norm) needed to coverA. Then for a universal constantK we haveα(logN ₂(A, α))^1/4≤KM. This inequality is best possible. We also show that forδ≥0, there exists a constantK(δ) such thatα(logN _t≤K(δ)M. Work partially supported by an N.S.F. grant.     

Littlewood–Paley theorem on spaces <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis>(<Emphasis Type="Italic">t</Emphasis>)</Superscript>(ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>)

We point out that if the Hardy–Littlewood maximal operator is bounded on the space L ^p(t)(ℝ), 1 < a ≤ p(t) ≤ b < ∞, t ∈ ℝ, then the well-known characterization of the spaces L ^p (ℝ), 1 < p < ∞, by the Littlewood–Paley theory extends to the space L ^p(t)(ℝ). We show that, for n > 1 , the Littlewood–Paley operator is bounded on L ^p(t) (ℝ ⁿ ), 1 < a ≤ p(t) ≤ b < ∞, t ∈ ℝ ⁿ , if and only if p(t) = const. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 12, pp. 1709–1715, December, 2008.     

On the multiplicity of the eigenvalues of a graph

Given a graph G with characteristic polynomial ϕ(t), we consider the ML-decomposition ϕ(t) = q ₁(t)q ₂(t)² ... q _m (t)^m, where each q _i (t) is an integral polynomial and the roots of ϕ(t) with multiplicity j are exactly the roots of q _j (t). We give an algorithm to construct the polynomials q _i (t) and describe some relations of their coefficients with other combinatorial invariants of G. In particular, we get new bounds for the energy E(G) = |λ_i| of G, where λ₁, λ₂, ..., λ_n are the eigenvalues of G (with multiplicity). Most of the results are proved for the more general situation of a Hermitian matrix whose characteristic polynomial has integral coefficients. This work was done during a visit of the second named author to UNAM.     

The nearness problems for symmetric centrosymmetric with a special submatrix constraint

We say that X=[x_ij]_i,j=1ⁿX=[x_{ij}]_{i,j=1}^n is symmetric centrosymmetric if x _ij  = x _ji and x _{n − j + 1,n − i + 1}, 1 ≤ i,j ≤ n. In this paper we present an efficient algorithm for minimizing ||AXA ^T  − B|| where ||·|| is the Frobenius norm, A ∈ ℝ ^m×n , B ∈ ℝ ^m×m and X ∈ ℝ ^n×n is symmetric centrosymmetric with a specified central submatrix [x _ij ] _{p ≤ i,j ≤ n − p} . Our algorithm produces a suitable X such that AXA ^T  = B in finitely many steps, if such an X exists. We show that the algorithm is stable any case, and we give results of numerical experiments that support this claim.     

Spectral invariance for pseudodifferential operators on weighted function spaces

The spectrum of each symmetric ψ DO of the symbol class S⁰ _{1, γ}, 0≤γ<1, acting on B³ _p,q(w(x)) and F³ _p,q(w(x)), is independent of the choice ofs ∈ℝ, 0<p≤∞ (p<∞ in the F-case), 0<q≤∞ and the weight w(x)∈W.     

The Helmholtz decomposition of weightedLq-spaces for Muckenhoupt weights

We consider weights of Muckenhoupt classA _q, 1<q<∞. For a bounded Lipschitz domain Ω⊂ℝⁿ we prove a compact embedding and a Poincaré inequality in weighted Sobolev spaces. These technical tools allow us to solve the weak Neumann problem for the Laplace equation in weighted spaces on ℝⁿ, ℝⁿ ₊, on bounded and on exterior domains Ω with boundary of classC ¹, which will yield the Helmholtz decomposition ofL _ω ^q(Ω)ⁿ for general ω∈A _q. This is done by transferring the method of Simader and Sohr [4] to the weighted case. Our result generalizes a result of Farwig and Sohr [2] where the Helmholtz decomposition ofL _ω ^p(Ω)ⁿ is proved for an exterior domain and weights of Muckenhoupt class without singularities or degeneracies in a neighbourhood of ϖΩ.

---

Sunto In questo lavoro consideriamo dei pesi della classe di MuckenhouptA _q, 1<q<∞. Per un dominio limitato lipschitziano Ω⊂ℝⁿ, dimostriamo una immersione compatta ed una disuguaglianza di Poincaré in spazi di Sobolev con peso. Questa tecnica ci consente di risolvere il problema debole di Neumann per l’equazione di Laplace in spazi pesati in ℝⁿ, ℝⁿ ₊ in domini limitati ed in domini esterni con frontiera di classeC ¹, che conduce alla decomposizione di Helmholtz diL _ω ^q(Ω)ⁿ per un qualsiasi ω∈A _q. Il risultato è ottenuto trasferendo il metodo di Simader e Sohr [4] al caso pesato. Quello qui presente estende un risultato di Farwig e Sohr [2] dove la decomposizione di Helmholtz diL _ω ^q(Ω)ⁿ è dimostrata per domini esterni e pesi della classe di Muckenhoupt privi di singolarità in un intorno di ϖΩ.

     

Kernels of Toeplitz operators,smooth functions,and Bernstein type inequalities

Let ϕ be a unimodular function on the unit circle and let K_p(ϕ) denote the kernel of the Toeplitz operator T_ϕ in the Hardy space H^p, p≥1; . Suppose K_p(ϕ)≠{0}. The problem is to find out how the smoothness of the symbol ϕ influences the boundary smoothness of functions in K_p(ϕ). One of the main results is as follows. Theorem 1 Let 1<p, q<+∞, 1<r≤+∞, q⁻¹=p⁻¹+r⁻¹. Suppose |ϕ|≡1 on and ϕ∈W _r ¹ (i.e., ). Then K_p(ϕ)⊂W _q ¹ . Moreover, for any f∈K_p(ϕ) we have ‖f′‖_q≤c(p, r)‖ϕ′‖_r ‖f‖. Bibliography: 19 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 201, 1992, pp. 5–21. Translated by K. M. D'yakonov.     

Systems of equations driven by stable processes

Let Z^j_t, j = 1, . . . , d, be independent one-dimensional symmetric stable processes of index α ∈ (0,2). We consider the system of stochastic differential equations where the matrix A(x)=(A_ij(x))1≤ i, j ≤ d is continuous and bounded in x and nondegenerate for each x. We prove existence and uniqueness of a weak solution to this system. The approach of this paper uses the martingale problem method. For this, we establish some estimates for pseudodifferential operators with singular state-dependent symbols. Let λ₂ > λ₁ > 0. We show that for any two vectors a, b∈ ℝ_d with |a|, |b| ∈ (λ₁, λ₂) and p sufficiently large, the L^p-norm of the operator whose Fourier multiplier is (|u · a|^α - |u · b|^α) / ∑_j=1^d |u_i|^α is bounded by a constant multiple of |a−b|^θ for some θ > 0, where u=(u₁ , . . . , u_d) ∈ ℝ^d. We deduce from this the L^p-boundedness of pseudodifferential operators with symbols of the form ψ(x,u)=|u · a(x)|^α / ∑_j=1^d |u_i|^α, where u=(u₁,...,u_d) and a is a continuous function on ℝ^d with |a(x)|∈ (λ₁, λ₂) for all x∈ ℝ^d. Research partially supported by NSF grant DMS-0244737. Research partially supported by NSF grant DMS-0303310.     

Hausdorff measures of different dimensions are not Borel isomorphic

We show that Hausdorff measures of different dimensions are not Borel isomorphic; that is, the measure spaces (ℝ, B, H ^s ) and (ℝ, B, H ^t ) are not isomorphic if s ≠ t, s, t ∈ [0, 1], where B is the σ-algebra of Borel subsets of ℝ and H ^d is the d-dimensional Hausdorff measure. This answers a question of B. Weiss and D. Preiss. To prove our result, we apply a random construction and show that for every Borel function ƒ: ℝ → ℝ and for every d ∈ [0, 1] there exists a compact set C of Hausdorff dimension d such that ƒ(C) has Hausdorff dimension ≤ d. We also prove this statement in a more general form: If A ⊂ ℝⁿ is Borel and ƒ: A → ℝ^m is Borel measurable, then for every d ∈ [0, 1] there exists a Borel set B ⊂ A such that dim B = d·dim A and dim ƒ(B) ≤ d·dim ƒ (A). Partially supported by the Hungarian Scientific Research Fund grant no. T 49786.