Local Comparability of Measures,Averaging and Maximal Averaging Operators

We study the boundedness properties of averaging and maximal averaging operators, under the following local comparability condition for measures: Intersecting balls of the same radius have comparable sizes. In geometrically doubling spaces, this property yields the weak type (1,1) of the uncentered maximal operator. We explore when local comparability implies doubling, and when it is more general. We also study the concrete case of the standard gaussian measure, where this property fails, but nevertheless averaging operators are uniformly bounded, with respect to the radius, in L ¹. However, such bounds grow exponentially fast with the dimension.     

Distances in orientations of graphs

We prove that there is a function h(k) such that every undirected graph G admits an orientation H with the following property: If an edge uv belongs to a cycle of length k in G, then uv or vu belongs to a directed cycle of length at most h(k) in H. Next, we show that every undirected bridgeless graph of radius r admits an orientation of radius at most r² + r, and this bound is best possible. We consider the same problem with radius replaced by diameter. Finilly, we show that the problem of deciding whether an undirected graph admits an orientation of diameter (resp. radius) 2 belongs to a class of problems called NP-hard.     

Boundedness of paraproduct operators on RD-spaces

Let(X,d,μ) be an RD-space with "dimension" n,namely,a space of homogeneous type in the sense of Coifman and Weiss satisfying a certain reverse doubling condition.Using the Calder'on reproducing formula,the authors hereby establish boundedness for paraproduct operators from the product of Hardy spaces H p(X) × H q(X) to the Hardy space H r(X),where p,q,r ∈(n/(n + 1),∞) satisfy 1/p + 1/q = 1/r.Certain endpoint estimates are also obtained.In view of the lack of the Fourier transform in this setting,the proofs are based on the derivation of appropriate kernel estimates.     

In Metric-measure Spaces Sobolev Embedding is Equivalent to a Lower Bound for the Measure

We study Sobolev inequalities on doubling metric measure spaces. We investigate the relation between Sobolev embeddings and lower bound for measure. In particular, we prove that if the Sobolev inequality holds, then the measure μ satisfies the lower bound, i.e. there exists b such that μ(B(x,r))≥b r ^α for r∈(0,1] and any point x from metric space.     

Hardy Spaces, Regularized BMO Spaces and the Boundedness of Calderón–Zygmund Operators on Non-homogeneous Spaces

One defines a non-homogeneous space (X,μ) as a metric space equipped with a non-doubling measure μ so that the volume of the ball with center x, radius r has an upper bound of the form r ⁿ for some n>0. The aim of this paper is to study the boundedness of Calderón–Zygmund singular integral operators T on various function spaces on (X,μ) such as the Hardy spaces, the L ^p spaces, and the regularized BMO spaces. This article thus extends the work of X. Tolsa (Math. Ann. 319:89–149, 2011) on the non-homogeneous space (? ⁿ ,μ) to the setting of a general non-homogeneous space (X,μ). Our framework of the non-homogeneous space (X,μ) is similar to that of Hytönen (2011) and we are able to obtain quite a few properties similar to those of Calderón–Zygmund operators on doubling spaces such as the weak type (1,1) estimate, boundedness from Hardy space into L ¹, boundedness from L ^∞ into the regularized BMO, and an interpolation theorem. Furthermore, we prove that the dual space of the Hardy space is the regularized BMO space, obtain a Calderón–Zygmund decomposition on the non-homogeneous space (X,μ), and use this decomposition to show the boundedness of the maximal operators in the form of a Cotlar inequality as well as the boundedness of commutators of Calderón–Zygmund operators and BMO functions.     

Tight sets and m-ovoids of finite polar spaces

An intriguing set of points of a generalised quadrangle was introduced in [J. Bamberg, M. Law, T. Penttila, Tight sets and m-ovoids of generalised quadrangles, Combinatorica, in press] as a unification of the pre-existing notions of tight set and m-ovoid. It was shown in [J. Bamberg, M. Law, T. Penttila, Tight sets and m-ovoids of generalised quadrangles, Combinatorica, in press] that every intriguing set of points in a finite generalised quadrangle is a tight set or an m-ovoid (for some m). Moreover, it was shown that an m-ovoid and an i-tight set of a common generalised quadrangle intersect in mi points. These results yielded new proofs of old results, and in this paper, we study the natural analogue of intriguing sets in finite polar spaces of higher rank. In particular, we use the techniques developed in this paper to give an alternative proof of a result of Thas [J.A. Thas, Ovoids and spreads of finite classical polar spaces, Geom. Dedicata 10 (1-4) (1981) 135-143] that there are no ovoids of H(2r,q²), Q⁻(2r+1,q), and W(2r−1,q) for r>2. We also strengthen a result of Drudge on the non-existence of tight sets in W(2r−1,q), H(2r+1,q²), and Q⁺(2r+1,q), and we give a new proof of a result of De Winter, Luyckx, and Thas [S. De Winter, J.A. Thas, SPG-reguli satisfying the polar property and a new semipartial geometry, Des. Codes Cryptogr. 32 (1-3) (2004) 153-166; D. Luyckx, m-Systems of finite classical polar spaces, PhD thesis, The University of Ghent, 2002] that an m-system of W(4m+3,q) or Q⁻(4m+3,q) is a pseudo-ovoid of the ambient projective space.     

Rearrangement invariance and relations among measures of smoothness

Relations between ?? ^r (f,t) _B and ?? ^r+1(f,t) _B of the sharp Marchaud and sharp lower estimate-type are shown to be satisfied for some Banach spaces of functions that are not rearrangement invariant. Corresponding results relating the rate of best approximation with ?? ^r (f,t) _B for those spaces are also given.     

Examples of proper k-ball contractive retractions in F-normed spaces

Assume X is an infinite dimensional F-normed space and let r be a positive number such that the closed ball Br(X) of radius r is properly contained in X. The main aim of this paper is to give examples of regular F-normed ideal spaces in which there is a 1-ball or a (1+ε)-ball contractive retraction of Br(X) onto its boundary with positive lower Hausdorff measure of noncompactness. The examples are based on the abstract results of the paper, obtained under suitable hypotheses on X.     

The Buchsbaum property of symbolic powers of Stanley-Reisner ideals of dimension 1

Let S be a polynomial ring and I be the Stanley-Reisner ideal of a simplicial complex Δ. The purpose of this paper is investigating the Buchsbaum property of S/I^(r) when Δ is pure dimension 1. We shall characterize the Buchsbaumness of S/I^(r) in terms of the graphical property of Δ. That is closely related to the characterization of the Cohen-Macaulayness of S/I^(r) due to the first author and N.V. Trung.     

T 1 theorem for Besov spaces on nonhomogeneous spaces

Suppose μ is a Radon measure on ℝ ^d , which may be non doubling. The only condition assumed on μ is a growth condition, namely, there is a constant C₀>0 such that for all x∈supp(μ) and r>0, μ(B(x, r))⪯C₀rⁿ, where 0<n⪯d. We prove T1 theorem for non doubling measures with weak kernel conditions. Our approach yields new results for kernels satisfying weakened regularity conditions, while recovering previously known Tolsa’s results. We also prove T1 theorem for Besov spaces on nonhomogeneous spaces with weak kernel conditions given in [7].     

Hardy and Carleson Measure Spaces Associated with Operators on Spaces of Homogeneous Type

Let (X, d, μ) be a metric measure space with doubling property. The Hardy spaces associated with operators L were introduced and studied by many authors. All these spaces, however, were first defined by L ²(X) functions and finally the Hardy spaces were formally defined by the closure of these subspaces of L ²(X) with respect to Hardy spaces norms. A natural and interesting question in this context is to characterize the closure. The purpose of this paper is to answer this question. More precisely, we will introduce \({CMO}_{L}^{p}(X)\), the Carleson measure spaces associated with operators L, and characterize the Hardy spaces associated with operators L via \(({CMO}_{L}^{p}(X))'\), the distributions of \({CMO}_{L}^{p}(X)\).     

Upper bound on the order of τ-critical hypergraphs

The right order of magnitude for the maximal number of vertices in an r-uniform τ-critical hypergraph H is achieved by obtaining an upper bound of O(τ(H)^r?1).     

Chromatic number and subtrees of graphs

Let G and H be two graphs. We say that G induces H if G has an induced subgraph isomorphic to H: A. Gyárfás and D. Sumner, independently, conjectured that, for every tree T. there exists a function f _T ; called binding function, depending only on T with the property that every graph G with chromatic number f _T (ω(G)) induces T. A. Gyárfás, E. Szemerédi and Z. Tuza confirmed the conjecture for all trees of radius two on triangle-free graphs, and H. Kierstead and S. Penrice generalized the approach and the conclusion of A. Gyárfás et al. onto general graphs. A. Scott proved an interesting topological version of this conjecture asserting that for every integer k and every tree T of radius r, every graph G with ω(G) ? k and sufficient large chromatic number induces a subdivision of T of which each edge is subdivided at most O(14 ^r-1(r - 1)!) times. We extend the approach of A. Gyárfás and present a binding function for trees obtained by identifying one end of a path and the center of a star. We also improve A. Scott's upper bound by modifying his subtree structure and partition technique, and show that for every integer k and every tree T of radius r, every graph with ω(G) ? k and sufficient large chromatic number induces a subdivision of T of which each edge is subdivided at most O(6 ^r?2) times.     

Enumeration of non-crossing pairings on bit strings

A non-crossing pairing on a bit string is a matching of 1s and 0s in the string with the property that the pairing diagram has no crossings. For an arbitrary bit-string w=p₁¹q₁⁰…pr¹qr⁰, let φ(w) be the number of such pairings. This enumeration problem arises when calculating moments in the theory of random matrices and free probability, and we are interested in determining useful formulas and asymptotic estimates for φ(w). Our main results include explicit formulas in the “symmetric” case where each pi=qi, as well as upper and lower bounds for φ(w) that are uniform across all words of fixed length and fixed r. In addition, we offer more refined conjectural expressions for the upper bounds. Our proofs follow from the construction of combinatorial mappings from the set of non-crossing pairings into certain generalized “Catalan” structures that include labeled trees and lattice paths.     

Nonlinear eigenvalue problems and spherical fibrations of Banach spaces

Eigenvalue problems of the form g′(v) = λh′(v) are considered, with the normalizations g(v) = r or h(v) = r, where g and h are real-valued C¹ functions on a real Banach space which are invariant under a periodic linear isometry. Theorems are proved on the existence of solutions λ(r), v(r), and on their dependence upon the normalization constant r > 0. In particular, the relation, as r → 0, of λ(r), v(r) to solutions of the linearized problem g″(0)v = λh″(0)v is discussed. The theorems are applied to elliptic problems for Euler-Lagrange operators corresponding to multiple integral functionals on closed subspaces of Sobolev spaces.     

Martingales with values in uniformly convex spaces

Using the techniques of martingale inequalities in the case of Banach space valued martingales, we give a new proof of a theorem of Enflo: every super-reflexive space admits an equivalent uniformly convex norm. Letr be a number in ]2, ∞[; we prove moreover that if a Banach spaceX is uniformly convex (resp. ifδ _x(?)/? ^r when? → 0) thenX admits for someq<∞ (resp. for someq<r) an equivalent norm for which the corresponding modulus of convexity satisfiesδ(?)/? ^q → ∞ when? → 0. These results have dual analogues concerning the modulus of smoothness. Our method is to study some inequalities for martingales with values in super-reflexive or uniformly convex spaces which are characteristic of the geometry of these spaces up to isomorphism.     

The fixed point property via dual space properties

A Banach space has the weak fixed point property if its dual space has a weak^∗ sequentially compact unit ball and the dual space satisfies the weak^∗ uniform Kadec-Klee property; and it has the fixed point property if there exists ε>0 such that, for every infinite subset A of the unit sphere of the dual space, A∪(−A) fails to be (2−ε)-separated. In particular, E-convex Banach spaces, a class of spaces that includes the uniformly nonsquare spaces, have the fixed point property.     

Some monotonicity properties of Schur powers of matrices and related inequalities

A series of inequalities are developed relating the spectral radius ?(A ° B) of the Schur product A ° B of two nonnegative matrices A and B with those of ?(A ° A) and ?(B ° B) yielding $\text{?(A ° B) ? [?(A ° A)?(B ° B)]}\text{1}\text{2}$. As a corollary it is proved that the spectral radius of the Schur powers ?_r = ?(A^[r]), A^[r] = A ° A °?°A (r factors) satisfies $\text{(}\text{1}\text{r}\text{)}\text{log}\text{?}{}_{\mathrm{r}}$ is decreasing while $\text{(}\text{1}\text{r?1}\text{)}\text{log}\text{?}{}_{\mathrm{r}}$ is increasing, the latter provided A is a stochastic matrix. The entropy of a finite stationary Markov chain is identified with $\text{d?}{}_{\mathrm{r}}\text{dr|}{}_{\mathrm{r=1}}$. A number of majorization comparisons for the spectral radius of Schur powers is given.     

Sharp upper bounds on the number of the scattering poles

We study the scattering poles of a compactly supported “black box” perturbations of the Laplacian in Rⁿ, n odd. We prove a sharp upper bound of the counting function N(r) modulo o(rⁿ) in terms of the counting function of the reference operator in the smallest ball around the black box. In the most interesting cases, we prove a bound of the type N(r)?A_nrⁿ+o(rⁿ) with an explicit A_n. We prove that this bound is sharp in a few special spherically symmetric cases where the bound turns into an asymptotic formula.