Showing papers in "Israel Journal of Mathematics in 2017"

An elementary proof of the A 2 bound

Abstract: A martingale transform T, applied to an integrable locally supported function f, is pointwise dominated by a positive sparse operator applied to |f|, the choice of sparse operator being a function of T and f. As a corollary, one derives the sharp Ap bounds for martingale transforms, recently proved by Thiele-Treil-Volberg, as well as a number of new sharp weighted inequalities for martingale transforms. The (very easy) method of proof (a) only depends upon the weak-L1 norm of maximal truncations of martingale transforms, (b) applies in the vector valued setting, and (c) has an extension to the continuous case, giving a new elementary proof of the A2 bounds in that setting.

Uo-convergence and its applications to Cesàro means in Banach lattices

Abstract: A net (x α ) in a vector lattice X is said to uo-converge to x if $$\left| {{x_\alpha } - x} \right| \wedge u\xrightarrow{o}0$$ for every u ≥ 0. In the first part of this paper, we study some functional-analytic aspects of uo-convergence. We prove that uoconvergence is stable under passing to and from regular sublattices. This fact leads to numerous applications presented throughout the paper. In particular, it allows us to improve several results in [27, 26]. In the second part, we use uo-convergence to study convergence of Cesaro means in Banach lattices. In particular, we establish an intrinsic version of Komlos’ Theorem, which extends the main results of [35, 16, 31] in a uniform way. We also develop a new and unified approach to Banach–Saks properties and Banach–Saks operators based on uo-convergence. This approach yields, in particular, short direct proofs of several results in [20, 24, 25].

Quantitative weighted estimates for rough homogeneous singular integrals

Abstract: We consider homogeneous singular kernels, whose angular part is bounded, but need not have any continuity. For the norm of the corresponding singular integral operators on the weighted space L 2(w), we obtain a bound that is quadratic in A 2 constant $${\left[ w \right]_{{A_2}}}$$ . We do not know if this is sharp, but it is the best known quantitative result for this class of operators. The proof relies on a classical decomposition of these operators into smooth pieces, for which we use a quantitative elaboration of Lacey's dyadic decomposition of Dini-continuous operators: the dependence of constants on the Dini norm of the kernels is crucial to control the summability of the series expansion of the rough operator. We conclude with applications and conjectures related to weighted bounds for powers of the Beurling transform.

Additive energy and the Hausdorff dimension of the exceptional set in metric pair correlation problems

Abstract: For a sequence of integers {a(x)} x≥1 we show that the distribution of the pair correlations of the fractional parts of {〈αa(x)〉} x≥1 is asymptotically Poissonian for almost all α if the additive energy of truncations of the sequence has a power savings improvement over the trivial estimate. Furthermore, we give an estimate for the Hausdorff dimension of the exceptional set as a function of the density of the sequence and the power savings in the energy estimate. A consequence of these results is that the Hausdorff dimension of the set of α such that {〈αx d 〉} fails to have Poissonian pair correlation is at most $$\frac{{d + 2}}{{d + 3}} < 1$$ . This strengthens a result of Rudnick and Sarnak which states that the exceptional set has zero Lebesgue measure. On the other hand, classical examples imply that the exceptional set has Hausdorff dimension at least $$\frac{2}{{d + 1}}$$ . An appendix by Jean Bourgain was added after the first version of this paper was written. In this appendix two problems raised in the paper are solved.

Upper tails for arithmetic progressions in random subsets

Abstract: We study the upper tail of the number of arithmetic progressions of a given length in a random subset of {1,..., n}, establishing exponential bounds which are best possible up to constant factors in the exponent. The proof also extends to Schur triples, and, more generally, to the number of edges in random induced subhypergraphs of ‘almost linear’ k-uniform hypergraphs.

Proof of the Erdős matching conjecture in a new range

Abstract: Let s > k ≧ 2 be integers. It is shown that there is a positive real e = e(k) such that for all integers n satisfying (s + 1)k ≦ n < (s + 1)(k + e) every k-graph on n vertices with no more than s pairwise disjoint edges has at most $$\left( {\begin{array}{*{20}{c}} {\left( {s + 1} \right)k - 1} \\ k \end{array}} \right)$$ edges in total. This proves part of an old conjecture of Erdős.

Random subgraphs of properly edge-coloured complete graphs and long rainbow cycles

Abstract: A subgraph of an edge-coloured complete graph is called rainbow if all its edges have different colours. In 1980 Hahn conjectured that every properly edge-coloured complete graph K n has a rainbow Hamiltonian path. Although this conjecture turned out to be false, it was widely believed that such a colouring always contains a rainbow cycle of length almost n. In this paper, improving on several earlier results, we confirm this by proving that every properly edge-coloured K n has a rainbow cycle of length n − O(n 3/4). One of the main ingredients of our proof, which is of independent interest, shows that a random subgraph of a properly edge-coloured K n formed by the edges of a random set of colours has a similar edge distribution as a truly random graph with the same edge density. In particular, it has very good expansion properties.

Orthogonal and unitary tensor decomposition from an algebraic perspective

Abstract: While every matrix admits a singular value decomposition, in which the terms are pairwise orthogonal in a strong sense, higher-order tensors typically do not admit such an orthogonal decomposition. Those that do have attracted attention from theoretical computer science and scientific computing. We complement this existing body of literature with an algebro-geometric analysis of the set of orthogonally decomposable tensors. More specifically, we prove that they form a real-algebraic variety defined by polynomials of degree at most four. The exact degrees, and the corresponding polynomials, are different in each of three times two scenarios: ordinary, symmetric, or alternating tensors; and real-orthogonal versus complex-unitary. A key feature of our approach is a surprising connection between orthogonally decomposable tensors and semisimple algebras—associative in the ordinary and symmetric settings and of compact Lie type in the alternating setting.

The euclidean distance degree of orthogonally invariant matrix varieties

Abstract: The Euclidean distance degree of a real variety is an important invariant arising in distance minimization problems. We show that the Euclidean distance degree of an orthogonally invariant matrix variety equals the Euclidean distance degree of its restriction to diagonal matrices. We illustrate how this result can greatly simplify calculations in concrete circumstances.

On Lipschitz extension from finite subsets

Abstract: We prove that for every n ∈ ℕ there exists a metric space (X, d X), an n-point subset S ⊆ X, a Banach space (Z, $${\left\| \right\|_Z}$$ ) and a 1-Lipschitz function f: S → Z such that the Lipschitz constant of every function F: X → Z that extends f is at least a constant multiple of $$\sqrt {\log n} $$ . This improves a bound of Johnson and Lindenstrauss [JL84]. We also obtain the following quantitative counterpart to a classical extension theorem of Minty [Min70]. For every α ∈ (1/2, 1] and n ∈ ℕ there exists a metric space (X, d X), an n-point subset S ⊆ X and a function f: S → l2 that is α-Holder with constant 1, yet the α-Holder constant of any F: X → l2 that extends f satisfies $${\left\| F \right\|_{Lip\left( \alpha \right)}} > {\left( {\log n} \right)^{\frac{{2\alpha - 1}}{{4\alpha }}}} + {\left( {\frac{{\log n}}{{\log \log n}}} \right)^{{\alpha ^2} - \frac{1}{2}}}$$ . We formulate a conjecture whose positive solution would strengthen Ball’s nonlinear Maurey extension theorem [Bal92], serving as a far-reaching nonlinear version of a theorem of Konig, Retherford and Tomczak-Jaegermann [KRTJ80]. We explain how this conjecture would imply as special cases answers to longstanding open questions of Johnson and Lindenstrauss [JL84] and Kalton [Kal04].

Packing spanning graphs from separable families

Abstract: Let $$\mathcal{G}$$ be a separable family of graphs. Then for all positive constants ϵ and Δ and for every sufficiently large integer n, every sequence G 1,..., G t ∈ $$\mathcal{G}$$ of graphs of order n and maximum degree at most Δ such that $$\left( {{G_1}} \right) + \cdots + e\left( {{G_t}} \right) \leqslant \left( {1 - \epsilon } \right)\left( {\begin{array}{*{20}{c}} n \\ 2 \end{array}} \right)$$ packs into K n . This improves results of Bottcher, Hladký, Piguet and Taraz when $$\mathcal{G}$$ is the class of trees and of Messuti, Rodl, and Schacht in the case of a general separable family. The result also implies approximate versions of the Oberwolfach problem and of the Tree Packing Conjecture of Gyarfas and Lehel (1976) for the case that all trees have maximum degree at most Δ. The proof uses the local resilience of random graphs and a special multi-stage packing procedure.

A cubical flat torus theorem and the bounded packing property

Abstract: We prove the bounded packing property for any abelian subgroup of a group acting properly and cocompactly on a CAT(0) cube complex. A main ingredient of the proof is a cubical flat torus theorem. This ingredient is also used to show that central HNN extensions of maximal free-abelian subgroups of compact special groups are virtually special, and to produce various examples of groups that are not cocompactly cubulated.

Sharp reversed Hardy–Littlewood–Sobolev inequality on R n

Abstract: This is the first in our series of papers that concerns Hardy–Littlewood–Sobolev (HLS) type inequalities. In this paper, the main objective is to establish the following sharp reversed HLS inequality in the whole space R n, $$\int {_{{R^n}}} \int {_{{R^n}}f\left( x \right)} {\left| {x - y} \right|^\lambda }g\left( y \right)dxdy \geqslant {\ell _{n,p,r}}{\left\| f \right\|_{{L^p}\left( {{R^n}} \right)}}{\left\| g \right\|_{{L^r}\left( {{R^n}} \right)}}$$ , for any non-negative functions f ∈ L p(R n), g ∈ L r(R n), and p, r ∈ (0, 1), λ > 0 such that 1/p+1/r −λ/n = 2. We will also explore some estimates for ln,p,r and the existence of optimal functions for the above inequality, which will shed light on some existing results in literature.

Dualities and derived equivalences for category O

Abstract: We determine the Ringel duals for all blocks in the parabolic versions of the BGG category $$\mathcal{O}$$ associated to a reductive finite-dimensional Lie algebra. In particular, we find that, contrary to the original category $$\mathcal{O}$$ and the specific previously known cases in the parabolic setting, the blocks are not necessarily Ringel self-dual. However, the parabolic category $$\mathcal{O}$$ as a whole is still Ringel self-dual. Furthermore, we use generalisations of the Ringel duality functor to obtain large classes of derived equivalences between blocks in parabolic and original category $$\mathcal{O}$$ . We subsequently classify all derived equivalence classes of blocks of category $$\mathcal{O}$$ in type A which preserve the Koszul grading.

The boundary of the outer space of a free product

Abstract: Let G be a countable group that splits as a free product of groups of the form G = G 1 *···* G k * F N , where F N is a finitely generated free group. We identify the closure of the outer space PO(G, {G 1,..., G k }) for the axes topology with the space of projective minimal, very small (G, {G 1,..., G k })-trees, i.e. trees whose arc stabilizers are either trivial, or cyclic, closed under taking roots, and not conjugate into any of the G i ’s, and whose tripod stabilizers are trivial. Its topological dimension is equal to 3N + 2k − 4, and the boundary has dimension 3N + 2k − 5. We also prove that any very small (G, {G 1,..., G k })-tree has at most 2N + 2k−2 orbits of branch points.

Nonlinear Roth type theorems in finite fields

Abstract: We obtain smoothing estimates for certain nonlinear convolution operators on prime fields, leading to quantitative nonlinear Roth type theorems. For instance, we produce triplets x, x + y, x + y 2 and x, x + y, \(x + \overline y \) in proportional subsets of F p .

The characterization of theta-distinguished representations of GL(n)

Abstract: Let θ and θ’ be a pair of exceptional representations in the sense of Kazhdan and Patterson [KP84], of a metaplectic double cover of GL n . The tensor θ ⊗ θ’ is a (very large) representation of GL n . We characterize its irreducible generic quotients. In the square-integrable case, these are precisely the representations whose symmetric square L-function has a pole at s = 0. Our proof of this case involves a new globalization result. In the general case these are the representations induced from distinguished data or pairs of representations and their contragredients. The combinatorial analysis is based on a complete determination of the twisted Jacquet modules of θ. As a corollary, θ is shown to admit a new “metaplectic Shalika model”.

The asymmetry of complete and constant width bodies in general normed spaces and the Jung constant

Abstract: In this paper we state a one-to-one connection between the maximal ratio of the circumradius and the diameter of a body (the Jung constant) in an arbitrary Minkowski space and the maximal Minkowski asymmetry of the complete bodies within that space. This allows to generalize and unify recent results on complete bodies and to derive a necessary condition on the unit ball of the space, assuming a given body to be complete. Finally, we state several corollaries, e.g. concerning the Helly dimension or the Banach–Mazur distance.

On the Lipschitz continuity of certain quasiregular mappings between smooth Jordan domains

Abstract: We first investigate the Lipschitz continuity of (K,K’)-quasiregular C 2 mappings between two Jordan domains with smooth boundaries, satisfying certain partial differential inequalities concerning Laplacian. Then two applications of the obtained result are given: As a direct consequence, we get the Lipschitz continuity of ρ-harmonic (K,K’)-quasiregular mappings, and as the other application, we study the Lipschitz continuity of (K,K’)- quasiconformal self-mappings of the unit disk, which are the solutions of the Poisson equation Δw = g. These results generalize and extend several recently obtained results by Kalaj, Mateljevic and Pavlovic.

A sharp eigenvalue theorem for fractional elliptic equations

Abstract: By using variational methods, in this paper we study a nonlinear elliptic problem defined in a bounded domain Ω ⊂ ℝ N , with smooth boundary ∂Ω, involving fractional powers of the Laplacian operator together with a suitable nonlinear term f. More precisely, we prove a characterization theorem on the existence of one weak solution for the elliptic problem $$\left\{ {\begin{array}{*{20}{c}} {{{( - \Delta )}^{\alpha /2}}\mu = \lambda f(\mu )in\Omega ,} \\ {u > 0in\Omega ,} \\ {u = 0in\partial \Omega ,} \end{array}} \right.$$ , where α ∈ (0, 2), N > α, λ > 0 and (−Δ)α/2 denotes the nonlocal fractional Laplacian operator. Our result extends to the nonlocal setting recent theorems for ordinary and classical elliptic equations, as well as a characterization for elliptic problems on certain non-smooth domains. To make the nonlinear methods work, some careful analysis of the fractional spaces involved is necessary.

On the Malle conjecture and the self-twisted cover

Abstract: For a large class of finite groups G, the number of Galois extensions E/Q of group G and discriminant |d E | ≤ y is shown to grow at least like a power of y, for some specified positive exponent. The groups G are the regular Galois groups over Q and the counted extensions E/Q are obtained by specializing a given regular Galois extension F/Q(T). The extensions E/Q can further be prescribed any unramified local behavior at each suitably large prime p ≤ log(y)/δ for some δ ≥ 1. This result is a step toward the Malle conjecture on the number of Galois extensions of given group and bounded discriminant. The local conditions further make it a notable constraint on regular Galois groups over Q. The method uses a new version of Hilbert’s irreducibility theorem that counts the specialized extensions and not just the specialization points. A main tool for it is the self-twisted cover that we introduce.

Reduced C*-algebras of Fell bundles over inverse semigroups

Abstract: We construct a weak conditional expectation from the section C*-algebra of a Fell bundle over a unital inverse semigroup to its unit fibre. We use this to define the reduced C*-algebra of the Fell bundle. We study when the reduced C*-algebra for an inverse semigroup action on a groupoid by partial equivalences coincides with the reduced groupoid C*-algebra of the transformation groupoid, giving both positive results and counterexamples.

Tracking rates of random walks

Abstract: We show that simple random walks on (non-trivial) relatively hyperbolic groups stay O(log(n))-close to geodesics, where n is the number of steps of the walk. Using similar techniques we show that simple random walks in mapping class groups stay $$O\left( {\sqrt {n\log \left( n \right)} } \right)$$ -close to geodesics and hierarchy paths. Along the way, we also prove a refinement of the result that mapping class groups have quadratic divergence. An application of our theorem for relatively hyperbolic groups is that random triangles in non-trivial relatively hyperbolic groups are O(log(n))-thin, random points have O(log(n))-small Gromov product and that in many cases the average Dehn function is subasymptotic to the Dehn function.

Multiple Commutator Formulas for Unitary Groups

Abstract: Let (A,Λ) be a formring such that A is quasi-finite R-algebra (i.e., a direct limit of module finite algebras) with identity. We consider the hyperbolic Bak’s unitary groups GU(2n, A, Λ), n ≥ 3. For a form ideal (I, Γ) of the form ring (A, Λ) we denote by EU(2n, I, Γ) and GU(2n, I, Γ) the relative elementary group and the principal congruence subgroup of level (I, Γ), respectively. Now, let (I i , Γ i ), i = 0,...,m, be form ideals of the form ring (A, Λ). The main result of the present paper is the following multiple commutator formula: [EU(2n, I 0, Γ 0),GU(2n, I 1, Γ 1), GU(2n, I 2, Γ 2),..., GU(2n, I m , Γ m )] =[EU(2n, I 0, Γ 0), EU(2n, I 1, Γ 1), EU(2n, I 2, Γ 2),..., EU(2n, I m , Γ m )], which is a broad generalization of the standard commutator formulas. This result contains all previous results on commutator formulas for classicallike groups over commutative and finite-dimensional rings.

Homogeneous Ricci almost solitons

Abstract: It is shown that a locally homogeneous proper Ricci almost soliton is either of constant sectional curvature or locally isometric to a product R×N(c), where N(c) is a space of constant curvature.

Exponential stability for the wave equation with degenerate nonlocal weak damping

Abstract: A damped nonlinear wave equation with a degenerate and nonlocal damping term is considered. Well-posedness results are discussed, as well as the exponential stability of the solutions. The degeneracy of the damping term is the novelty of this stability approach.

On the well-posedness of degenerate fractional differential equations in vector valued function spaces

Abstract: We characterize completely the well-posedness on the vector-valued Holder and Lebesgue spaces of the degenerate fractional differential equation D α (Mu)(t) = Au(t) + f(t), t ∈ ℝ by using vector-valued multiplier results in the spaces C γ (ℝ;X) and L p (ℝ;X), where A and M are closed linear operators defined on the Banach space X, 0 < γ < 1, 1 < p < ∞, the fractional derivative is understood in the sense of Caputo and α is positive.

Optimal comparison of the p -norms of Dirichlet polynomials

Abstract: Let 1 ≤ p < q < ∞. We show that $$\sup \frac{{{{\left\| D \right\|}_{{H_q}}}}}{{{{\left\| D \right\|}_{{H_q}}}}} = \exp \left( {\frac{{\log x}}{{\log \log x}}\left( {\log \sqrt {\frac{q}{p}} + O\left( {\frac{{\log \log \log x}}{{\log \log x}}} \right)} \right)} \right),$$ where the supremum is taken over all non-zero Dirichlet polynomials of the form D(s) = Σ n≤x a n n −s and $${\left\| D \right\|_{{H_q}}} = \mathop {\lim }\limits_{T \to \infty } {\left( {\frac{1}{{2T}}\int_{ - T}^T | \sum\limits_{n \leqslant x} {{a_n}{n^{ - it}}\left| {^pdt} \right.} } \right)^{1/p}}.$$ An application is given to the study of multipliers between Hardy spaces H p of Dirichlet series.

How far can we go with Amitsur’s theorem in differential polynomial rings?

Abstract: A well-known theorem by S. A. Amitsur shows that the Jacobson radical of the polynomial ring R[x] equals I[x] for some nil ideal I of R. In this paper, however, we show that this is not the case for differential polynomial rings, by proving that there is a ring R which is not nil and a derivation D on R such that the differential polynomial ring R[x;D] is Jacobson radical. We also show that, on the other hand, the Amitsur theorem holds for a differential polynomial ring R[x;D], provided that D is a locally nilpotent derivation and R is an algebra over a field of characteristic p > 0. The main idea of the proof introduces a new way of embedding differential polynomial rings into bigger rings, which we name platinum rings, plus a key part of the proof involves the solution of matrix theory-based problems.

The number of Hamiltonian decompositions of regular graphs

Abstract: A Hamilton cycle in a graph Γ is a cycle passing through every vertex of Γ. A Hamiltonian decomposition of Γ is a partition of its edge set into disjoint Hamilton cycles. One of the oldest results in graph theory is Walecki’s theorem from the 19th century, showing that a complete graph K n on an odd number of vertices n has a Hamiltonian decomposition. This result was recently greatly extended by Kuhn and Osthus. They proved that every r-regular n-vertex graph Γ with even degree r = cn for some fixed c > 1/2 has a Hamiltonian decomposition, provided n = n(c) is sufficiently large. In this paper we address the natural question of estimating H(Γ), the number of such decompositions of Γ. Our main result is that H(Γ) = r (1+o(1))nr/2. In particular, the number of Hamiltonian decompositions of K n is $${n^{\left( {1 + o\left( 1 \right)} \right){n^2}/2}}$$ .