On a problem due to Littlewood concerning polynomials with unimodular coefficients

Littlewood raised the question of how slowly $\lVert f_{n}\rVert_{4}^{4}-\lVert f_{n}\rVert_{2}^{4}$ (where $\lVert.\rVert _{r}$ denotes the L ^r norm on the unit circle) can grow for a sequence of polynomials f _n with unimodular coefficients and increasing degree. The results of this paper are the following. For $$g_n(z)=\sum_{k=0}^{n-1}e^{\pi ik^2/n} z^k $$ the limit of $(\lVert g_{n}\rVert_{4}^{4}-\lVert g_{n}\rVert_{2}^{4})/\lVert g_{n}\rVert_{2}^{3}$ is 2/π, which resolves a mystery due to Littlewood. This is however not the best answer to Littlewood’s question: for the polynomials $$h_n(z)=\sum_{j=0}^{n-1}\sum _{k=0}^{n-1} e^{2\pi ijk/n} z^{nj+k} $$ the limit of $(\lVert h_{n}\rVert_{4}^{4}-\lVert h_{n}\rVert_{2}^{4})/\lVert h_{n}\rVert_{2}^{3}$ is shown to be 4/π ². No sequence of polynomials with unimodular coefficients is known that gives a better answer to Littlewood’s question. It is an open question as to whether such a sequence of polynomials exists.     

Invertible Factorization over Multiplier Algebras

Let ${\mathcal{A}}$ denote the multiplier algebra of an E-valued reproducing kernel Hilbert space, ${H_E^2(k)}$ . Then when H ²(k) is nice, we give necessary and sufficient conditions that T > 0 factors as A*A, where A and ${A^{-1} \in \mathcal{A}}$ . Such nice spaces include the Bergman and Hardy spaces on the unit polydisk and unit ball in ${\mathbb{C}^d}$ .     

The Littlewood-Paley g-function Associated with the Spherical Mean Operator

First we study the Gauss and Poisson semigroups connected with the spherical mean operator. Next, we define and study the Littlewood– Paley g-function associated with the spherical mean operator for which we prove the L ^p -boundedness for ${p \in]1, 2]}$ .     

On a result of Hardy and Littlewood concerning Fourier series

Let f ∈ L ¹( $ \mathbb{T} $ ) and assume that $$ f\left( t \right) \sim \frac{{a_0 }} {2} + \sum\limits_{k = 1}^\infty {\left( {a_k \cos kt + b_k \sin kt} \right)} $$ Hardy and Littlewood [1] proved that the series $ \sum\limits_{k = 1}^\infty {\frac{{a_k }} {k}} $ converges if and only if the improper Riemann integral $$ \mathop {\lim }\limits_{\delta \to 0^ + } \int_\delta ^\pi {\frac{1} {x}} \left\{ {\int_{ - x}^x {f(t)dt} } \right\}dx $$ exists. In this paper we prove a refinement of this result.     

Dyadic sets, maximal functions and applications on ax + b-groups

Let S be the Lie group ${{\mathbb R}^n\ltimes {\mathbb R}}$ , where ${{\mathbb R}}$ acts on ${{\mathbb R}^n}$ by dilations, endowed with the left-invariant Riemannian symmetric space structure and the right Haar measure ρ, which is a Lie group of exponential growth. Hebisch and Steger in [Math. Z. 245: 37–61, 2003] proved that any integrable function on (S, ρ) admits a Calderón–Zygmund decomposition which involves a particular family of sets, called Calderón–Zygmund sets. In this paper, we show the existence of a dyadic grid in the group S, which has nice properties similar to the classical Euclidean dyadic cubes. Using the properties of the dyadic grid, we prove a Fefferman–Stein type inequality, involving the dyadic Hardy–Littlewood maximal function and the dyadic sharp function. As a consequence, we obtain a complex interpolation theorem involving the Hardy space H ¹ and the space BMO introduced in [Collect. Math. 60: 277–295, 2009].     

A Triangulation of ?P 3 as Symmetric Cube of S 2

The symmetric group $\operatorname{Sym}(d)$ acts on the Cartesian product (S ²) ^d by coordinate permutation, and the quotient space $(S^{2})^{d}/\operatorname{Sym}(d)$ is homeomorphic to the complex projective space ?P ^d . We used the case d=2 of this fact to construct a 10-vertex triangulation of ?P ² earlier. In this paper, we have constructed a 124-vertex simplicial subdivision $(S^{2})^{3}_{124}$ of the 64-vertex standard cellulation $(S^{2}_{4})^{3}$ of (S ²)³, such that the $\operatorname{Sym}(3)$ -action on this cellulation naturally extends to an action on $(S^{2})^{3}_{124}$ . Further, the $\operatorname{Sym}(3)$ -action on $(S^{2})^{3}_{124}$ is ??good??, so that the quotient simplicial complex $(S^{2})^{3}_{124}/\operatorname{Sym}(3)$ is a 30-vertex triangulation $\mathbb{C}P^{3}_{30}$ of ?P ³. In other words, we have constructed a simplicial realization $(S^{2})^{3}_{124} \to\mathbb{C} P^{3}_{30}$ of the branched covering (S ²)³???P ³.     

A Note on Variable Exponent Hörmander Spaces

In this paper we introduce the variable exponent Hörmander spaces and we study some of their properties. In particular, it is shown that ${{(\mathcal{B}_{p_{(\cdot)}}^{c}(\Omega))^{\prime}}}$ is isomorphic to ${{\mathcal{B}^{loc}_{\widetilde{p^\prime(\cdot)}(\Omega)}}}$ (Ω open set in ${{\mathbb{R}^n, p? > 1}}$ and the Hardy–Littlewood maximal operator M is bounded in ${L_p(\cdot))}$ extending a Hörmander’s result to our context. As a consequence, a number of results on sequence space representations of variable exponent Hörmander spaces are given.     

Separable forms of {{\mathbb{G}}_m} -ACTIONS ON {{\mathbb{A}}^3}

Let k be a field of characteristic zero. We consider k-forms of $ {\mathbb G} $ _m -actions on $ {\mathbb A} $ ³ and show that they are linearizable. In particular, $ {\mathbb G} $ _m -actions on $ {\mathbb A} $ ³ are linearizable, and k-forms of $ {\mathbb A} $ ³ that admit an effective action of an infinite reductive group are trivial.     

On the L p -distortion of finite quotients of amenable groups

We study the L ^p -distortion of finite quotients of amenable groups. In particular, for every ${2\leq p < \infty}$ , we prove that the ? ^p -distortions of the groups ${C_2\wr C_n}$ and ${C_{2^n}\rtimes C_n}$ are in ${\Theta((\log n)^{1/p}),}$ and that the ? ^p -distortion of ${C_n^2 \rtimes_A \mathbf{Z}}$ , where A is the matrix ${{\left({\small\begin{array}{cc}2 & 1 \\ 1 & 1 \end{array}} \right)}}$ is in ${\Theta((\log \log n)^{1/p}).}$      

Nil-Manifolds Cannot be Immersed as Hypersurfaces in Euclidean Spaces

We prove that the $2n + 1$ -dimensional Heisenberg group H _n and the 4-manifolds $Nil^4 $ and $Nil^3 \times \mathbb{R}$ endowed with an arbitrary left-invariant metric admit no C ³-regular immersions into Euclidean spaces $\mathbb{R}^{2n + 2} $ and $\mathbb{R}^5 $ , respectively.     

The universal Glivenko–Cantelli property

Let ${\mathcal{F}}$ be a separable uniformly bounded family of measurable functions on a standard measurable space ${(X, \mathcal{X})}$ , and let ${N_{[]}(\mathcal{F}, \varepsilon, \mu)}$ be the smallest number of ${\varepsilon}$ -brackets in L ¹(μ) needed to cover ${\mathcal{F}}$ . The following are equivalent:

1. ${\mathcal{F}}$ is a universal Glivenko–Cantelli class.
2. ${N_{[]}(\mathcal{F},\varepsilon,\mu) < \infty}$ for every ${\varepsilon > 0}$ and every probability measure μ.
3. ${\mathcal{F}}$ is totally bounded in L ¹(μ) for every probability measure μ.
4. ${\mathcal{F}}$ does not contain a Boolean σ-independent sequence.

It follows that universal Glivenko–Cantelli classes are uniformity classes for general sequences of almost surely convergent random measures.     

An example related to Gregory’s Theorem

In this paper, we give an example of a complete computable infinitary theory T with countable models ${\mathcal{M}}$ and ${\mathcal{N}}$ , where ${\mathcal{N}}$ is a proper computable infinitary extension of ${\mathcal{M}}$ and T has no uncountable model. In fact, ${\mathcal{M}}$ and ${\mathcal{N}}$ are (up to isomorphism) the only models of T. Moreover, for all computable ordinals α, the computable ${\Sigma_\alpha}$ part of T is hyperarithmetical. It follows from a theorem of Gregory (JSL 38:460–470, 1972; Not Am Math Soc 17:967–968, 1970) that if T is a Π ₁ ¹ set of computable infinitary sentences and T has a pair of models ${\mathcal{M}}$ and ${\mathcal{N}}$ , where ${\mathcal{N}}$ is a proper computable infinitary extension of ${\mathcal{M}}$ , then T would have an uncountable model.     

Mixed Invariant Subspaces over the Bidisk

It is known that the structure of invariant subspaces I of the Hardy space H ² over the bidisk is extremely complicated. One reason is that it is difficult to describe infinite dimensional wandering spaces ${I\ominus zI}$ completely. In this paper, we study the structure of nontrivial closed subspaces N of H ² with ${T_zN\subset N}$ and ${T^*_wN\subset N}$ , which are called mixed invariant subspaces under T _z and ${T^*_w}$ . We know that the dimension of ${N\ominus zN}$ ranges from 1 to ??. If ${T^*_w(N\ominus zN)\subset N\ominus zN}$ , we may describe N completely. If ${T^*_w(N\ominus zN)\not\subset N\ominus zN}$ , it seems difficult to describe N generally. So we study N under the condition ${dim\,(N\ominus zN)=1}$ . Write ${M=H^2\ominus N}$ . We describe ${M\ominus wM}$ precisely. We give a characterization of N for which there is a nonzero function ${\varphi}$ in ${M\ominus wM}$ satisfying ${z^k\varphi\in M\ominus wM}$ for every k ?? 0. We also see that the space ${M\ominus wM}$ has a deep connection with the de Branges?CRovnyak spaces studied by Sarason.     

Length of Geodesics and Quantitative Morse Theory on Loop Spaces

Let M ⁿ be a closed Riemannian manifold of diameter d. Our first main result is that for every two (not necessarily distinct) points ${p,q \in M^n}$ and every positive integer k there are at least k distinct geodesics connecting p and q of length ${\leq 4nk^2d}$ . We demonstrate that all homotopy classes of M ⁿ can be represented by spheres swept-out by “short” loops unless the length functional has “many” “deep” local minima of a “small” length on the space ${\Omega_{pq}M^n}$ of paths connecting p and q. For example, one of our results implies that for every positive integer k there are two possibilities: Either the length functional on ${\Omega_{pq} M^n}$ has k distinct non-trivial local minima with length ${\leq 2kd}$ and “depth” ${\geq 2d}$ ; or for every m every map of S ^m into ${\Omega_{pq}M^n}$ is homotopic to a map of S ^m into the subspace ${\Omega_{pq}^{4(k+2)(m+1)d}M^n}$ of ${\Omega_{pq}M^n}$ that consists of all paths of length ${\leq 4(k+2)(m+1)d}$ .     

On the topology of semi-algebraic functions on closed semi-algebraic sets

We consider a closed semi-algebraic set ${X \subset \mathbb{R}^n}$ and a C ² semi-algebraic function ${f : \mathbb{R}^n \rightarrow\mathbb{R}}$ such that ${f_{\vert X}}$ has a finite number of critical points. We relate the topology of X to the topology of the sets ${X \cap \{ f * \alpha \}}$ , where ${* \in \{\le,=,\ge \}}$ and ${\alpha \in \mathbb{R}}$ , and the indices of the critical points of ${f_{\vert X}}$ and ${-f_{\vert X}}$ . We also relate the topology of X to the topology of the links at infinity of the sets ${X \cap \{ f * \alpha\}}$ and the indices of these critical points. We give applications when ${X=\mathbb{R}^n}$ and when f is a generic linear function.     

On additive decompositions of the set of primitive roots modulo p

It is conjectured that the set ${\mathcal {G}}$ of the primitive roots modulo p has no decomposition (modulo p) of the form ${\mathcal {G}= \mathcal {A} +\mathcal {B}}$ with ${|\mathcal {A}|\ge 2}$ , ${|\mathcal {B} |\ge 2}$ . This conjecture seems to be beyond reach but it is shown that if such a decomposition of ${\mathcal {G}}$ exists at all, then ${|\mathcal {A} |}$ , ${|\mathcal {B} |}$ must be around p ^1/2, and then this result is applied to show that ${\mathcal {G}}$ has no decomposition of the form ${\mathcal {G} =\mathcal {A} + \mathcal {B} + \mathcal {C}}$ with ${|\mathcal {A} |\ge 2}$ , ${|\mathcal {B} |\ge 2}$ , ${|\mathcal {C} |\ge 2}$ .     

Traces of monotone Sobolev functions

In this paper we prove that ifu: ${\mathbb{B}}^n \to {\mathbb{R}}$ , where ${\mathbb{B}}^n $ is the unit ball in ? ⁿ , is a monotone function in the Sobolev space W^1·p ( ${\mathbb{B}}^n $ ), andn ? 1 <p ≤n, thenu has nontangential limits at all the points of $\partial {\mathbb{B}}^n $ except possibly on a set ofp-capacity zero. The key ingredient in the proof is an extension of a classical theorem of Lindelöf to monotone functions in W^1·p ( ${\mathbb{B}}^n $ ),n ? 1 <p ≤n.     

A new optimal quaternary sequence family of length 2(2 n − 1) obtained from the orthogonal transformation of Families {\mathcal{B}} and {\mathcal{C}}

In this paper, a general orthogonal transformation on the optimal quaternary sequence Families ${\mathcal{B}}$ and ${\mathcal{C}}$ is presented. Consequently, the known optimal Family ${\mathcal{D}}$ and a new optimal Family ${\mathcal{E}}$ are produced in a uniform method. In contrast to the known optimal Family ${\mathcal{D}}$ , the new Family ${\mathcal{E}}$ has the same parameters such as the sequence length 2(2 ⁿ ? 1), the family size 2 ⁿ , and the maximal nontrivial correlation value ${2^{\frac{n+1}{2}}+2}$ , where n is a positive integer, but with a different correlation function.     

Homology of Littlewood complexes

Let $V$ be a symplectic vector space of dimension $2n$ . Given a partition $\lambda $ with at most $n$ parts, there is an associated irreducible representation $\mathbf{{S}}_{[\lambda ]}(V)$ of $\mathbf{{Sp}}(V)$ . This representation admits a resolution by a natural complex $L^{\lambda }_{\bullet }$ , which we call the Littlewood complex, whose terms are restrictions of representations of $\mathbf{{GL}}(V)$ . When $\lambda $ has more than $n$ parts, the representation $\mathbf{{S}}_{[\lambda ]}(V)$ is not defined, but the Littlewood complex $L^{\lambda }_{\bullet }$ still makes sense. The purpose of this paper is to compute its homology. We find that either $L^{\lambda }_{\bullet }$ is acyclic or it has a unique nonzero homology group, which forms an irreducible representation of $\mathbf{{Sp}}(V)$ . The nonzero homology group, if it exists, can be computed by a rule reminiscent of that occurring in the Borel–Weil–Bott theorem. This result can be interpreted as the computation of the “derived specialization” of irreducible representations of $\mathbf{{Sp}}(\infty )$ and as such categorifies earlier results of Koike–Terada on universal character rings. We prove analogous results for orthogonal and general linear groups. Along the way, we will see two topics from commutative algebra: the minimal free resolutions of determinantal ideals and Koszul homology.