<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis>,<Emphasis Type="Italic">q</Emphasis></Subscript>-Cohomology and Normal Solvability

We consider some problems concerning the L _p,q -cohomology of Riemannian manifolds. In the first part, we study the question of the normal solvability of the operator of exterior derivation on a surface of revolution M considered as an unbounded linear operator acting from L_p^k (M) into L^k+1_q (M). In the second part, we prove that the first L _p,q-cohomology of the general Heisenberg group is nontrivial, provided that p < q. Received: 17 January 2006 Supported by INTAS (Grant 03–51–3251) and the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grants NSh 311.2003.1, NSh 8526.2006.1).     

A weighted estimate for the square function on the unit ball in ℂ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

We show that the Luzin area integral or the square function on the unit ball of ℂ ⁿ , regarded as an operator in the weighted space L ²(w) has a linear bound in terms of the invariant A ₂ characteristic of the weight. We show a dimension-free estimate for the “area-integral” associated with the weighted L ²(w) norm of the square function. We prove the equivalence of the classical and the invariant A ₂ classes.     

Finite presentability of <Emphasis Type="Italic">SL</Emphasis><Subscript>1</Subscript>(<Emphasis Type="Italic">D</Emphasis>)

Let D be a finite dimensional division algebra over a local field of characteristic p and let SL ₁(D) denote the group of elements of reduced norm 1 in D. In this paper we prove that SL ₁(D) is finitely presented as a profinite group. This work is part of the author’s Ph.D. Thesis at Yale University.     

Generalization of the Kneser theorem on zeros of solutions of the equation <Emphasis Type="Italic">y</Emphasis>″ + <Emphasis Type="Italic">p</Emphasis>(<Emphasis Type="Italic">t</Emphasis>)<Emphasis Type="Italic">y</Emphasis> = 0

We establish conditions for the oscillation of solutions of the equation y″ + p(t)Ay = 0 in a Banach space, where A is a bounded linear operator and p: ℝ+ → ℝ+ is a continuous function. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 4, pp. 571–576, April, 2007.     

Zeros in Tables of Characters for the Groups <Emphasis Type="Italic">S</Emphasis><Subscript>n</Subscript> and <Emphasis Type="Italic">A</Emphasis><Subscript>n</Subscript>. II

Let P(n) be the set of all partitions of a natural number n. In the representation theory of symmetric groups, for every partition α ∈ P(n), the partition h(α) ∈ P(n) is defined so as to produce a certain set of zeros in the character table for S_n. Previously, the analog f(α) of h(α) was obtained pointing out an extra set of zeros in the table mentioned. Namely, h(α) is greatest (under the lexicographic ordering ≤) of the partitions β of n such that χ^α(g_β) ≠ 0, and f(α) is greatest of the partitions γ of n that are opposite in sign to h(α) and are such that χ^α(g_γ) ≠ 0, where χ^α is an irreducible character of S_n, indexed by α, and g_β is an element in the conjugacy class of S_n, indexed by β. For α ∈ P(n), under some natural restrictions, here, we construct new partitions h′(α) and f′(α) of n possessing the following properties. (A) Let α ∈ P(n) and n ⩾ 3. Then h′(α) is identical is sign to h(α), χ^α(g_h′(α)) ≠ 0, but χ^α(g_γ) = 0 for all γ ∈ P(n) such that the sign of γ coincides with one of h(α), and h′(α) < γ < h(α). (B) Let α ∈ P(n), α ≠ α′, and n ⩾ 4. Then f′(α) is identical in sign to f(α), χ^α(g_f′(α)) ≠ 0, but χ^α(g_γ) = 0 for all γ ∈ P(n) such that the sign of γ coincides with one of f(α), and f′(α) < γ < f(α). The results obtained are then applied to study pairs of semiproportional irreducible characters in A_n. Supported by RFBR grant No. 04-01-00463. __________ Translated from Algebra i Logika, Vol. 44, No. 6, pp. 643–663, November–December, 2005.     

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.     

The Fine Spectra of the Difference Operator △ Over the Sequence Space bvp, （1 ≤ p 〈 ∞）^＊

The sequence space bvp consisting of all sequences （xk） such that （xk -xk-1） belongs to the space gp has recently been introduced by Basar and Altay [Ukrainian Math. J., 55（1）, 136-147（2003）]; where 1 ≤ p ≤ ∞. In the present paper, some results concerning with the continuous dual and f-dual, and the AD-property of the sequence space bvp have been given and the norm of the difference operator A acting on the sequence space bvp has been found. The fine spectrum with respect to the Goldberg＇s classification of the difference operator △ over the sequence space bvp has been determined, where 1≤p〈∞.     

Lacunary Strong (<Emphasis Type="Italic">A</Emphasis><Subscript><Emphasis Type="Italic">σ</Emphasis></Subscript><Emphasis Type="Italic">, p</Emphasis>)-Convergence

The definition of lacunary strongly convergence is extended to the definition of lacunary strong (A _σ , p)-convergence with respect to invariant mean when A is an infinite matrix and p = (p _i ) is a strictly positive sequence. We study some properties and inclusion relations.     

On noncommutative weighted local ergodic theorems on <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-spaces

In the present paper we consider a von Neumann algebra M with a faithful normal semi-finite trace τ, and {α _t }, a strongly continuous extension to L ^p (M, τ) of a semigroup of absolute contractions on L ¹(M, τ). By means of a non-commutative Banach Principle we prove for a Besicovitch function b and x ∊ L ^p (M, τ), that the averages 1/T ∫₀ ^T b(t)α _t (x)dt converge bilateral almost uniformly in L ^p (M, τ) as T → 0. Communicated by Dénes Petz     

Two sided norm estimate of the Bergman projection on <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> spaces

We give some explicit values of the constants C ₁ and C ₂ in the inequality C ₁/sin(π/p) ⩽ |P| _p ⩽ C ₂/sin(π/p) where |P| _p denotes the norm of the Bergman projection on the L ^p space. Partially supported by MNZZS Grant No. ON144010.     

Pseudo-coefficients des séries <Emphasis Type="Italic">κ</Emphasis>-discrètes de GL(<Emphasis Type="Italic">n,F</Emphasis>)

Let F be a non-Archimedean locally compact field, n be an integer ≥ 2, κ be a character of F ^× of finite order dividing n, and G be the group GL(n, F). We prove that if π is an irreducible κ-discrete series of G, then there exist some pseudo-coefficients for the twisted character trace(π ∘ A) where A denotes a non-zero intertwining operator between π ⊗ (κ ∘ det) and π.     

Modulus Semigroups and Perturbation Classes for Linear Delay Equations in <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

In this paper we study C₀-semigroups on X × L_p( − h, 0; X) associated with linear differential equations with delay, where X is a Banach space. In the case that X is a Banach lattice with order continuous norm, we describe the associated modulus semigroup, under minimal assumptions on the delay operator. Moreover, we present a new class of delay operators for which the delay equation is well-posed for p in a subinterval of [1,∞). Dedicated to the memory of H. H. Schaefer     

On the <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> norm of spectral clusters for compact manifolds with boundary

We use microlocal and paradifferential techniques to obtain L ⁸ norm bounds for spectral clusters associated with elliptic second-order operators on two-dimensional manifolds with boundary. The result leads to optimal L ^q bounds, in the range 2⩽q⩽∞, for L ² - normalized spectral clusters on bounded domains in the plane and, more generally, for two-dimensional compact manifolds with boundary. We also establish new sharp L ^q estimates in higher dimensions for a range of exponents q̅_n⩽q⩽∞. The authors were supported by the National Science Foundation, Grants DMS-0140499, DMS-0099642, and DMS-0354668.     

The <Emphasis Type="Italic">p</Emphasis>-rank of Ext<Subscript>ℤ</Subscript>(<Emphasis Type="Italic">G</Emphasis>, ℤ) in certain models of <Emphasis Type="Italic">ZFC</Emphasis>

We prove that if the existence of a supercompact cardinal is consistent with ZFC, then it is consistent with ZFC that the p-rank of Ext _ℤ(G, ℤ) is as large as possible for every prime p and for any torsion-free Abelian group G. Moreover, given an uncountable strong limit cardinal μ of countable cofinality and a partition of Π (the set of primes) into two disjoint subsets Π₀ and Π₁, we show that in some model which is very close to ZFC, there is an almost free Abelian group G of size 2^μ = μ⁺ such that the p-rank of Ext _ℤ(G, ℤ) equals 2^μ = μ⁺ for every p ∈ Π₀ and 0 otherwise, that is, for p ∈ Π₁. Number 874 in Shelah’s list of publications. Supported by the German-Israeli Foundation for Scientific Research & Development project No. I-706-54.6/2001. Supported by a grant from the German Research Foundation DFG. __________ Translated from Algebra i Logika, Vol. 46, No. 3, pp. 369–397, May–June, 2007.     

Maximal Subsemigroups of Finite Transformation Semigroups K（n, r）

Let Tn be the full transformation semigroup on the n-element set Xn. For an arbitrary integer r such that 2 ≤ r ≤ n-1, we completely describe the maximal subsemigroups of the semigroup K(n, r) = {α∈Tn : |im α| ≤ r}. We also formulate the cardinal number of such subsemigroups which is an answer to Problem 46 of Tetrad in 1969, concerning the number of subsemigroups of Tn.     

Sign change of the function <Emphasis Type="Italic">S</Emphasis>(<Emphasis Type="Italic">t</Emphasis>) on short intervals

A theorem for the sign variation of the argument of the Riemann zeta function S(t) in the interval (t − A, t + A) with A = 4.39 ln ln ln ln T for each t, T ≤ t ≤ T + H excluding values from the set E with the measure mes(E) = O(H(ln ln T)⁻¹(ln ln ln T)^−0,5) is proved.     

Generalized convex kernels of simply connected <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript> sets in the plane

Let D be a compact, simply connected subset of ℜ². Assume that every two points of D can be connected by a polygonal line with at most n edges within D. Then there is a point q ∈ D that can be connected to any other point in D by a polygonal line with at most n edges. This is best possible for all n.     

Composite Integers n for Which φ（n）｜n- 1

In this note, we show that the number of composite integers n ≤ x such that φ（n）｜n - 1 is at most O（x^1/2（loglog x）^1/2）, thus improving earlier results by Pomerance and by Shan.     

The flatness properties of <Emphasis Type="Italic">S</Emphasis>-poset <Emphasis Type="Italic">A</Emphasis>(<Emphasis Type="Italic">I</Emphasis>) and Rees factor <Emphasis Type="Italic">S</Emphasis>-posets

Let S be a pomonoid and I a proper right ideal of S. In a previous paper, using the amalgamated coproduct A(I) of two copies of _S S over I, we were able to solve one of the problems posed in S. Bulman-Fleming et al. (Commun. Algebra 34:1291–1317, 2006). In the present paper, we investigate further flatness properties of A(I). We also solve another problem stated in the paper cited above. Namely, we determine the condition under which Rees factor S-posets have property (P _w ). Research supported by nwnu-kjcxgc-03-18.