Haar type and Carleson constants

For a collection of dyadic intervals, a Banach space X, andp(1, 2], we assume the upper ^p estimates where x_IX, and h_I denotes the L normalized Haar function supportedon I. We determine the minimal requirement on the size of suchthat these estimates imply that X is of Haar type p. The characterizationis given in terms of the Carleson constant of .     

On the Location of Zeros of Certain Classes of Polynomials with Applications to Numerical Analysis

A polynomial is said to be of type (p₁, p₂, p₃) relative tothe unit circle if it has p₁ zeros interior to, p₂ on, and p₃exterior to the unit circle. Stability criteria frequently arisewhere a polynomial or a family of polynomials must be shownto be of type (p₁, p₂, 0) or of type (p₁, 0, 0). Here we reconsiderthe practical problem of showing that a polynomial is of oneor other of these types, and we show that the testing of a polynomialof degree n may always be reduced to the testing of one of degreen–1. The simplicity of the method is illustrated by itsapplication to several well known difference schemes for partialdifferential equations.     

Markov and Bernstein type Inequalities in Lp for Classes of Polynomials with Constraints

The Markov-type inequality is proved for all real algebraic polynomials f of degree atmost n having at most k, with 0 k n, zeros (counting multiplicities)in the open unit disk of the complex plane, and for all p >0, where c(p) = c^{p + 1}(l + p^–2) with some absolute constantc > 0. This inequality has been conjectured since 1983 whenthe L case of the above result was proved. It improves and generalizesmany earlier results. Up to the multiplicative constant c(p)>0 the above inequality is sharp. A sharp Bernstein-type analoguefor real trigonometric polynomials is also established, whichis interesting on its own, and plays a key role in the proofof the Markov-type inequality.     

Maximal Functions Associated with The Jacobi Transform

In this paper, we define a Hardy–Littlewood maximal functionoperator M associated with the Jacobi transform, and prove thatM is of weak (1, 1) and strong (p, p) (p > 1) type. 1991Mathematics Subject Classification 42B25, 44A15.     

Toward a Characterization of Conway's Group Co3

In this paper, Conway's group, Co₃, is characterized among simplegroups of even type with e(G) = 3, by a restriction on the p-localstructure for some odd prime p for which m_2,p(G) = 3. 2000 MathematicsSubject Classification 20D05.     

A p Interpolation in the Unit Ball

We study interpolating sequences in the unit ball for A^–pwith p > 0, the Banach space of holomorphic functions f with(1 – |z|²)^p |f(z)| bounded. The finite unions of A^–p-interpolatingsequences are characterized by a Carleson type condition.     

Embedding Properties of Metabelian Lie Algebras and Metabelian Discrete Groups

We show that for every natural number m a finitely generatedmetabelian group G embeds in a quotient of a metabelian groupof type FP_m. Furthermore, if m 4, the group G can be embeddedin a metabelian group of type FP_m. For L a finitely generatedmetabelian Lie algebra over a field K and a natural number mwe show that, provided the characteristic p of K is 0 or p >m, then L can be embedded in a metabelian Lie algebra of typeFP_m. This result is the best possible as for 0 < p m everymetabelian Lie algebra over K of type FP_m is finite dimensionalas a vector space.     

On the mixed Cauchy problem with data on singular conics

We consider a problem of mixed Cauchy type for certain holomorphicpartial differential operators with the principal part Q_2p(D)essentially being the (complex) Laplace operator to a power,^p. We provide inital data on a singular conic divisor givenby P = 0, where P is a homogeneous polynomial of degree 2p.We show that this problem is uniquely solvable if the polynomialP is elliptic, in a certain sense, with respect to the principalpart Q_2p(D).     

An essentially saturated surface not of Kahler type

It is shown that if X is an Inoue surface of type S_M then theirreducible components of the Douady space of Xⁿ are compact,for all n 0. This gives an example, sought by Moosa [J. ReineAngew. Math. 586 (2005) 1–20], of an essentially saturatedcompact complex manifold (in the sense of model theory) thatis not of Kähler type. Among the known compact complexsurfaces without curves, these are the only examples.     

Mod p Reducibility of Unramified Representations of Finite Groups of Lie Type

Dedicated to the memory of Professor A. I. Kostrikin The main problem under discussion is to determine, for quasi-simplegroups of Lie type G, irreducible representations of G thatremain irreducible under reduction modulo the natural primep. The method is new. It works only for p >3 and for representations that can be realized over an unramified extension of Q_p, thefield of p -adic numbers. Under these assumptions, the mainresult says that the trivial and the Steinberg representationsof G are the only representations in question provided G isnot of type A1. This is not true for G=SL(2, p). The paper containsa result of independent interest on infinitesimally irrreduciblerepresentations of G over an algebraically closed field ofcharacteristic p. Assuming that G is not of rank 1 and G G₂(5),it is proved that either the Jordan normal form of a root elementcontains a block of size d with 1<d<p -1 or the highestweight of is equal to p -1 times the sum of the fundamentalweights. 2000 Mathematical Subject Classification: 20C33, 20G15.     

Strong Type Inequalities and an Almost-Orthogonality Principle for Families of Maximal Operators Along Directions in R2

The paper proves an almost-orthogonality principle for maximaloperators over arbitrary sets of directions in R². Namely, theL^p-bounds for an operator of this type are obtained from thecorresponding L^p-bounds of the maximal functions associatedto a certain partition of the set of directions, and from theparticular structure of this partition. Applications to severaltypes of maximal operators are provided.     

Some non-uniformly homeomorphic spaces

in this paper we prove that for 0<p, q≤1 the real F-spacesL _q[0, 1] and ℓ _p are not uniformly homeomorphic. The particular casep=q=1 is due to Enflo and our work is motivated by his. Partially supported by NSF grant #DMS-9104040. The author would like to dedicate this paper to Nancy and Bernie Weston on the occasion of their 40th wedding anniversary.     

Two Examples Concerning Martingales in Banach Spaces

The analytic concepts of martingale type p and cotype q of aBanach space have an intimate relation with the geometric conceptsof p-concavity and q-convexity of the space under consideration,as shown by pisier. In particular, for a banach space X, havingmartingale type p for some p > 1 implies that X has martingalecotype q for some q < . The generalisation of these concepts to linear operators wasstudied by the author, and it turns out that the duality aboveonly holds in a weaker form. An example is constructed showingthat this duality result is best possible. So-called random martingale unconditionality estimates, introducedby Garling as a decoupling of the unconditional martingale differences(UMD) inequality, are also examined. It is shown that the random martingale unconditionality constantof for martingales of length n asymptotically behaves like n. This improves previous estimatesby Geiss, who needed martingales of length 2ⁿ to show this asymptotic.At the same time the order in the paper is the best that canbe expected.     

Mackey formula in type A

Let G be a connected reductive group defined over a finite fieldwith q elements and let F: G G be the corresponding Frobeniusendomorphism. We prove that the Mackey formula for Lusztig inductionand restriction holds in G^F if all the simple components ofG are of type A. Our result holds for every Frobenius endomorphism(split or non-split) and for every q. 1991 Mathematics SubjectClassification: 20G05, 20G40.     

Harmonic and Logarithmic Summability of Orthogonal Series are Equivalent up to a Set of Measure Zero

We prove Tauberian theorems from J_p-summability methods of powerseries type to ordinary convergence, respectively M_p-summabilitymethods of weighted means. Particular cases are the Abel andCesàro, as well as logarithmic and harmonic summability.Besides numerical series, we also consider orthogonal serieswith coefficients from L₂. In the latter case, it turns outthat one of our Tauberian conditions is satisfied almost everywhereon the underlying measure space, thereby proving the claim statedin the title.     

Vitali's and Harnack's type results for vector-valued functions

In this brief note, we extend Vitali's theorem for holomorphic functions obtained by Arendt and Nikolski to nets of functions of sheaves of smooth vector-valued functions. As a consequence we also extend a Harnack's theorem for compact operator-valued harmonic functions recently obtained by Enflo and Smithies to bounded operator-valued harmonic functions, avoiding the assumption that the Hilbert space H where the operators are defined is separable.     

Constructions of stable equivalences of Morita type for finite-dimensional algebras III

In this paper, we provide a new method to produce stable equivalencesof Morita type. Our main results can be stated as follows. LetA and B be two finite-dimensional k-algebras over a field k.Suppose that two bimodules _AM_B and _BN_A define a stable equivalenceof Morita type between A and B and that R is a generator forA-modules. Then there is a stable equivalence of Morita typedefined by X and Y between the endomorphism algebra End_A(R)of the module R and the endomorphism algebra End_B(N_AR) of themodule N_AR. If M and N satisfy the property that both (N_A–,M_B–) and (M_B–, N_A–) are adjoint pairs of functors,then so do the modules X and Y. Moreover, we show that the self-injectivedimension and the Gorenstein property are invariant under stableequivalences of Morita type with the above-mentioned adjointproperty.     

Maximal Subgroups of Large Rank in Exceptional Groups of Lie Type

Let G = G(q) be a finite almost simple exceptional group ofLie type over the field of q elements, where q = p^a and p isprime. The main result of the paper determines all maximal subgroupsM of G(q) such that M is an almost simple group which is alsoof Lie type in characteristic p, under the condition that rank(M)> rank(G). The conclusion is that either M is a subgroupof maximal rank, or it is of the same type as G over a subfieldof F_q, or (G, M) is one of (, F₄(q)), (, C₄(q)), (E₇(q),³D₄(q)). This completes work of the first author with Saxl andTesterman, in which the same conclusion was obtained under someextra assumptions.     

Representations of compact linear operators in Banach spaces and nonlinear eigenvalue problems

Let X and Y be reflexive Banach spaces with strictly convexduals, and let T be a compact linear map from X to Y. It isshown that a certain nonlinear equation, involving T and itsadjoint, has a normalised solution (an ‘eigenvector’)corresponding to an ‘eigenvalue’, and that the sameis true for each member of a countable family of similar equationsinvolving the restrictions of T to certain subspaces of X. Theaction of T can be described in terms of these ‘eigenvectors’.There are applications to the p-Laplacian, the p-biharmonicoperator and integral operators of Hardy type.     

Semisimple Modules for Finite Groups of Lie Type

Let k be an algebraically closed field of characteristic p >0, and let G be a connected, reductive algebraic group overk. In [8] and [11], conditions on the dimension of rationalG modules were seen to imply semisimplicity of these modules.In [8], certain of these conditions were extended to cover thefinite groups of Lie type. In this paper, we extend some ofthe results of [11] to cover these finite Lie type groups. Themain such extension is the following result.