Maximal estimates for the -dimensional Walsh-Fourier series

The -dimensional dyadic martingale Hardy spaces are introduced and it is proved that the maximal operator of the means of a Walsh-Fourier series is bounded from to and is of weak type , provided that the supremum in the maximal operator is taken over a positive cone. As a consequence we obtain that the means of a function converge a.e. to the function in question. Moreover, we prove that the means are uniformly bounded on whenever . Thus, in case , the means converge to in norm. The same results are proved for the conjugate means, too.

     

The Hausdorff operator is bounded on the real Hardy space

We prove that the Hausdorff operator generated by a function is bounded on the real Hardy space . The proof is based on the closed graph theorem and on the fact that if a function in is such that its Fourier transform equals for (or for ), then .

     

Lomonosov's theorem cannot be extended to chains of four operators

We show that the celebrated Lomonosov theorem cannot be improved by increasing the number of commuting operators. Specifically, we prove that if is the operator without a non-trivial closed invariant subspace constructed by C. J. Read, then there are three operators , and (non-multiples of the identity) such that commutes with , commutes with , commutes with , and is compact. It is also shown that the commutant of contains only series of .

     

Invertible completions of upper triangular operator matrices

In this note we prove that if

is a upper triangular operator matrix acting on the Banach space , then is invertible for some if and only if and satisfy the following conditions:

(i)

is left invertible;

(ii)

is right invertible;

(iii)

.

Furthermore we show that , where is the union of certain of the holes in which happen to be subsets of .

     

A note on -bases of rings

Let be a tower of rings of characteristic . Suppose that is a finitely presented -module. We give necessary and sufficient conditions for the existence of -bases of over . Next, let be a polynomial ring where is a perfect field of characteristic , and let be a regular noetherian subring of containing such that . Suppose that is a free -module. Then, applying the above result to a tower of rings, we shall show that a polynomial of minimal degree in is a -basis of over .

     

Completely continuous multilinear operators on spaces

Given a -linear operator from a product of spaces into a Banach space , our main result proves the equivalence between being completely continuous, having an -valued separately continuous extension to the product of the biduals and having a regular associated polymeasure. It is well known that, in the linear case, these are also equivalent to being weakly compact, and that, for , being weakly compact implies the conditions above but the converse fails.

     

Some corollaries of Frobenius' normal -complement theorem

For a prime divisor of the order of a finite group , we present the set of -subgroups generating . In particular, we present the set of primary subgroups of generating the last member of the lower central series of . The proof is based on the Frobenius Normal -Complement Theorem and basic properties of minimal nonnilpotent groups. Let be a group and a group-theoretic property inherited by subgroups and epimorphic images such that all minimal non--subgroups (-subgroups) of are not nilpotent. Then (see the lemma), if is generated by all -subgroups of it follows that is a -group.

     

Normalizers of the congruence subgroups of the Hecke group II

Let . Let be an ideal of and let be the maximal ideal of such that . Then . In particular, if is square free, then is self-normalized in .

     

Conjugate -connections and holonomy groups

In this article we show that when the structure group of the reducible principal bundle is and is an -subbundle of , the rank of the holonomy group of a connection which is gauge equivalent to its conjugate connection is less than or equal to , and use the estimate to show that for all odd prime , if the holonomy group of the irreducible connection as above is simple and is not isomorphic to , , or , then it is isomorphic to .

     

Open covers and partition relations

An open cover of a topological space is said to be an -cover if there is for each finite subset of the space a member of the cover which contains the finite set, but the space itself is not a member of the cover. We prove theorems which imply that a set of real numbers has Rothberger's property if, and only if, for each positive integer , for each -cover of , and for each function from the two-element subsets of , there is a subset of such that is constant on , and each element of belongs to infinitely many elements of (Theorem 1). A similar characterization is given of Menger's property for sets of real numbers (Theorem 6).

     

No submaximal topology on a countable set is -complementary

Two -topologies and given on the same set , are called transversal if their union generates the discrete topology on . The topologies and are -complementary if they are transversal and their intersection is the cofinite topology on . We establish that for any connected Tychonoff topology there exists a connected Tychonoff transversal one. Another result is that no -complementary topology exists for the maximal topology constructed by van Douwen on the rational numbers. This gives a negative answer to Problem 162 from Open Problems in Topology (1990).

     

Annihilating a subspace of with the sign of a continuous function

Let be a -finite, nonatomic, Baire measure space. Let be a finite dimensional subspace of . There is a bounded, continuous function, , defined on , such that

(1) for all , and (2) almost everywhere.

     

On pointed Hopf algebras of dimension

In this note we describe nonsemisimple Hopf algebras of dimension with coradical isomorphic to , abelian of order , over an algebraically closed field of characteristic zero. If is cyclic or , then we also determine the number of isomorphism classes of such Hopf algebras.

     

Universal -lattices of minimal rank

Let be the minimal rank of -universal -lattices, by which we mean positive definite -lattices which represent all positive -lattices of rank . It is a well known fact that for . In this paper, we determine and find all -universal lattices of rank for .

     

The first occurrence for the irreducible modules of general linear groups in the polynomial algebra

Let be a prime number and let be the group of all invertible matrices over the prime field . It is known that every irreducible -module can occur as a submodule of , the polynomial algebra with variables over . Given an irreducible -module , the purpose of this paper is to find out the first value of the degree of which occurs as a submodule of , the subset of consisting of homogeneous polynomials of degree . This generalizes Schwartz-Tri's result to the case of any prime .

     

Remarks about Schlumprecht space

Let denote the Schlumprecht space. We prove that

(1) is finitely disjointly representable in ;

(2) contains an -spreading model;

(3) for any sequence of natural numbers, is isomorphic to the space .

     

Perfect cliques and colorings of Polish spaces

A coloring of a set is any subset of , where 1$"> is a natural number. We give some sufficient conditions for the existence of a perfect -homogeneous set, in the case where is and is a Polish space. In particular, we show that it is sufficient that there exist -homogeneous sets of arbitrarily large countable Cantor-Bendixson rank. We apply our methods to show that an analytic subset of the plane contains a perfect -clique if it contains any uncountable -clique, where is a natural number or (a set is a -clique in if the convex hull of any of its -element subsets is not contained in ).

     

Iwasawa invariants and class numbers of quadratic fields for the prime

Let be a square-free integer with and . Put and . For the cyclotomic -extension of , we denote by the -th layer of over . We prove that the -Sylow subgroup of the ideal class group of is trivial for all integers if and only if the class number of is not divisible by the prime . This enables us to show that there exist infinitely many real quadratic fields in which splits and whose Iwasawa -invariant vanishes.

     

On approximately convex functions

A real valued function defined on a real interval is called -convex if it satisfies

The main results of the paper offer various characterizations for -convexity. One of the main results states that is -convex for some positive and if and only if can be decomposed into the sum of a convex function, a function with bounded supremum norm, and a function with bounded Lipschitz-modulus. In the special case , the results reduce to that of Hyers, Ulam, and Green obtained in 1952 concerning the so-called -convexity.

     

On compact subsets in coechelon spaces of infinite order

For coechelon spaces of infinite order it is proved that every compact subset of is contained in a closed absolutely convex hull of some null sequence if and only if the matrix is regularly decreasing.