Trudinger type inequalities in and their best exponents

We study Trudinger type inequalities in and their best exponents . We show for , ( is the surface area of the unit sphere in ), there exists a constant such that

for all . Here is defined by

It is also shown that with is false, which is different from the usual Trudinger's inequalities in bounded domains.

     

A factorization theorem for the derivative of a function in

We show that a function is the derivative of a function in the Hardy space of the unit disk for if and only if where and . Here, can be chosen to be non-vanishing, , and . As an application, we characterize positive measures on the unit disk such that the operator is bounded from the tent space to , where .

     

A description of Hilbert -modules in which all closed submodules are orthogonally closed

Let , be -algebras and a full Hilbert --bimodule such that every closed right submodule is orthogonally closed, i.e., . Then there are families of Hilbert spaces , such that and are isomorphic to -direct sums , resp. , and is isomorphic to the outer direct sum .

     

The maximal ideal space of with respect to the Hadamard product

It is shown that the space of all regular maximal ideals in the Banach algebra with respect to the Hadamard product is isomorphic to The multiplicative functionals are exactly the evaluations at the -th Taylor coefficient. It is a consequence that for a given function in and for a function holomorphic in a neighborhood of with and for all the function is in

     

The distribution of solutions of the congruence

For a cube of size , we obtain a lower bound on so that is nonempty, where is the algebraic subset of defined by

a positive integer and an integer not divisible by . For we obtain that is nonempty if , for we obtain that is nonempty if , and for we obtain that is nonempty if . Using the assumption of the Grand Riemann Hypothesis we obtain is nonempty if .

     

On the range and the kernel of the operator

Let denote the algebra of (bounded linear) operators on the separable complex Hilbert space , and let denote a norm ideal in . For , let the derivation be defined by , and let be defined by . The main result of this paper is to show that if , are contractions, then for every operator such that , then for all .

     

The Wiener transform on the Besicovitch spaces

In his fundamental research on generalized harmonic analysis, Wiener proved that the integrated Fourier transform defined by is an isometry from a nonlinear space of functions of bounded average quadratic power into a nonlinear space of functions of bounded quadratic variation. We consider this Wiener transform on the larger, linear, Besicovitch spaces defined by the norm . We prove that maps continuously into the homogeneous Besov space for and , and is a topological isomorphism when .

     

Global bifurcation in generic systems of nonlinear Sturm-Liouville problems

We consider the system of coupled nonlinear Sturm-Liouville boundary value problems where , are real spectral parameters. It will be shown that if the functions and are `generic' then for all integers , there are smooth 2-dimensional manifolds , , of `semi-trivial' solutions of the system which bifurcate from the eigenvalues , , of , , respectively. Furthermore, there are smooth curves , , along which secondary bifurcations take place, giving rise to smooth, 2-dimensional manifolds of `non-trivial' solutions. It is shown that there is a single such manifold, , which `links' the curves , . Nodal properties of solutions on and global properties of are also discussed.

     

A Liouville-type theorem on halfspaces for the Kohn Laplacian

Let be the Kohn Laplacian on the Heisenberg group and let be a halfspace of whose boundary is parallel to the center of . In this paper we prove that if is a non-negative -superharmonic function such that

then in .

     

On covering multiplicity

Let be a system of arithmetic sequences which forms an -cover of (i.e. every integer belongs at least to members of ). In this paper we show the following surprising properties of : (a) For each there exist at least subsets of with such that . (b) If forms a minimal -cover of , then for any there is an such that for every there exists an for which and

     

Spectral multiplier theorem for spaces associated with some Schrödinger operators

Let be the semigroup of linear operators generated by a Schrödinger operator , where is a nonnegative polynomial. We say that is an element of if the maximal function belongs to . A criterion on functions which implies boundedness of the operators on is given.

     

The growth theorem of convex mappings on the unit ball in

Let be an arbitrary norm on . Let be a normalized biholomorphic convex mapping on the unit ball in with respect to the norm . We will give an upper bound of the growth of .

     

Matrix presentations of braids and applications

We show that there exists a one-to-one correspondence between the class of certain block tridiagonal matrices with the entries or and the free monoid generated by generators and relation and give some applications for braids. In particular, we give new formulation of the reduced Alexander matrices for closed braids.

     

On the product of two generalized derivations

Two elements and in a ring determine a generalized derivation on by setting for any in . We characterize when the product is a generalized derivation in the cases when the ring is the algebra of all bounded operators on a Banach space , and when is a -algebra . We use these characterizations to compute the commutant of the range of .

     

On the irrationality of a certain series

We prove that if is an integer greater than one, and are any positive rationals such that for all integers , then

is irrational and is not a Liouville number.

     

Character degrees and local subgroups of -separable groups

Let be a finite -solvable group for different primes and . Let and be such that . We prove that every of -degree has -degree if and only if and .

     

Compact operators whose real and imaginary parts are positive

Let be a compact operator on a Hilbert space such that the operators and are positive. Let be the singular values of and the eigenvalues of , all enumerated in decreasing order. We show that the sequence is majorised by . An important consequence is that, when is less than or equal to , and when this inequality is reversed.

     

Fixed point iteration for pseudocontractive maps

Let be a compact convex subset of a real Hilbert space, ; a continuous pseudocontractive map. Let and be real sequences in [0,1] satisfying appropriate conditions. For arbitrary define the sequence iteratively by where are arbitrary sequences in . Then, converges strongly to a fixed point of . A related result deals with the convergence of to a fixed point of when is Lipschitz and pseudocontractive. Our theorems also hold for the slightly more general class of continuous hemicontractive nonlinear maps.

     

A finiteness theorem for a class of exponential congruences

For given elements and belonging to the ring of integers of a number field we consider the set of all tuples in for which divides for any and prove under some reasonable assumptions that the set of solutions is finite.

     

Nonexpansive, -continuous antirepresentations have common fixed points

Let be a closed convex subset of a Banach (dual Banach) space . By we denote an antirepresentation of a semitopological semigroup as nonexpansive mappings on . Suppose that the mapping is jointly continuous when has the weak (weak*) topology and the Banach space of bounded right uniformly continuous functions on has a right invariant mean. If is weakly compact (for some the set is weakly* compact) and norm separable, then has a common fixed point in .