Commutator inequalities associated with the polar decomposition

Let be a polar decomposition of an complex matrix . Then for every unitarily invariant norm , it is shown that

where denotes the operator norm. This is a quantitative version of the well-known result that is normal if and only if . Related inequalities involving self-commutators are also obtained.

     

The expected norm of random polynomials

The results of this paper concern the expected norm of random polynomials on the boundary of the unit disc (equivalently of random trigonometric polynomials on the interval ). Specifically, for a random polynomial

let

Assume the random variables , are independent and identically distributed, have mean 0, variance equal to 1 and, if 2$">, a finite moment . Then

and

as .

In particular if the polynomials in question have coefficients in the set (a much studied class of polynomials), then we can compute the expected norms of the polynomials and their derivatives

and

This complements results of Fielding in the case, Newman and Byrnes in the case, and Littlewood et al. in the case.

     

Explicit continued fractions with expected partial quotient growth

For let be the continued fraction expansion of . Write

We construct some numbers 's with

     

Gateaux derivative of norm

We prove that for Hilbert space operators and , it follows that

,\end{displaymath}">

where . Using the concept of -Gateaux derivative, we apply this result to characterize orthogonality in the sense of James in , and to give an easy proof of the characterization of smooth points in .

     

Mixed-mean inequalities for subsets

For let and denote the arithmetic mean and geometric mean of elements of , respectively. It is proved that if is an integer in , then

with equality if and only if . Furthermore, as a generalization of this inequality, a mixed power-mean inequality for subsets is established.

     

An elementary proof of the law of quadratic reciprocity over function fields

Let and be relatively prime monic irreducible polynomials in (). In this paper, we give an elementary proof for the following law of quadratic reciprocity in :

where is the Legendre symbol.

     

Sharp estimates for the maximum over minimum modulus of rational functions

Let 1$">, and be a rational function with numerator, denominator of degree , respectively. In several applications, one needs to know the size of the set such that for ,

In an earlier paper, we showed that

where denotes linear Lebesgue measure. Here we obtain, for each , the sharp version of this inequality in terms of condenser capacity. In particular, we show that as ,

     

Multiple solutions for elliptic problems with singular and sublinear potentials

For certain positive numbers and we establish the multiplicity of solutions to the problem

where is a bounded open domain in containing the origin with smooth boundary while is continuous, superlinear at zero and sublinear at infinity.

     

Large time behavior of small solutions to subcritical derivative nonlinear Schrödinger equations

We consider the derivative nonlinear Schrödinger equations

where the coefficient satisfies the time growth condition

is a sufficiently small constant and the nonlinear interaction term consists of cubic nonlinearities of derivative type

where and . We suppose that the initial data satifsfy the exponential decay conditions. Then we prove the sharp decay estimate , for all , where . Furthermore we show that for there exist the usual scattering states, when and the modified scattering states, when

     

Exact estimates for integrals involving Dirichlet series with nonnegative coefficients

We consider the Dirichlet series

with coefficients for all . Among others, we prove exact estimates of certain weighted -norms of on the unit interval for any , in terms of the coefficients . Our estimation is based on the close relationship between Dirichlet series and power series. This enables us to derive exact estimates for integrals involving the former one by relying on exact estimates for integrals involving the latter one. As a by-product, we obtain an analogue of the Cauchy-Hadamard criterion of (absolute) convergence of the more general Dirichlet series

with complex coefficients .

     

Sharp maximal inequality for stochastic integrals

Let be a nonnegative supermartingale and be a predictable process with values in . Let denote the stochastic integral of with respect to . The paper contains the proof of the sharp inequality

where . A discrete-time version of this inequality is also established.

     

The limiting case of the Marcinkiewicz integral: growth for convex sets

The Marcinkiewicz integral

plays a well-known and prominent role in harmonic analysis. In this paper, we estimate the growth of it in the limiting case . Throughout, we assume that is convex; it is interesting that this condition cannot be dropped.

     

Asymptotic estimates for a two-dimensional problem with polynomial nonlinearity

In this paper we give asymptotic estimates of the least energy solution of the functional

as goes to infinity. Here is a smooth bounded domain of . Among other results we give a positive answer to a question raised by Chen, Ni, and Zhou (2000) by showing that .

     

An improved Mordell type bound for exponential sums

For a sparse polynomial , with and , we show that

thus improving upon a bound of Mordell. Analogous results are obtained for Laurent polynomials and for mixed exponential sums.

     

The Molchanov-Vainberg Laplacian

It is well known that the Green function of the standard discrete Laplacian on ,

exhibits a pathological behavior in dimension . In particular, the estimate

fails for . This fact complicates the study of the scattering theory of discrete Schrödinger operators. Molchanov and Vainberg suggested the following alternative to the standard discrete Laplacian,

and conjectured that the estimate

holds for all . In this paper we prove this conjecture.

     

On criteria for extremality of Teichmüller mappings

Let be a Teichmüller self-mapping of the unit disk corresponding to a holomorphic quadratic differential . If satisfies the growth condition (as ), for any given 0$">, then is extremal, and for any given , there exists a subsequence of such that

is a Hamilton sequence. In addition, it is shown that there exists with bounded Bers norm such that the corresponding Teichmüller mapping is not extremal, which gives a negative answer to a conjecture by Huang in 1995.

     

The yellow cake

In this paper we consider the following property:

For every function there are functions

(for ) such that

We show that, despite some expectation suggested by S. Shelah (1997), does not imply . Next, we introduce cardinal characteristics of the continuum responsible for the failure of .

     

Weak norms of random sums

Let denote a sequence of measurable functions on , and let denote the weak norm. It is shown that

where is a sequence of independent random variables taking on values and with equal probability. Moreover, it is shown that

The paper concludes by providing an example indicating that, if , then the estimate

is the best possible.

     

Weak weighted inequalities for a dyadic one-sided maximal function in

In this note we introduce a dyadic one-sided maximal function defined as

where is a certain cube associated with the dyadic cube and . We characterize the pair of weights for which the maximal operator applies into weak- for .

     

On a Littlewood-Paley type inequality

The following is proved: If is a function harmonic in the unit ball and if then the inequality

holds, where is the nontangential maximal function of This improves a recent result of Stoll. This inequality holds for polyharmonic and hyperbolically harmonic functions as well.