Sobolev estimates for operators given by averages over cones

We prove a result related to work by A. Greenleaf and G. Uhlmann concerning Sobolev estimates for operators given by averages over cones. This is done using the almost orthogonality lemma of Cotlar and Stein, and the van der Corput lemma on oscillatory integrals.

     

On the discrepancy of (0,1)-sequences

We give bounds for the L^p-discrepancy, , of the van der Corput sequence in base 2. Further, we give a best possible upper bound for the star discrepancy of (0,1)-sequences and show that this bound is attained for the van der Corput sequence. Finally, we give a (0,1)-sequence with essentially smaller star discrepancy than for the van der Corput sequence.     

Pointwise van der Corput lemma for functions of several variables

A multidimensional version of the well-known van der Corput lemma is presented. A class of phase functions is described for which the corresponding oscillatory integrals satisfy a multidimensional decay estimate. The obtained estimates are uniform with respect to parameters on which the phases and amplitudes may depend.     

Precise distribution properties of the van der Corput sequence and related sequences

The discrepancy is a quantitative measure for the irregularity of distribution of sequences in the unit interval. This article is devoted to the precise study of L_p–discrepancies of a special class of digital (0,1)–sequences containing especially the van der Corput sequence. We show that within this special class of digital (0,1)–sequences over ℤ₂ the van der Corput sequence is the worst distributed sequence with respect to L₂–discrepancy. Further we prove that the L_p–discrepancies of the van der Corput sequence satisfy a central limit theorem and we study the discrepancy function of (0,1)–sequences pointwise.     

A thorough analysis of the discrepancy of shifted Hammersley and van der Corput point sets

We study the star discrepancy of Hammersley nets and van der Corput sequences which are important examples of low-dimensional quasi-Monte Carlo point sets. By a so-called digital shift, the quality of distribution of these point sets can be improved. In this paper, we advance and extend existing bounds on digitally shifted Hammersley and van der Corput point sets and establish criteria for the choice of digital shifts leading to optimal results. Our investigations are partly based on a close analysis of certain sums of distances to the nearest integer. Mathematics Subject Classi cation (2000) 11K38; 11K09     

Van der Corput sets with respect to compact groups

We study the notion of van der Corput sets with respect to general compact groups.     

Van der Corput Sequences and Linear Permutations

This extended abstract is concerned with the irregularities of distribution of one-dimensional permuted van der Corput sequences that are generated from linear permutations. We show how to obtain upper bounds for the discrepancy and diaphony of these sequences, by relating them to Kronecker sequences and applying earlier results of Faure and Niederreiter.     

Pseudo-randomness of van der Corput’s sequences

The aim of this paper is to find main terms of the star D _N ^* and extremal D _N discrepancies of the two dimensional sequence (x _n , x _n+1), n = 0, 1, 2, ..., N − 1, where x _n , n = 0, 1, 2, ..., is the van der Corput sequence. This give a quantitative form of a well-known result that van der Corput sequence is not pseudorandom. This research was supported by the Slovak Academy of Sciences Vega Grant No. 2/7138/27.     

Uniform distribution of prime powers and sets of recurrence and van der Corput sets in ℤ k

We establish new results on sets of recurrence and van der Corput sets in ? ^k which refine and unify some of the previous results obtained by Sárk?zy, Furstenberg, Kamae and Mèndes France, and Bergelson and Lesigne. The proofs utilize a general equidistribution result involving prime powers which is of independent interest.     

On the distribution of the van der Corput sequences

Archiv der Mathematik - For an integer $$p\ge 2$$ , let $$\{x_n\}_{n\in {\mathbb {N}}}\subset {\mathbb {T}}$$ be the p-adic van der Corput sequence. For intervals $$[0,\alpha )\subset {\mathbb...     

Ruzsa’s problem on sets of recurrence

The main purpose of this paper is to prove the existence of Poincaré sequences of integers which are not van der Corput sets. This problem was considered in I. Ruzsa’s expository article [R₁] (1982–83) on correlative and intersective sets. Thus the existence is shown of a positive non-continuous measureμ on the circle which Fourier transform vanishes on a set of recurrence, i.e.S={n ∈Z; (n)=0} is a set of recurrence but not a van der Corput set. The method is constructive and involves some combinatorial considerations. In fact, we prove that the generic density condition for both properties are the same.     

Fourier transforms of irregular mixed homogeneous hypersurface measures

With the help of Van der Corput lemmas, decay estimates are proven for Fourier transforms of mixed homogeneous hypersurface measures with densities that can be quite irregular. The primary results are local in nature, but can be extended to global theorems in an appropriate sense. The estimates are sharp for a certain range of indices in the theorems.     

Regularities of the Distribution of β-adic van der Corput Sequences

For Pisot numbers β with irreducible β-polynomial, we prove that the discrepancy function D(N, [0,y)) of the β-adic van der Corput sequence is bounded if and only if the β-expansion of y is finite or its tail is the same as that of the expansion of 1. If β is a Parry number, then we can show that the discrepancy function is unbounded for all intervals of length y ? \Bbb Q(b) y \notin {\Bbb Q}(\beta) . We give explicit formulae for the discrepancy function in terms of lengths of iterates of a reverse β-substitution.     

Regularities of the Distribution of β-adic van der Corput Sequences

For Pisot numbers β with irreducible β-polynomial, we prove that the discrepancy function D(N, [0,y)) of the β-adic van der Corput sequence is bounded if and only if the β-expansion of y is finite or its tail is the same as that of the expansion of 1. If β is a Parry number, then we can show that the discrepancy function is unbounded for all intervals of length . We give explicit formulae for the discrepancy function in terms of lengths of iterates of a reverse β-substitution.     

Order of the Epstein Zeta-Function in the Critical Strip

Let Q(x₁,...,x_k) be a positive quadratic form of k2 variables and let (s,Q) be the Epstein zeta-function of the form Q. The growth rate of (s,Q) on the line Re s = (k–1)/2 is investigated. For k4 and for an integral form Q, the problem is reduced to a similar problem on the behavior of the Dirichlet L-series on the line Re s = 1/2. In the case k=3, the diagonal form over is investigated by the van der Corput method. For k=2, the known result due to Titchmarsh is re-proved by using a variant of the van der Corput method given by Heath-Brown. Bibliography: 9 titles.     

A van der Corput lemma for the -adic numbers

We prove a version of van der Corput's lemma for polynomials over the -adic numbers.

     

Weak Hopf lemma for the invariant Laplacian and related elliptic operators

We obtain a weak version of the Hopf lemma for the invariant Laplacian on the unit ball of the complex n$n$-space. We also show that our result is sharp in some sense. Motivated by this result, we also consider a class of degenerate elliptic operators with the degeneracy depending on the distance to the boundary of the domain. We study the dependence of the validity of Hopf lemma on the degree of degeneracy of the operator. We show that Hopf lemma holds if the degeneracy is small and fails in general if the degeneracy is large. What is more interesting is the critical case for which we show that certain weak version of Hopf lemma holds.     

Estimates of oscillatory integrals with stationary phase and singular amplitude: Applications to propagation features for dispersive equations

In this paper, we study time‐asymptotic propagation phenomena for a class of dispersive equations on the line by exploiting precise estimates of oscillatory integrals. We propose first an extension of the van der Corput Lemma to the case of phases which may have a stationary point of real order and amplitudes allowed to have an integrable singular point. The resulting estimates provide optimal decay rates which show explicitly the influence of these two particular points. Then we apply these abstract results to solution formulas of a class of dispersive equations on the line defined by Fourier multipliers. Under the hypothesis that the Fourier transform of the initial data has a compact support or an integrable singular point, we derive uniform estimates of the solutions in space‐time cones, describing their motions when the time tends to infinity. The method permits also to show that symbols having a restricted growth at infinity may influence the dispersion of the solutions: we prove the existence of a cone, depending only on the symbol, in which the solution is time‐asymptotically localized. This corresponds to an asymptotic version of the notion of causality for initial data without compact support.     

An Approximation Algorithm for a Problem of Partitioning a Sequence into Clusters with Constraints on Their Cardinalities

We give a slight refinement to the process by which estimates for exponential sums are extracted from bounds for Vinogradov’s mean value. Coupling this with the recent works of Wooley, and of Bourgain, Demeter and Guth, providing optimal bounds for the Vinogradov mean value, we produce a powerful new kth derivative estimate. Roughly speaking, this improves the van der Corput estimate for k ≥ 4. Various corollaries are given, showing for example that \(\zeta \left( {\sigma + it} \right){ \ll _\varepsilon }{t^{{{\left( {1 - \sigma } \right)}^{3/2}}/2 + \varepsilon }}\) for t ≥ 2 and 0 ≤ σ ≤ 1, for any fixed ε > 0.     

Some problems of the theory of quantum statistical systems with an energy spectrum of the fractional-power type

We consider the problem of the effective interaction potential in a quantum many-particle system leading to the fractional-power dispersion law. We show that passing to fractional-order derivatives is equivalent to introducing a pair interparticle potential. We consider the case of a degenerate electron gas. Using the van der Waals equation, we study the equation of state for systems with a fractional-power spectrum. We obtain a relation between the van der Waals constant and the phenomenological parameter ??, the fractional-derivative order. We obtain a relation between energy, pressure, and volume for such systems: the coefficient of the thermal energy is a simple function of ??. We consider Bose??Einstein condensation in a system with a fractional-power spectrum. The critical condensation temperature for 1 < ?? < 2 is greater in the case under consideration than in the case of an ideal system, where ?? = 2.