Wave equations with time-dependent dissipation II. Effective dissipation

This article is intended to present a construction of structural representations of solutions to the Cauchy problem for wave equations with time-dependent dissipation above scaling. These representations are used to give estimates of the solution and its derivatives based on Lq(Rn), q?2.The article represents the second part within a series. In [Jens Wirth, Wave equations with time-dependent dissipation I. Non-effective dissipation, J. Differential Equations 222 (2) (2006) 487-514] weak dissipations below scaling were discussed.     

Estimates for some convolution operators with singularities in their kernels on a sphere with applications

We study convolution operators whose kernels have singularities on the unit sphere. For these operators we obtainH ^p -H ^q estimates, where p is less or equal q, and prove their sharpness. To this end, we develop a new method that uses special representations for the symbol of such operators as sums of certain oscillatory integrals and applies the stationary phase method and A. Miyachi results for model oscillating multipliers. Moreover, we also obtain estimates for operators from L ^p to BMO and those from BMO to BMO.     

Estimates for Green function and Poisson kernels of higher-order Dirichlet boundary value problems

Pointwise estimates are derived for the kernels associated to the polyharmonic Dirichlet problem on bounded smooth domains. As a consequence, one obtains optimal weighted L^p-L^q-regularity estimates for weights involving the distance function.     

Perturbation estimates for the nonlinear matrix equation X−A * X q A=Q (0<q<1)

The nonlinear matrix equation X?A ^* X ^q A=Q with 0<q<1 is investigated. Two perturbation estimates for the unique positive definite solution of the equation are derived. The theoretical results are illustrated by numerical examples.     

Potential operators associated with Hankel and Hankel-Dunkl transforms

We study Riesz and Bessel potentials in the settings of Hankel transform, modified Hankel transform and Hankel-Dunkl transform. We prove sharp or qualitatively sharp pointwise estimates of the corresponding potential kernels. Then we characterize those 1 ≤ p, q≤∞, for which the potential operators satisfy L ^p -L ^q estimates. In case of the Riesz potentials, we also characterize those 1 ≤ p, q ≤ ∞, for which two-weight L ^p -L ^q estimates, with power weights involved, hold. As a special case of our results, we obtain a full characterization of two power-weight L ^p -L ^q bounds for the classical Riesz potentials in the radial case. This complements an old result of Rubin and its recent reinvestigations by De Nápoli, Drelichman and Durán, and Duoandikoetxea.     

Sharp Weak Type Inequalities for the Dyadic Maximal Operator

We obtain sharp estimates for the localized distribution function of $\mathcal{M}\phi $ , when ? belongs to L ^p,∞ where $\mathcal{M}$ is the dyadic maximal operator. We obtain these estimates given the L ¹ and L ^q norm, q<p and certain weak-L ^p conditions.In this way we refine the known weak (1,1) type inequality for the dyadic maximal operator. As a consequence we prove that the inequality 0.1 is sharp allowing every possible value for the L ¹ and the L ^q norm for a fixed q such that 1<q<p, where ∥?∥ _p,∞ is the usual quasi norm on L ^p,∞.     

Global estimates of fundamental solutions for higher-order Schr?dinger equations

In this paper we first establish global pointwise time-space estimates of the fundamental solution for Schr?dinger equations, where the symbol of the spatial operator is a real non-degenerate elliptic polynomial. Then we use such estimates to establish related L ^p ?CL ^q estimates on the Schr?dinger solution. These estimates extend known results from the literature and are sharp. This result was lately already generalized to a degenerate case (cf. [4]).     

A scale-invariant Klein–Gordon model with time-dependent potential

The goal of the present paper is to derive statements about energy estimates as well as L ^p ?L ^q decay estimates for a Klein?CGordon model with a particular time-dependent mass. The study of this special case of a scale-invariant model is an important step within a systematic investigation of Klein?CGordon models with time-dependent mass.     

On the discrepancy of some special sequences

We obtain estimates for the discrepancy of the sequence (xs^(d)(q;n))_n=0^∞, where s^(d)(q;n) denotes the sum of the dth powers of the q-ary digits of the nonnegative integer n and x is an irrational number of finite approximation type. Furthermore metric results for a similar type of sequences are given.     

Compactness of composition operators on a Hilbert space of Dirichlet series

In this paper new L_α^p→L_β^q estimates are proved for translation-invariant Radon transforms along curves for α?β and p<q. For a fixed α and β, if p is sufficiently close to 2 the best possible q is obtained, up to ε. The method is related to that of [5].     

Intrinsic Ultracontractivity for Schrödinger Operators Based on Fractional Laplacians

We study the Feynman-Kac semigroup generated by the Schrödinger operator based on the fractional Laplacian ??(???Δ)^α/2???q in R ^d , for q?≥?0, α?∈?(0,2). We obtain sharp estimates of the first eigenfunction φ ₁ of the Schrödinger operator and conditions equivalent to intrinsic ultracontractivity of the Feynman-Kac semigroup. For potentials q such that lim_{|x| →?∞?} q(x)?=?∞ and comparable on unit balls we obtain that φ ₁(x) is comparable to (|x|?+?1)^???d???α (q(x)?+?1)^???1 and intrinsic ultracontractivity holds iff lim_{|x| →?∞?} q(x)/log|x|?=?∞. Proofs are based on uniform estimates of q-harmonic functions.     

On approximation of real numbers by algebraic numbers of bounded degree

Dirichlet proved that for any real irrational number ξ there exist infinitely many rational numbers p/q such that |ξ−p/q|<q⁻². The correct generalization to the case of approximation by algebraic numbers of degree ?n, n>2, is still unknown. Here we prove a result which improves all previous estimates concerning this problem for n>2.     

Metrics and smoothing of translation-invariant Radon transforms along curves

In this paper new L_α^p→L_β^q estimates are proved for translation-invariant Radon transforms along curves for α?β and p<q. For a fixed α and β, if p is sufficiently close to 2 the best possible q is obtained, up to ε. The method is related to that of Greenblatt (Ph.D. Thesis, Princeton University, 1998).     

Estimates for the Hill operator, II

Consider the Hill operator Ty=-y^″+q(t)y in L²(R), where the real potential q is 1-periodic and q,q^′∈L²(0,1). The spectrum of T consists of spectral bands separated by gaps γ_n,n?1 with length |γ_n|?0. We obtain two-sided estimates of the gap lengths ∑n²|γ_n|² in terms of . Moreover, we obtain the similar two-sided estimates for spectral data (the height of the corresponding slit on the quasimomentum domain, action variables for the KdV equation and so on). In order prove this result we use the analysis of a conformal mapping corresponding to quasimomentum of the Hill operator. That makes it possible to reformulate the problems for the differential operator as the problems of the conformal mapping theory. Then the proof is based on the analysis of the conformal mapping, the embedding theorems and the identities. Furthermore, we obtain the similar two-sided estimates for potentials which have p?2 derivatives.     

On exact order estimates of N-widths of classes of functions analytic in a simply connected domain

In the spaces E _q(Ω), 1 < q < ∞, introduced by Smirnov, we obtain exact order estimates of projective and spectral n-widths of the classes W ^r E _p(Ω) and W ^r E _p(Ω)Ф in the case where p and q are not equal. We also indicate extremal subspaces and operators for the approximative values under consideration.     

Approximation of classes of functions of many variables by their orthogonal projections onto subspaces of trigonometric polynomials

In the spaceL _q, 1<q<∞ we establish estimates for the orders of the best approximations of the classes of functions of many variablesB _1,θ ^r andB _p,α ^r by orthogonal projections of functions from these classes onto the subspaces of trigonometric polynomials. It is shown that, in many cases, the estimates obtained in the present work are better in order than in the case of approximation by polynomials with harmonics from the hyperbolic cross.     

L q theory for a generalized Stokes System

Regularity properties of solutions to the stationary generalized Stokes system are studied. The extra stress tensor is assumed to have a growth given by some N-function, which includes the situation of p-growth. We show results about differentiability of weak solutions. As a consequence we obtain the gradient L ^q estimates for the problem. These estimates are applied to the stationary generalized Navier Stokes equations.     

Sur le taux de dissipation d'entropie sans troncature angulaire

We show that, if a function has a bounded entropic dissipation rate, then it satisfies some regularity like estimates; this is done in linear and nonlinear 3D cases, without angular cutoff, and for power laws as 1/r^s, with s > 2.     

The lengths of Hermitian self-dual extended duadic codes

Duadic codes are a class of cyclic codes that generalize quadratic residue codes from prime to composite lengths. For every prime power q, we characterize integers n such that there is a duadic code of length n over F_q² with a Hermitian self-dual parity-check extension. We derive asymptotic estimates for the number of such n as well as for the number of lengths for which duadic codes exist.     

Multilinear interpolation between adjoint operators

Multilinear interpolation is a powerful tool used in obtaining strong-type boundedness for a variety of operators assuming only a finite set of restricted weak-type estimates. A typical situation occurs when one knows that a multilinear operator satisfies a weak L^q estimate for a single index q (which may be less than one) and that all the adjoints of the multilinear operator are of similar nature, and thus they also satisfy the same weak L^q estimate. Under this assumption, in this note we give a general multilinear interpolation theorem which allows one to obtain strong-type boundedness for the operator (and all of its adjoints) for a large set of exponents. The key point in the applications we discuss is that the interpolation theorem can handle the case q?1. When q>1, weak L^q has a predual, and such strong-type boundedness can be easily obtained by duality and multilinear interpolation (cf. Interpolation Spaces, An Introduction, Springer, New York, 1976; Math. Ann. 319 (2001) 151; in: Function Spaces and Applications (Lund, 1986), Lecture Notes in Mathematics, Vol. 1302, Springer, Berlin, New York, 1988; J. Amer. Math. Soc. 15 (2002) 469; Proc. Amer. Math. Soc. 21 (1969) 441).