On Lp estimates for the wave equation

New and more elementary proofs are given of two results due to W. Littman: (1) Let $\text{n ? 2, p ?}\text{2n}\text{(n ? 1)}$. The estimate $\text{∫∫ (¦▽u¦}{}^{\mathrm{p}}\text{+ ¦u}{}_{\mathrm{t}}\text{¦}{}^{\mathrm{p}}\text{) dx dt ? C ∫∫ ¦□u¦}{}^{\mathrm{p}}\text{dx dt}$ cannot hold for all u?C₀^∞(Q), Q a cube in $\text{R}{}^{\mathrm{<mi\; mathvariant="italic"\; is="true">n</mi>}}\text{× R}$, some constant C. (2) Let n ? 2, p ≠ 2. The estimate $\text{∫ (¦▽(t)¦}{}^{\mathrm{p}}\text{+ ¦u}{}_{\mathrm{t}}\text{(t)¦}{}^{\mathrm{p}}\text{) dx ? C(t) ∫ (¦▽u(0)¦}{}^{\mathrm{p}}\text{+ ¦u}{}_{\mathrm{t}}\text{(0)¦}{}^{\mathrm{p}}\text{) dx}$ cannot hold for all C^∞ solutions of the wave equation □u = 0 in $\text{R}{}^{\mathrm{n}}\text{x R}$; all t ?$\text{R}$; some function C: $\text{R}$ → $\text{R}$.     

Second-order optimality of randomized estimation and test procedures

Let P_{(Θ, τ)} 6 $\text{A}$, θ ∈ Θ ? $\text{R}$, τ ∈ T ? $\text{R}$^p denote a family of probability measures, where τ denotes the vector of nuisance parameters. Starting from randomized asymptotic maximum likelihood (as. m. l.) estimators for (θ, τ) we construct randomized estimators which are asymptotically median unbiased up to $\text{o(n}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}\text{)}$ resp. test procedures which are as. similar of level $\text{α + o(n}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}\text{)}$ (for testing θ = θ₀, τ ∈ T against one sided alternatives). The estimation procedures are second-order efficient in the class of estimators which are median unbiased up to $\text{o(n}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}\text{)}$ and the test procedures are second-order efficient in the class of tests which are as. of level $\text{α + o(n}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}\text{)}$. These results hold without any continuity condition on the family of probability measures.     

About some problems of spectral synthesis

For a given pair $\text{(A,b)∈}\text{R}{}^{\mathrm{n\times n}}\text{×}\text{R}{}^{\mathrm{n\times 1}}$ such that A is cyclic and b is a cyclic generator (with respect to A) of $\text{R}{}^{\mathrm{n\times 1}}$, it is shown that for every nonnegative integer m we can find a nonnegative integer t and a sequence $\text{{f}{}_{\mathrm{j\}}}{}^{\mathrm{t}}{}_{\mathrm{j=0}}\text{,f}{}_{\mathrm{j}}\text{∈}\text{R}{}^{\mathrm{1\times n}}$,so that a the zeros of the rational function det P(z), where $\text{P(z) = zI ? A ? ∑}{}^{\mathrm{t}}{}_{\mathrm{j=0}}\text{z}{}^{\mathrm{-(m+j)}}\text{b?}$_f, lie in the open unit disc in the complex plane. The result is directly applicable to a stabilizability problem for linear systems with a time delay in control action.     

Asymptotic behavior for an almost periodic,strongly dissipative wave equation

This paper deals with asymptotic behavior for (weak) solutions of the equation $\text{u}{}_{\mathrm{tt}}\text{? Δu + β(u}{}_{\mathrm{t}}\text{) ? ?(t, x)}$, on $\text{R}$⁺ × Ω; u(t, x) = 0, on $\text{R}$⁺ × ?Ω. If $\text{?∈L∞(R}{}^{\mathrm{+}}\text{,L}{}^2\text{(Ω))}$ and β is coercive, we prove that the solutions are bounded in the energy space, under weaker assumptions than those used by G. Prouse in a previous work. If in addition $\text{?}{}_{\mathrm{t}}\text{∈S}{}^2\text{(R}{}^{\mathrm{+}}\text{,L}{}^2\text{(Ω))}$ and ? is srongly almost-periodic, we prove for strongly monotone β that all solutions are asymptotically almost-periodic in the energy space. The assumptions made on β are much less restrictive than those made by G. Prouse: mainly, we allow β to be multivalued, and in the one-dimensional case β need not be defined everywhere.     

Positive solutions and large time behaviors of Schrödinger semigroups,Simon's problem

Let H = ?Δ + V, where V is a multiplication operator by a real-valued function V(x) on Rⁿ which is uniformly Hölder continuous and $\text{(1 + ¦x¦}{}^2\text{)}{}^{\mathrm{<mtext>?</mtext><mtext>2</mtext>}}\text{V(x) ∈ L}{}_{\mathrm{\infty }}\text{(R}{}^{\mathrm{n}}\text{)}$ for some ? > 4. The relationship between existence of positive solutions, with growth conditions, of Hg = 0 and asymptotic behaviors as t → ∞ of e^?th is established. Using it B. Simon's problem for H on R² is solved.     

Capacitance and the conformal module of quadrilaterals

Let ? be defined on T^r and have an absolutely convergent Fourier series

. Set $\text{∥?∥ = ∑ ¦?}{}_{\mathrm{k}}\text{¦}$. In this paper the problem of determining the limit of $\text{∥?}{}^{\mathrm{n}}\text{∥}$, as n → ∞, is studied.     

The number of different distances determined by n points in the plane

A classical problem in combinatorial geometry is that of determining the minimum number f(n) of different distances determined by n points in the Euclidean plane. In 1952, L. Moser proved that $\text{f(n)>n}{}^{\mathrm{<mtext>2</mtext><mtext>3</mtext>}}\text{(2)}\text{9}\text{3}$ and this has remained for 30 years as the best lower bound known for f(n). It is shown that $\text{f(n) > cn}{}^{\mathrm{<mtext>5</mtext><mtext>7</mtext>}}$ for some fixed constant c.     

On the enumeration of self-complementary m-placed relations

A formula for the number a_m(2n) of self-complementary m-placed relations is given. Then we obtain from this formula asymptotic results, e.g.$\text{a}{}_{\mathrm{m}}\text{(2n) ～ (2(}{}^{\mathrm{2n}}\text{)}{}^{\mathrm{<mtext>m</mtext><mtext>2?n</mtext>}}\text{)/n}$     

Sublinear perturbations of the differential equation y(n) = 0 and of the analogous difference equation

According to a result of A. Ghizzetti, for any solution y(t) of the differential equation where $\text{y}{}^{\mathrm{(n)}}\text{(t)+}\text{∑}\text{i=0}\text{n?1}\text{g}{}_{\mathrm{i}}\text{(t) y}{}^{\mathrm{i}}\text{(t)=0}\text{(t ? 1)}$, $\text{∝}{}_1{}^{\mathrm{\infty }}\text{¦g}{}_{\mathrm{i}}\text{(x)¦x}{}^{\mathrm{n?I?1}}\text{dx < ∞}$ (0 ?i ? n ?1, either y(t) = 0 for t ? 1 or there is an integer r with 0 ? r ? n ? 1 such that $\text{lim}\text{t → ∞}\text{y(t)}\text{t}{}^{\mathrm{r}}$ exists and ≠0. Related results are obtained for difference and differential inequalities. A special case of the former has interesting applications in the study of orthogonal polynomials.     

Self-correcting approximate solution by the iterative method for linear and nonlinear stochastic differential equations

The iterative method of the first author makes possible a self-correcting scheme for the approximate solution obtained. This scheme accelerates the convergence and substantially decreases the number of terms necessary in computation for applications involving either linear or nonlinear stochastic systems. A “feedback” term compensates for the approximation of the system inverse operator by a partial sum. Further, errors are determined in calculating the mean solution 〈y〉 and the correlation $\text{R}{}^{\mathrm{y}}\text{(t}{}^1\text{,t}{}^2\text{) = 〈y(t}{}^1\text{)}\text{y}\text{1}\text{(t}{}^2\text{)〉}$ by using the approximations 〈φ_n〉 and 〈φⁿ(t¹)φⁿ(t²)〉 , where φ_n represents n terms of the solution by the iterative method for stochastic differential equations.     

The Dirichlet problem for the Kohn Laplacian on the Heisenberg group,I

For (x,y,t)∈$\text{R}$ⁿ × $\text{R}$ⁿ × $\text{R}$, denote $\text{X}{}_{\mathrm{j}}\text{=}\text{?}\text{?x}{}_{\mathrm{j}}\text{+ 2y}{}_{\mathrm{j}}\text{?}\text{?t}\text{, y}{}_{\mathrm{j}}\text{=}\text{?}\text{?y}{}_{\mathrm{j}}\text{? 2x}{}_{\mathrm{j}}\text{?}\text{?t}$ and $\text{L}{}_{\mathrm{\alpha }}\text{=?}\text{1}\text{4}\text{∑}\text{j=1}\text{n}\text{X}{}_{\mathrm{j}}{}^2\text{+ Y}{}_{\mathrm{j}}{}^2\text{+}\text{?}\text{?t}$. When α = n ? 2q, $\text{L}$_a represents the action of the Kohn Laplacian □_b on q-forms on the Heisenberg group. For ?n < α < n, we construct a parametrix for the Dirichlet problem in smooth domains D near non-characteristic points of ?D. A point w of ?D is non-characteristic if one of X₁,…, X_n, Y₁,…, Y_n is transverse to ?D at w. This yields sharp local estimates in the Dirichlet problem in the appropriate non-isotropic Lipschitz classes. The main new tool is a “convolution calculus” of pseudo-differential operators that can be applied to the relevant layer potentials, for which the usual asymptotic composition formula is false. Characteristic points are treated in Part II.     

Arnold's canonical matrices and the asymptotic simplification of ordinary differential equations

Let A(x,ε) be an n×n matrix function holomorphic for |x|?x₀, 0<ε?ε₀, and possessing, uniformly in x, an asymptotic expansion $\text{A(x,ε)?Σ}{}^{\mathrm{\infty }}{}_{\mathrm{r=0}}\text{A}{}_{\mathrm{r}}\text{(x) ε}{}^{\mathrm{r}}$, as ε→0⁺. An invertible, holomorphic matrix function P(x,ε) with an asymptotic expansion $\text{P(x,ε)?Σ}{}^{\mathrm{\infty }}{}_{\mathrm{r=0}}\text{P}{}_{\mathrm{r}}\text{(x)ε}{}^{\mathrm{r}}$, as ε→0⁺, is constructed, such that the transformation y = P(x,ε)z takes the differential equation $\text{ε}{}^{\mathrm{h}}\text{dy}\text{dx}\text{= A(x,ε)y,h}$ a positive integer, into $\text{ε}{}^{\mathrm{h}}\text{dz}\text{dx}\text{= B(x,ε)z}$, where B(x,ε) is asymptotically equal, to all orders, to a matrix in a canonical form for holomorphic matrices due to V.I. Arnold.     

Combinatorial sums,identities and trace identities of the 2 × 2 matrices

An asymptotic formula, involving integrals, is given for certain combinatorial sums. By evaluating a multi-integral it is then found that as n → ∞, the codimensions c_n(F₂) and the trace codimensions t_n(F₂) of F₂, the 2 × 2 matrices, are asymptotically equal: $\text{c}{}_{\mathrm{n}}\text{(F}{}_2\text{) ? t}{}_{\mathrm{n}}\text{(F}{}_2\text{)}$.     

Decay of solutions of nonlinear wave equations in three space dimensions

Given a solution of the Cauchy problem for nonlinear wave equations of the type $\text{?}{}^2\text{u}\text{?t}{}^2\text{? Δu + f(u) = 0}$ in three space dimensions the asymptotic behaviour in time is considered. It is shown that for nonlinearities which behave like powers $\text{¦u¦}{}^{\mathrm{\sigma \; ?\; 1}}\text{u}$ uniform decay holds with a certain rate depending on σ if $\text{5 > σ >}\text{1}\text{2}\text{+}\text{1}\text{2}\text{√13}$, and moreover scattering states exist if σ is not too small. This improves former results of W. A. Strauss (J. Funct. Anal. 2 (1968), 409–457).     

General logarithmic Sobolev inequalities and Orlicz imbeddings

We obtain, for a large class of measures μ, general inequalities of the form $\text{∫R}{}^{\mathrm{n}}\text{|u|}{}^{\mathrm{p}}\text{A(}\text{log}{}^1\text{|u|) dμ ? K(6u : W}{}^{\mathrm{m,p}}\text{(R}{}^{\mathrm{n}}\text{,dμ)6}{}^{\mathrm{p}}\text{+ 6 u 6}{}^{\mathrm{p}}\text{A(}\text{log}{}^1\text{6 u 6))}$, where $\text{6u6 = 6 u: L}{}^{\mathrm{p}}\text{(R}{}^{\mathrm{n}}\text{,dμ)6}{}^{\mathrm{p}}\text{,}\text{log}{}^1\text{t =}\text{max}\text{{1,}\text{log}\text{t}}$, and the function A depends in an appropriate way on μ. Our results extend similar results obtained by Rosen for the case p = 2, A(t) = t^s. We also investigate some implications of these inequalities for the imbedding of Sobolev spaces into Orlicz spaces.     

Derivations of Witt vectors with application to K2 of truncated polynomial rings and Laurent series

The absolute Kähler module $\text{Ω}{}_{\mathrm{w}}{}_{\mathrm{n}}{}^{\mathrm{(k)}}$ of the truncated generalized Witt vectors of a field k of positive characteristic is zero if and only if k is perfect. This recovers known information on $\text{K}{}_2\text{(}\text{k[t]}\text{(t}{}^{\mathrm{n}}\text{)}\text{)}$ with which the structure of K₂(k((t))) can be studied.     

Bernstein's asymptotic best bound for the Kth derivative of a polynomial

A complete proof, which yields an error term showing a dependence on n and x, is given for Bernstein's result that the best bound on the kth derivative on polynomials of degree less than or equal to n, at a point x in ( ?1, 1), is asymptotic to $\text{(}\text{n}\text{√1 ? x}{}^2\text{)}{}^{\mathrm{k}}$ as n tends to infinity.