An integral operator connected with the Helmholtz equation

Let Lu be the integral operator defined by $\text{(L}{}_{\mathrm{k?}}\text{)(x, y) = ∝ s ∝ ?(x′, y′)(}\text{e}{}^{\mathrm{ik?}}\text{?}\text{) dx′ dy′, (x, y) ? S}$ where S is the interior of a smooth, closed Jordan curve in the plane, k is a complex number with Re k ? 0, Im k ? 0, and ?² = (x ?x′)² + (y ? y′)². We define $\text{q(x, y) = [}\text{dist}\text{((x, y), ?S)]}\text{1}\text{2}\text{, (x, y) ? S; L}{}_2\text{(q, S) = {? : ∝ s ∝ ¦ ?(x, y)¦}{}^2\text{q(x, y) dx dy < ∞}; W}{}_2{}^1\text{(q, S) = {? : ? ? L}{}_2\text{(q, S),}\text{??}\text{?x}\text{,}\text{?f}\text{?y}\text{? L}{}_2\text{(q, S)}}$, where in the definition of W₂¹(q, S) the derivatives are taken in the sense of distributions. We prove that L_k is a continuous 1-l mapping of L₂(q, S) onto W₂¹(q, S).     

Trace formulas for a class of Toeplitz-like operators II

Let P_T denote the orthogonal projection of L²(R¹, dΔ) onto the space of entire functions of exponential type ? T which are square summable on the line with respect to the measure $\text{dΔ(γ) = ¦ h(γ)¦}{}^2\text{dγ}$, and let G denote the operator of multiplication by a suitably restricted complex valued function g. It is shown that if $\text{(γ}{}^2\text{+ 1)}{}^{\mathrm{?1}}\text{log}\text{¦ h(γ)¦}$ is summable, if $\text{¦ h ¦}{}^{\mathrm{?2}}$ is locally summable, and if $\text{h}\text{h}{}^{\mathrm{\#}}$ belongs to the span in L^∞ of e^?iyTH^∞:T ? 0, in which h is chosen to be an outer function and h^#(γ) agrees with the complex conjugate of h(γ) on the line, then

$\text{lim trace}\text{T↑∞}\text{{(P}{}_{\mathrm{T}}\text{GP}{}_{\mathrm{T}}\text{)}{}^{\mathrm{n}}\text{? P}{}_{\mathrm{T}}\text{G}{}^{\mathrm{n}}\text{P}{}_{\mathrm{T}}\text{}}$

exists and is independent of h for every positive integer n. This extends the range of validity of a formula due to Mark Kac who evaluated this limit in the special case h = 1 using a different formalism. It also extends earlier results of the author which were established under more stringent conditions on h. The conclusions are based in part upon a preliminary study of a more general class of projections.     

Loeb-measurable solutions to 1finite games

Let ¹M be a denumerately comprehensive enlargement of a set-theoretic structure sufficient to model R. If F is an internal ¹finite subset of ¹N such that $\text{F = {1,…,γ}, γ?}{}^1\text{N?N}$, we define a class of ¹finite cooperative games having the form $\text{Γ}{}_{\mathrm{F}}\text{(}{}^1\text{ν) = 〈F,A(F),}{}^1\text{ν〉}$, where A(F) is the internal algebra of the internal subsets of F, and ${}^1\text{ν}$ is a set-function with $\text{Dom}{}^1\text{ν=A(F)}$, $\text{Rng}{}^1\text{ν =}{}^1\text{R}{}_{\mathrm{+}}$, and ${}^1\text{ν(Ø) = 0}$. If $\text{SI(}{}^1\text{ν)}$ is the space of S-imputations of a game Γ_F(¹ν) such that ${}^1\text{ν(F)<η}$, for some $\text{η?}{}^1\text{N}$, then we prove that $\text{SI(}{}^1\text{ν)}$ contains two nonempty subsets: $\text{QK(Γ}{}_{\mathrm{F}}\text{(}{}^1\text{ν))}$ and $\text{SM}{}_1\text{(Γ}{}_{\mathrm{F}}\text{(}{}^1\text{ν))}$, termed the quasi-kernel and S-bargaining set, respectively. Both $\text{QK(Γ}{}_{\mathrm{F}}\text{(}{}^1\text{ν))}$ and $\text{SM}{}_1\text{(Γ}{}_{\mathrm{F}}\text{(}{}^1\text{ν))}$ are external solution concepts for games of the form Γ_F (¹ν) and are defined in terms of predicates that are approximate in infinitesimal terms. Furthermore, if L(Θ) is the Loeb space generated by the ¹finitely additive measure space 〈F, A(F), U_F〉, and if a game Γ_F(¹ν) has a nonatomic representation $\text{ψ(}{}^1\text{ν}{}^{\mathrm{?0}}\text{)}$ on L(Θ) with respect to S-bounded transformations, then the standard part of any element in $\text{QK(Γ}{}_{\mathrm{F}}\text{(}{}^1\text{ν))}$ is Loeb-measurable and belongs to the quasi-kernel of $\text{ψ(}{}^1\text{ν}{}^{\mathrm{?0}}\text{)}$ defined in standard terms.     

On the probability that k positive integers are relatively prime II

Given a set S of positive integers let $\text{Z}{}_{\mathrm{k}}{}^{\mathrm{S}}\text{(t)}$ denote the number of k-tuples 〈m₁, …, m_k〉 for which $\text{m}{}_{\mathrm{i}}\text{∈ S ? [1, t]}$ and (m₁, …, m_k) = 1. Also let $\text{P}{}_{\mathrm{k}}{}^{\mathrm{S}}\text{(n)}$ denote the probability that k integers, chosen at random from $\text{S ? [1, n]}$, are relatively prime. It is shown that if P = {p₁, …, p_r} is a finite set of primes and S = {m : (m, p₁ … p_r) = 1}, then $\text{Z}{}_{\mathrm{k}}{}^{\mathrm{S}}\text{(t) = (td(S))}{}^{\mathrm{k}}\text{Π}{}_{\mathrm{\nu ?P}}\text{(1 ?}\text{1}\text{p}{}^{\mathrm{k}}\text{) + O(t}{}^{\mathrm{k?1}}\text{)}$ if k ≥ 3 and $\text{Z}{}_2{}^{\mathrm{S}}\text{(t) = (td(S))}{}^2\text{Π}{}_{\mathrm{p?P}}\text{(1 ?}\text{1}\text{p}{}^2\text{) + O(t}\text{log}\text{t)}$ where d(S) denotes the natural density of S. From this result it follows immediately that $\text{P}{}_{\mathrm{k}}{}^{\mathrm{S}}\text{(n) → Π}{}_{\mathrm{p?P}}\text{(1 ?}\text{1}\text{p}{}^{\mathrm{k}}\text{) = (ζ(k))}{}^{\mathrm{?1}}\text{Π}{}_{\mathrm{p\in P}}\text{(1 ?}\text{1}\text{p}{}^{\mathrm{k}}\text{)}{}^{\mathrm{?1}}$ as n → ∞. This result generalizes an earlier result of the author's where $\text{P = ?}$ and S is then the whole set of positive integers. It is also shown that if S = {p₁^x₁ … p_r^x_r : x_i = 0, 1, 2,…}, then $\text{P}{}_{\mathrm{k}}{}^{\mathrm{S}}\text{(n) → 0}$ as n → ∞.     

Spectral representation for Schrödinger operators with long-range potentials

A spectral representation for the self-adjoint Schrödinger operator H = ?Δ + V(x), x? R³, is obtained, where V(x) is a long-range potential: $\text{V(x) = O(¦ x ¦}{}^{\mathrm{<mtext>?</mtext><mtext>(1</mtext><mtext>2)</mtext><mtext>?\delta </mtext>}}\text{)}$, grad $\text{V(x) = O(¦ x ¦}{}^{\mathrm{<mtext>?</mtext><mtext>(3</mtext><mtext>2)</mtext><mtext>?\delta </mtext>}}\text{)}$, $\text{ΛV(x) = O(¦ x}\text{s}\text{?}{}^{\mathrm{?\delta }}\text{) (δ > 0), Λ}$ being the Laplace-Beltrami operator on the unit sphere Ω. Namely, we shall construct a unitary operator $\text{F}$ from PL₂(R³) onto $\text{L}{}_2\text{((0, ∞); L}{}_2\text{(Ω)), P}$ being the orthogonal projection onto the absolutely continuous subspace for H, such that for any Borel function α(λ),

$\text{(α(H)(Pf,g)=}\text{∫}\text{0}\text{∞}\text{(α(λ)(Ff)(λ),(Fg)(λ))L}{}_2\text{(ω) dλ}$

.     

On maximum principles for a class of nonlinear second-order elliptic equations

In a recent paper [3] the authors derived maximum principles which involved $\text{u(x)}\text{and}\text{q = ¦}\text{grad}\text{u¦}$, where u(x) is a classical solution of an alliptic differential equation of the form ($\text{g(q}{}^2\text{)u}{}_{\mathrm{,i}}\text{)}{}_{\mathrm{,i}}\text{+ ?(u) ?(q}{}^2\text{) = 0}$. In this paper these results are extended to the more general case in which $\text{g = g(u, q}{}^2\text{)}\text{and}\text{?(u) ?(q}{}^2\text{)}$ is replaced by h(u, q²).     

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.     

On the absolute continuity of Schrödinger operators with spherically symmetric,long-range potentials,II

Let H = ?Δ + V, where the potential V is spherically symmetric and can be decomposed as a sum of a short-range and a long-range term, V(r) = V_S(r) + V_L. Let λ = lim sup_r→∞V_L(r) < ∞ (we allow λ = ? ∞) and set λ⁺ = max(λ, 0). Assume that for some r₀, V_L(r) ?C^2k(r₀, ∞) and that there exists δ > 0 such that $\text{(}\text{d}\text{dr}\text{)}{}^{\mathrm{j}}\text{V}{}_{\mathrm{<mtext>L</mtext>}}\text{(r) · (λ}{}^{\mathrm{+}}\text{? V}{}_{\mathrm{<mtext>L</mtext>}}\text{(r) + 1)}{}^{\mathrm{?1}}\text{= O(r}{}^{\mathrm{?j\delta }}\text{), j = 1,…, 2k,}\text{as}\text{r → ∞}$. Assume further that $\text{∝}{}_1{}^{\mathrm{\infty }}\text{(}\text{dr}\text{¦ V}{}_{\mathrm{<mtext>L</mtext>}}\text{(r)¦}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{) = ∞}$ and that 2kδ > 1. It is shown that: (a) The restriction of H to C^∞(Rⁿ) is essentially self-adjoint, (b) The essential spectrum of H contains the closure of (λ, ∞). (c) The part of H over (λ, ∞) is absolutely continuous.     

Operator properties of Sobolev imbeddings over unbounded domains

For an open set Ω ? $\text{R}$^N, 1 ? p ? ∞ and λ ∈ $\text{R}$⁺, let $\text{W}\text{?}{}_{\mathrm{p}}{}^{\mathrm{\lambda }}\text{(Ω)}$ denote the Sobolev-Slobodetzkij space obtained by completing $\text{C}{}_0{}^{\mathrm{\infty }}\text{(Ω)}$ in the usual Sobolev-Slobodetzkij norm (cf. A. Pietsch, “r-nukleare Sobol. Einbett. Oper., Ellipt. Dgln. II,” Akademie-Verlag, Berlin, 1971, pp. 203–215). Choose a Banach ideal of operators $\text{U}$, 1 ? p, q ? ∞ and a quasibounded domain Ω ? $\text{R}$^N. Theorem 1 of the note gives sufficient conditions on λ such that the Sobolev-imbedding map $\text{W}\text{?}{}_{\mathrm{p}}{}^{\mathrm{\lambda }}\text{(Ω) λ L}{}_{\mathrm{q}}\text{(Ω)}$ exists and belongs to the given Banach ideal $\text{U}$: Assume the quasibounded domain fulfills condition C_k^l for some l > 0 and 1 ? k ? N. Roughly this means that the distance of any $\text{x ? Ω}$ to the boundary ?Ω tends to zero as $\text{O(¦ x ¦}{}^{\mathrm{?l}}\text{)}$ for $\text{¦ x ¦ → ∞}$, and that the boundary consists of sufficiently smooth ?(N ? k)-dimensional manifolds. Take, furthermore, 1 ? p, q ? ∞, p > k. Then, if μ, ν are real positive numbers with λ = μ + v ∈ $\text{N}$, μ > λ _S($\text{U}$; p,q:N) and v > N/l · λ_D($\text{U}$;p,q), one has that $\text{W}\text{?}{}_{\mathrm{p}}{}^{\mathrm{\lambda }}\text{(Ω) λ L}{}_{\mathrm{q}}\text{(Ω)}$ belongs to the Banach ideal $\text{U}$. Here λ_D($\text{U}$;p,q;N)∈$\text{R}$⁺ and λ_S($\text{U}$;p,q;N)∈$\text{R}$⁺ are the D-limit order and S-limit order of the ideal $\text{U}$, introduced by Pietsch in the above mentioned paper. These limit orders may be computed by estimating the ideal norms of the identity mappings l_pⁿ → l_qⁿ for n → ∞. Theorem 1 in this way generalizes results of R. A. Adams and C. Clark for the ideals of compact resp. Hilbert-Schmidt operators (p = q = 2) as well as results on imbeddings over bounded domains.Similar results over general unbounded domains are indicated for weighted Sobolev spaces.As an application, in Theorem 2 an estimate is given for the rate of growth of the eigenvalues of formally selfadjoint, uniformly strongly elliptic differential operators with Dirichlet boundary conditions in $\text{L}{}_2\text{(Ω)}$, where Ω fulfills condition C₁^l.For an open set Ω in $\text{R}$^N, let $\text{W}\text{?}{}_{\mathrm{p}}{}^{\mathrm{\lambda }}\text{(Ω)}$ denote the Sobolev-Slobodetzkij space obtained by completing $\text{C}{}_0{}^{\mathrm{\infty }}\text{(Ω)}$ in the usual Sobolev-Slobodetzkij norm, see below. Taking a fixed Banach ideal of operators and 1 ? p, q ? ∞, we consider quasibounded domains Ω in $\text{R}$^N and give sufficient conditions on λ such that the Sobolev imbedding operator $\text{W}\text{?}{}_{\mathrm{p}}{}^{\mathrm{\lambda }}\text{(Ω) λ L}{}_{\mathrm{q}}\text{(Ω)}$ exists and belongs to the Banach ideal. This generalizes results of C. Clark and R. A. Adams for compact, respectively, Hilbert-Schmidt operators (p = q = 2) to general Banach ideals of operators, as well as results on imbeddings over bounded domains. Similar results over general unbounded domains may be proved for weighted Sobolev spaces. As an application, we give an estimate for the rate of growth of the eigenvalues of formally selfadjoint, uniformly strongly elliptic differential operators with Dirichlet boundary conditions in $\text{L}{}_2\text{(Ω)}$, where Ω is a quasibounded open set in $\text{R}$^N.     

Resolvent estimates for a certain class of Schrödinger operators with exploding potentials

Let $\text{H = ?Δ + V}{}_{\mathrm{<mtext>E</mtext>}}\text{(¦x¦)+ V(x)}$ be a Schrödinger operator in Rⁿ. Here $\text{V}{}_{\mathrm{<mtext>E</mtext>}}\text{(¦x¦)}$ is an “exploding” radially symmetric potential which is at least C² monotone nonincreasing and O(r²) as r → ∞. V is a general potential which is short range with respect to V_E. In particular, V_E  0 leads to the “classical” short-range case (V being an Agmon potential). Let Λ = lim_{r → ∞}V_E(r) and R(z) = (H ? z)^?1, 0 < Im z, Λ < Re z < ∞. It is shown that R(z) can be extended continuously to Im z = 0, except possibly for a discrete subset $\text{N}$?(Λ, ∞), in a suitable operator topology $\text{B(L, L}{}^1\text{)}$. And L ? L²(Rⁿ) is a weighted L²-space; H is then absolutely continuous over (Λ, ∞), except possibly for a discrete set of eigenvalues. The corresponding eigenfunctions are shown to be rapidly decreasing.     

Generalized Bell numbers and zeros of successive derivatives of an entire function

Six different formulations equivalent to the statement that, for n ? 2, the sum ∑_{k = 1}ⁿ (?1)^kS(n, k) ≠ 0, where the S(n, k) are Stirling numbers of the second kind, are shown to hold. Using number-theoretic methods, a sufficient condition for the above statement to be true for a set of positive integers n having density 1 is then obtained. It remains open whether it is true for all n > 2. The equivalent statements then yield information on the irreducibility of the polynomials ∑_{k = 1}ⁿS(n, k)t^{k = 1} over the rationals, the nonreal zeros for successive derivatives $\text{(}\text{d}\text{dz}\text{)}{}^{\mathrm{n}}\text{exp}\text{(e}{}^{\mathrm{iz}}\text{)}$, a gap theorem for the nonzero coefficients of exp(?e^z), and the continuous solution of the differential-difference equation $\text{?(x) = 1, 0 ? x < 1, ?′(x) = ?¦x¦?(x ? 1), 1 ? x < ∞}$, where ∥ denotes the greatest integer function.     

Hermite series estimates of a probability density and its derivatives

The following estimate of the pth derivative of a probability density function is examined: $\text{Σ}{}_{\mathrm{k\; =\; 0}}{}^{\mathrm{N}}\text{a}\text{?}{}_{\mathrm{k}}\text{h}{}_{\mathrm{k}}\text{(x)}$, where h_k is the kth Hermite function and $\text{a}\text{?}{}_{\mathrm{k}}\text{= (}\text{(?1)}{}^{\mathrm{p}}\text{n}\text{)}$Σ_{i = 1}ⁿh_k^(p)(X_i) is calculated from a sequence X₁,…, X_n of independent random variables having the common unknown density. If the density has r derivatives the integrated square error converges to zero in the mean and almost completely as rapidly as O(n^?α) and O(n^?α log n), respectively, where $\text{α =}\text{2(r ? p)}\text{(2r + 1)}$. Rates for the uniform convergence both in the mean square and almost complete are also given. For any finite interval they are O(n^?β) and $\text{O(n}{}^{\mathrm{<mtext>?\beta </mtext><mtext>2</mtext>}}\text{log}\text{n)}$, respectively, where $\text{β =}\text{(2(r ? p) ? 1)}\text{(2r + 1)}$.     

On exponential bases for the Sobolev spaces over an interval

We suppose that K is a countable index set and that $\text{Λ = {λ}{}_{\mathrm{k}}\text{¦ k ? K}}$ is a sequence of distinct complex numbers such that forms a Riesz (strong) basis for L²[a, b], a < b. Let Σ = {σ₁, σ₂,…, σ_m} consist of m complex numbers not in Λ. Then, with p(λ) = Π_{k = 1}^m (λ ? σ_k), forms a Riesz (strong) bas Sobolev space H^m[a, b]. If we take σ₁, σ₂,…, σ_m to be complex numbers already in Λ, then, defining p(λ) as before, forms a Riesz (strong) basis for the space H^?m[a, b]. We also discuss the extension of these results to “generalized exponentials” tⁿe^λ_kt.     

On the closability of some positive definite symmetric differential forms on c0∞(ω)

Let $\text{S}$ be a Dirichlet form in $\text{L}{}^2\text{(Ω; m)}$, where Ω is an open subset of $\text{R}$ⁿ, n ? 2, and m a Radon measure on Ω; for each integer k with 1 ? k < n, let $\text{S}$_k be a Dirichlet form on some k-dimensional submanifold $\text{Ω}{}_{\mathrm{k}}$ of Ω. The paper is devoted to the study of the closability of the forms E with domain $\text{C}{}_0{}^{\mathrm{\infty }}\text{(Ω)}$ and defined by: $\text{(?,g)=}$$\text{E}$$\text{(?, g)+}\text{∑}\text{i}\text{p}\text{=1}$$\text{E}$k_i where 1 ? k_p < ? < n, and where , g^k_i denote restrictions of ?, g in $\text{C}{}_0{}^{\mathrm{\infty }}\text{(Ω)}$ to . Conditions are given for E to be closable if, for each i = 1,…, p, one has k_i = n ? i. Other conditions are given for E to be nonclosable if, for some i, k_i < n ? i.     

Minimax estimators for a multinormal precision matrix

Let S_p×p ～ Wishart (Σ, k), Σ unknown, k > p + 1. Minimax estimators of Σ^?1 are given for L₁, an Empirical Bayes loss function; and L₂, a standard loss function (R_i ≡ E(L_i ∣ Σ), i = 1, 2). The estimators are $\text{Σ}\text{?}{}^{\mathrm{?1}}\text{=}\text{a}\text{S}{}^{\mathrm{?1}}\text{+}\text{br}\text{(S)I}{}_{\mathrm{p\times p}}$, a, b ≥ 0, r(·) a functional on $\text{R}{}^{\mathrm{<mtext>p(p+2)</mtext><mtext>2</mtext>}}$. Stein, Efron, and Morris studied the special cases $\text{Σ}{}_{\mathrm{<mtext>a</mtext>}}{}^{\mathrm{?1}}\text{=}\text{a}\text{S}{}^{\mathrm{?1}}\text{(E}\text{Σ}\text{?}{}_{\mathrm{<mtext>k?p?1</mtext>}}{}^{\mathrm{?1}}\text{= Σ}{}^{\mathrm{?1}}\text{)}$ and $\text{Σ}\text{?}{}_1{}^{\mathrm{?1}}\text{=}\text{a}\text{S}{}^{\mathrm{?1}}\text{+ (b/tr S)I}$, for certain, a, b. From their work $\text{R}{}_1\text{(Σ}{}^{\mathrm{?1}}\text{,}\text{Σ}\text{?}{}_1{}^{\mathrm{?1}}\text{; S) ≤ R}{}_1\text{(Σ}{}^{\mathrm{?1}}\text{,}\text{Σ}\text{?}{}_{\mathrm{<mtext>a</mtext>}}{}^{\mathrm{?1}}\text{; S) (?Σ)}$, a = k ? p ? 1, b = p² + p ? 2; whereas, we prove $\text{R}{}_2\text{(Σ}{}^{\mathrm{?1}}\text{Σ}\text{?}{}_{\mathrm{<mtext>a</mtext>}}{}^{\mathrm{?1}}\text{; S) ≤}\text{R}{}_2\text{(Σ}{}^{\mathrm{?1}}\text{,}\text{Σ}\text{?}{}_1{}^{\mathrm{?1}}\text{; S) (?Σ)}$. The reversal is surprising because $\text{L}{}_1\text{(Σ}{}^{\mathrm{?1}}\text{,}\text{Σ}\text{?}{}_1{}^{\mathrm{?1}}\text{; S) →}\text{L}{}_2\text{(Σ}{}^{\mathrm{?1}}\text{,}\text{Σ}\text{?}{}_1{}^{\mathrm{?1}}\text{; S)}$ a.e. (for a particular L₂). Assume $\text{R}$ (compact) ? $\text{S}$, $\text{S}$ the set of p × p p.s.d. matrices. A “divergence theorem” on functions F_p×p : $\text{R}$ → $\text{S}$ implies identities for R_i, i = 1, 2. Then, conditions are given for $\text{R}{}_{\mathrm{i}}\text{(Σ}{}^{\mathrm{?1}}\text{,}\text{Σ}\text{?}{}^{\mathrm{?1}}\text{; S) ≤}\text{R}{}_{\mathrm{i}}\text{(Σ}{}^{\mathrm{?1}}\text{,}\text{Σ}\text{?}{}_1{}^{\mathrm{?1}}\text{; S) ≤}\text{R}{}_{\mathrm{i}}\text{(Σ}{}^{\mathrm{?1}}\text{,}\text{Σ}\text{?}{}_{\mathrm{<mtext>a</mtext>}}{}^{\mathrm{?1}}\text{; S) (?Σ)}$, i = 1, 2. Most of our results concern estimators with r(S) = t(U)/tr(S), U = p ∣S∣^1/p/tr(S).     

Concrete model theory for a class of operators with unitary part

Let m and v_t, 0 ? t ? 2π be measures on T = [0, 2π] with m smooth. Consider the direct integral $\text{H}$ = ⊕L²(v^t) dm(t) and the operator $\text{(L?)(t, λ) = e}{}^{\mathrm{?i\lambda }}\text{?(t, λ) ? 2e}{}^{\mathrm{?i\lambda }}\text{∫}{}_{\mathrm{t}}{}^{\mathrm{2\pi }}\text{∫}{}_{\mathrm{T}}\text{?(s, x) e(s, t) dv}{}_{\mathrm{s}}\text{(x) dm(s)}$ on $\text{H}$, where e(s, t) = exp ∫_s^t ∫_Tdv_λ(θ) dm(λ). Let μ_t be the measure defined by $\text{∫}{}_{\mathrm{T}}\text{?(x) dμ}{}_{\mathrm{t}}\text{(x) = ∫}{}_0{}^{\mathrm{t}}\text{∫}{}_{\mathrm{T}}\text{?(x) dv}{}_{\mathrm{s}}\text{dm(s)}$ for all continuous ?, and let ?_t(z) = exp[?∫ (e^iθ + z)(e^iθ ? z)^?1dμ_t(gq)]. Call {v_t} regular iff for all $\text{t, ¦?}{}_{\mathrm{t}}\text{(e}{}^{\mathrm{i\theta }}\text{)¦ = ¦?}{}_{\mathrm{2\pi }}\text{(e}{}^{\mathrm{i\theta }}\text{)¦}$ for 1 a.e.     

Un analogue d'un théorème de Hardy pour la transformation de DunklAn analogue of Hardy's theorem for the Dunkl transform

In this Note we give a generalization of Hardy's theorem for the Dunkl transform $\text{F}{}_{\mathrm{D}}$ on $\text{R}{}^{\mathrm{d}}$. More precisely, for all a>0, b>0 and p,q∈[1,+∞], we determine the measurable functions f such that and , where $\text{L}{}_{\mathrm{k}}{}^{\mathrm{p}}\text{(}\text{R}{}^{\mathrm{d}}\text{)}$ are the L^p spaces associated with the Dunkl transform. To cite this article: L. Gallardo, K. Trimèche, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 849–854.     

Szegö limit theorems in quantum mechanics

We prove a Szegö-type theorem for some Schrödinger operators of the form $\text{H =}\text{?1}\text{2Δ}\text{+ V}$ with V smooth, positive and growing like $\text{V}{}_0\text{¦x¦}{}^{\mathrm{k}}\text{, k > 0}$. Namely, let π_λ be the orthogonal projection of L² onto the space of the eigenfunctions of H with eigenvalue ?λ; let A be a 0th order self-adjoint pseudo-differential operator relative to Beals-Fefferman weights $\text{?(x, ξ) = 1, Φ(x, ξ) = (1 + ¦ξ¦}{}^2\text{+ V(x))}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ and with total symbol a(x, ξ); and let f∈C($\text{R}$). Then

$\text{lim}\text{λ→∞}\text{1}\text{rank}\text{π}{}_{\mathrm{\lambda }}\text{tr}\text{f(π}{}_{\mathrm{\lambda }}\text{Aπ}{}_{\mathrm{\lambda }}\text{)=}\text{lim}\text{λ→∞}\text{1}\text{vol}\text{(H(x, ξ)?λ)}\text{∫}\text{H?λ}\text{f(a(x, ξ))dxdξ}$

(assuming one limit exists).     

Lp estimates for the wave equation

Let u(x, t) be the solution of u_tt ? Δ_xu = 0 with initial conditions $\text{u(x, 0) = g(x)}\text{and}\text{u}{}_{\mathrm{t}}\text{(x, 0) = ?;(x)}$. Consider the linear operator $\text{T: ?; → u(x, t)}$. (Here g = 0.) We prove for t fixed the following result. Theorem 1: T is bounded in L^p if and only if . Theorem 2: If the coefficients are variables in C and constant outside of some compact set we get: (a) If n = 2k the result holds for $\text{¦ p}{}^{\mathrm{?1}}\text{? 2}{}^{\mathrm{?1}}\text{¦ < (n ? 1)}{}^{\mathrm{?1}}$. (b) If n = 2k ? 1, the result is valid for $\text{¦ p}{}^{\mathrm{?1}}\text{? 2}{}^{\mathrm{?1}}\text{¦ ? (n ? 1)}$. This result are sharp in the sense that for p such that $\text{¦ p}{}^{\mathrm{?1}}\text{? 2}{}^{\mathrm{?1}}\text{¦ > (n ? 1)}{}^{\mathrm{?1}}$ we prove the existence of $\text{?; ? L}{}^{\mathrm{P}}$ in such a way that $\text{T?; ? L}{}^{\mathrm{P}}$. Several applications are given, one of them is to the study of the Klein-Gordon equation, the other to the completion of the study of the family of multipliers $\text{m(ξ) = ψ(ξ) e}{}^{\mathrm{i¦\xi ¦}}\text{¦ ξ ¦}{}^{\mathrm{?b}}$ and finally we get that the convolution against the kernel $\text{K(x) = ?(x)(1 ? ¦ x ¦)}{}^{\mathrm{?1}}$ is bounded in H¹.