On a pair of simultaneous functional equations

For each p in the simplex P of $\text{R}$^k we introduce convex subsets of P, Π_I(p) and Π_II(p). For f a real function on P we define Cav₁f to be the smallest function greater than f on P and concave on Π₁(p) for each p in P (and similarly Vex_IIf). Given u a continuous real function on P we prove that the following problems:

$\text{Minimize}\text{f;f:→R, f?}\text{Cav}{}_{\mathrm{I}}\text{Vex}{}_{\mathrm{II}}\text{max}\text{{u,f}}$

$\text{Minimize}\text{f;f:→R, f?}\text{Vex}{}_{\mathrm{II}}\text{Cav}{}_{\mathrm{I}}\text{min}\text{{u,f}}$

have the same solution which is also the only solution of f = Vex₁₁ max{u,f} = Cav₁ min{u,f}. This is an extension of a former proof by Mertens and Zamir for the case where P is a. product of convex R and S with Π_I(p) = r × S and Π_II(p) = R × s.     

Linear operators preserving the (p,q)-numerical range

Let $\text{C}{}_{\mathrm{n\times n}}$ and $\text{H}{}_{\mathrm{n}}$ denote respectively the space of n×n complex matrices and the real space of n×n hermitian matrices. Let p,q,n be positive integers such that p?q?n. For $\text{A?}\text{C}{}_{\mathrm{n\times n}}$, the (p,q)-numerical range of A is the set

$\text{W}{}_{\mathrm{p,q}}\text{(A)={trC}{}_{\mathrm{p}}\text{(J}{}_{\mathrm{q}}\text{UAU}{}^1\text{):U}\text{unitary}\text{}}$

, where C_p(X) is the pth compound matrix of X, and Jq is the matrix I_q?O_n-q. Let $\text{L}$ denote $\text{H}$_n or $\text{C}{}_{\mathrm{n\times n}}$. The problem of determining all linear operators T: $\text{L}$→$\text{L}$ such that

$\text{W}{}_{\mathrm{p,q}}\text{(T(A))=W}{}_{\mathrm{p,q}}\text{(A)}\text{for all}\text{A?}\text{L}$

is treated in this paper.     

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.     

Superexponential decay of solutions of a semilinear wave equation

Consider a smooth solution of $\text{u}{}_{\mathrm{tt}}\text{? Δu + q(x) ¦ u ¦}{}^{\mathrm{p?1}}\text{u = 0 x ? R}{}^3\text{, q ? 0}$ and is C¹, and 1 < p < 5. Assume that the initial data decay sufficiently rapidly at infinity, $\text{q(x) ? a}\text{exp}\text{(?b ¦ x ¦}{}^{\mathrm{c}}\text{), a, b > 0, c > 1}$, and for simplicity, q_r ? 0. Then the local energy decays faster than exponentially.     

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: $\text{R}$ → $\text{R}$.     

The virial theorem in relativistic quantum mechanics

For a class of potentials including the Coulomb potential q = μr^?1 with ¦ μ ¦ < 1 (1) (i.e., atomic numbers Z ? 137), the virial theorem $\text{(u, α ·}\text{p}\text{u) = (u, r(}\text{?q}\text{?r}\text{)u)}$ is shown to hold, u being an eigenfunction of the operator

$\text{Hu = TU : = (α ·}\text{p}\text{+ β + q)u}$

,

$\text{D(H) = {u ¦ u ∈ [H}{}_{\mathrm{loc}}{}^1\text{(}\text{R}\text{+}{}^3\text{)]}{}^4\text{, r}{}^{\mathrm{<mtext>?</mtext><mtext>1</mtext><mtext>2</mtext>}}\text{u, TU ∈ [L}{}^2\text{(R)}{}^3\text{]}{}^4\text{}}$

($\text{R}$₊³ := $\text{R}$?{0}). The result implies in particular that H with (1) does not have any eigenvalues embedded in the continuum. The proof uses a scale transformation.     

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.     

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).     

Multiplicativity of lp norms for matrices. II

For 1 ? p ? ∞, let

$\text{|A|}{}_{\mathrm{p}}\text{=}\text{Σ}\text{i=1}\text{m}\text{Σ}\text{j=1}\text{n}\text{, |α}{}_{\mathrm{ij}}\text{|}{}^{\mathrm{p}}{}^{\mathrm{<mtext>1</mtext><mtext>p</mtext>}}$

, be the l_p norm of an m × n complex A = (α_ij) ?C_{m × n}. The main purpose of this paper is to find, for any p, q ? 1, the best (smallest) possible constants τ(m, k, n, p, q) and σ(m, k, n, p, q) for which inequalities of the form

$\text{|AB|}{}_{\mathrm{p}}\text{? τ(m, k, n, p, q) |A|}{}_{\mathrm{p}}\text{|B|}{}_{\mathrm{q}}\text{, |AB|}{}_{\mathrm{p}}\text{? σ (m, k, n, p, q)|A|}{}_{\mathrm{q}}\text{|B|}{}_{\mathrm{p}}$

hold for all A?C_{m × k}, B?C_{k × n}. This leads to upper bounds for inner products on C^k and for ordinary l_p operator norms on C_{m × n}.     

Abstract kinetic equations relevant to supercritical media

The abstract Hilbert space equation $\text{(T?)′(x) = ?(A?)(x)}$, x∈$\text{R}$₊, is studied with a partial range boundary condition $\text{(Q}{}_{\mathrm{+}}\text{?)(0) = ?}{}_{\mathrm{+}}\text{?}\text{Ran}\text{Q}{}_{\mathrm{+}}$. Here T is bounded, injective and self-adjoint, A is Fredholm and self-adjoint, with finite-dimensional negative part, and Q₊ is the orthogonal projection onto the maximal T-positive T-invariant subspace. This models half-space stationary transport problems in supercritical media. A complete existence and uniqueness theory is developed.     

Bifurcation at multiple eigenvalues and stability of bifurcating solutions

If K is a bounded linear operator from the real Banach space U into the real Banach space V and $\text{?:U×}$$\text{R}$→V has the value zero at (0, 0), the existence and linear stability of the equilibrium solutions of the dynamical system

$\text{K}\text{du}\text{dt}\text{= ?(u, α)}$

which are close to the origin in U×$\text{R}$ are studied. It is assumed that $\text{?}{}_{\mathrm{u}}\text{(0, 0): U → V}$ is a Freholm operator of index zero. The only restriction on the dimension of the null space of $\text{?}{}_{\mathrm{u}}\text{(0, 0)}$ and the order of vanishing, at (0, 0), of ? restricted to the null space of $\text{D?(0,0)}$:U×$\text{R}$→V, is that they both be finite positive integers. The main result gives conditions under which the equation, which determines the equilibrium solutions in a neighborhood of the origin, also determines the stability of these equilibrium solutions.     

Votes and a half-binomial

In two party elections with popular vote ratio $\text{p}\text{q}\text{,}\text{1}\text{2}\text{≤p=1 ?q}$, a theoretical model suggests replacing the so-called MacMahon cube law approximation $\text{(}\text{p}\text{q}\text{)}{}^3$, for the ratio $\text{P}\text{Q}$ of candidates elected, by the ratio $\text{?}{}_{\mathrm{k}}\text{(p)}\text{?}{}_{\mathrm{k}}\text{(q)}$ of the two half sums in the binomial expansion of (p+q)^2k+1 for some k. This ratio is nearly $\text{(}\text{p}\text{q}\text{)}{}^3$ when k = 6. The success probability $\text{g}{}_{\mathrm{k}}\text{(p)=(}\text{p}{}^{\mathrm{a}}\text{(p}{}^{\mathrm{a}}\text{+q}{}^{\mathrm{a}}\text{)}$ for the power law $\text{(}\text{p}\text{q}\text{)}{}^{\mathrm{a}}\text{?}\text{P}\text{Q}$ is shown to so closely approximate $\text{?}{}_{\mathrm{k}}\text{(p)=Σ}{}_0{}^{\mathrm{k}}\text{(}{}_{\mathrm{r}}{}^{\mathrm{2k+1}}\text{)p}{}^{\mathrm{2k+1?r}}\text{q}{}^{\mathrm{r}}$, if we choose $\text{a = a}{}_{\mathrm{k}}\text{=}\text{(2k+1)!}\text{4}{}^{\mathrm{k}}\text{k!k!}$, that $\text{1≤}\text{?}{}_{\mathrm{k}}\text{(p)}\text{g}{}_{\mathrm{k}}\text{(p)}\text{≤1.01884086}$ for $\text{k≥1}\text{if}\text{1}\text{2}\text{≤p≤1}$. Computationally, we avoid large binomial coefficients in computing $\text{?}{}_{\mathrm{k}}\text{(p)}$ for k>22 by expressing $\text{2?}{}_{\mathrm{k}}\text{(p)?1}$ as the sum $\text{(p?q) Σ}{}_0{}^{\mathrm{k}}\text{(4pq)}{}^{\mathrm{s}}\text{a}{}_{\mathrm{s}}\text{(2s+1)}$, whose terms decrease by the factors $\text{(4pq)(1?}\text{1}\text{2s}\text{)}$. Setting K = 4k+3, we compute a_k for the large k using a continued fraction $\text{πa}{}_{\mathrm{k}}{}^2\text{=}\text{K+1}{}^2\text{(}\text{2K+3}{}^2\text{(}\text{2K+5}{}^2\text{(2K+…)}\text{)}\text{)}$ derived from the ratio of π to the finite Wallis product approximation.     

Global existence for a delay differential equation

An elastic-plastic bar with simply connected cross section Q is clamped at the bottom and given a twist at the top. The stress function u, at a prescribed cross section, is then the solution of the variational inequality (0.1) $\text{min}{}_{\mathrm{<mtext>v</mtext><mtext>?</mtext><mtext>K</mtext>}}\text{{∝}{}_{\mathrm{Q}}\text{¦}\text{▽}\text{v¦}{}^2\text{? 2θ}{}_1\text{∝}{}_{\mathrm{Q}}\text{v} = ∝}{}_{\mathrm{Q}}\text{¦}\text{▽}\text{u¦}{}^2\text{? 2θ}{}_1\text{∝}{}_{\mathrm{Q}}\text{u, u}\text{?}\text{K,}\text{where}\text{(0.2) K = {v}\text{?}\text{H}{}_0{}^1\text{(Q), ¦}\text{▽}\text{v¦ ? 1}\text{a.e.}\text{}}\text{and}\text{θ}{}_1$ is equal to the angle of the twist (after normalizing the units). Introducing the Lagrange multiplier λ_θ1, the unloading problem consists in solving the variational inequality (0.3) is the twisting angle for the unloaded bar; θ₂ < θ₁. Let (0.4) $\text{K}{}^1\text{= {v}\text{?}\text{H}{}_0{}^1\text{(Q), ?d(x) ? v(x) ? d(x)},}\text{where}\text{d(x) =}\text{dist}\text{.(x, ?Q)}$, and denote by $\text{u}{}^1\text{, w}{}^1$ the solutions of (0.1), (0.3), respectively, when K is replaced by $\text{K}{}^1$. The following results are well known for the loading problem (0.1):(0.5) $\text{u = u}{}^1$; (0.6) the plastic set $\text{P = (X}\text{?}\text{Q}\text{?}\text{; ¦}\text{▽}\text{u(x)¦ = 1}}$ is connected to the boundary. In this paper we show that, in general, (0.7) $\text{w ≠ w}{}^1$; (0.8) the plastic set $\text{P}\text{?}\text{= {x}\text{?}\text{Q}\text{?}\text{; ¦}\text{▽}\text{w(x)¦ = 1}}$ is not connected to the boundary. That is, we construct domains Q for which (0.7) and (0.8) hold for a suitable choice of θ₁, θ₂.     

Parametrization of causal actions of universal covering groups and global hyperbolicity

The transformation group of the universal covering of the real projective line, obtained by lifting ordinary projective transformations, is given explicitly in terms of canonical coordinates. A similar formulation is given of the action of the universal covering of SU(2, 2) upon the universal covering of the ?hilov boundary of its associated bounded Hermitian symmetric domain, structured as R¹ × SU(2). The former group $\text{G}\text{?}$, isomorphic to $\text{S}\text{?}\text{U(1, 1)}$, has a unique and continuous bi-invariant global partial ordering ? (similar to that expressing space-time causality relations) corresponding to its bivariant Lorentzian metric; the partial ordering is the same as that induced by the ordering of the real line which the transformation group preserves. As an application, the compactness of the intervals $\text{[g}{}_1\text{, g}{}_2\text{] = {g ?}\text{G}\text{?}\text{: g}{}_1\text{? g ? g}{}_2\text{}}\text{for}\text{g}{}_1\text{, g}{}_2\text{?}\text{G}\text{?}$, necessary for global hyperbolicity of the metric, is studied. It is shown that [g₁, g₂] is compact if and only if g ? ζg₁ for all g in a neighborhood of g₂, where ζ is the generator of the center of $\text{G}\text{?}$ satisfying ζ ? e. In particular, the interior of [e, ζ] is a maximal open global hyperbolic submanifold; $\text{S}\text{?}\text{U(1, 1)}$ is not globally hyperbolic.     

Estimées Ck pour l'équation ∂̄b sur un convexe de type fini de Cn

Let q=1,…,n?1 and D be a bounded convex domain in $\text{C}{}^{\mathrm{n}}$ of finite type m. We construct two integral operators T_q and $\text{T}{}_{\mathrm{q}}$ such that for all $\text{p∈}\text{N}\text{,}\text{T}{}_{\mathrm{q}}\text{,}\text{T}{}_{\mathrm{q}}\text{:C}{}^{\mathrm{p}}{}_{\mathrm{0,q}}\text{(bD)→C}{}^{\mathrm{p+1/m}}{}_{\mathrm{0,q?1}}\text{(bD)}$ are continuous, and for all (0,q)-forms h continuous on bD with $\text{?}{}_{\mathrm{b}}\text{h}$ continuous on bD too, with the additional hypothesis when q=n?1 that ∫_bDh∧φ=0 for all φ∈C^∞_n,0(bD) $\text{?}\text{?}{}_{\mathrm{b}}$-fermée, we show $\text{h=}\text{?}\text{?}{}_{\mathrm{b}}\text{(T}{}_{\mathrm{q}}\text{?}\text{T}{}_{\mathrm{q}}\text{)h+(T}{}_{\mathrm{q+1}}\text{?}\text{T}{}_{\mathrm{q+1}}\text{)}\text{?}\text{?}{}_{\mathrm{b}}\text{h}$. For this construction, we use the Diederich–Fornæss support function of Alexandre (Publ. IRMA Lille 54 (III) (2001)). To prove the continuity of T_q, we integrate by parts and take care of the tangential derivatives. The normal component in z of the kernel of $\text{T}{}_{\mathrm{q}}$ will have a bad behaviour, so, in order to find a good representative of its equivalence class, we isolate the tangential component of the kernel and then integrate by parts again. To cite this article: W. Alexandre, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Asymptotic properties of general symmetric hyperbolic systems

Results on partition of energy and on energy decay are derived for solutions of the Cauchy problem $\text{?u}\text{?t}\text{+ ∑}{}_{\mathrm{j\; =\; 1}}{}^{\mathrm{n}}\text{A}{}_{\mathrm{j}}\text{?u}\text{?x}{}_{\mathrm{j}}\text{= 0, u(0, x) = ?(x)}$. Here the A_j's are constant, k × k Hermitian matrices, x = (x₁,…, x_n), t represents time, and u = u(t, x) is a k-vector. It is shown that the energy of Mu approaches a limit $\text{E}{}_{\mathrm{M}}\text{(?)}\text{as}\text{¦ t ¦ → ∞}$, where M is an arbitrary matrix; that there exists a sufficiently large subspace of data ?, which is invariant under the solution group U₀(t) and such that $\text{U}{}_0\text{(t)? = 0}\text{for}\text{¦ x ¦ ? a ¦ t ¦ ? R, a}\text{and}\text{R}$ depending on ? and that the local energy of nonstatic solutions decays as $\text{¦ t ¦ → ∞}$. More refined results on energy decay are also given and the existence of wave operators is established, considering a perturbed equation $\text{E(x)}\text{?u}\text{?t}\text{+ ∑}{}_{\mathrm{j\; =\; 1}}{}^{\mathrm{n}}\text{A}{}_{\mathrm{j}}\text{?u}\text{?x}{}_{\mathrm{j}}\text{= 0,}\text{where}\text{¦ E(x) ? I ¦ = O(¦ x ¦}{}^{\mathrm{?1\; ?\; ?}}\text{)}$ at infinity.     

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).     

Product formulas for semigroups with elliptic generators

Real constant coefficient nth order elliptic operators, Q, which generate strongly continuous semigroups on L²($\text{R}$^k) are analyzed in terms of the elementary generator,

$\text{A = (?n)}{}^{\mathrm{<mtext>(</mtext><mtext>n</mtext><mtext>2</mtext><mtext>\; ?\; 1</mtext>}}\text{)(n!)}{}^{\mathrm{?1}}\text{∑}\text{k}\text{j = 1}\text{?}{}^{\mathrm{n}}\text{?x}{}_{\mathrm{j}}{}^{\mathrm{n}}$

, for n even. Integral operators are defined using the fundamental solutions p_n(x, t) to u_t = Au and using real polynomials q_l,…, q_k on $\text{R}$^m by the formula, for q = (q_l,…, q_k),

$\text{(F(t)?)(x) = ∫}$

_$\text{R}$m

$\text{?(x + q(z)) P}{}_{\mathrm{n}}\text{(z, t)dz}$

. It is determined when, strongly on L²($\text{R}$^k),

$\text{e}{}^{\mathrm{tQ}}\text{=}\text{lim}\text{j → ∞}\text{F}\text{t}\text{j}{}^{\mathrm{j}}$

. If n = 2 or k = 1, this can always be done. Otherwise the symbol of Q must have a special form.