By Maxson C.J.
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Example text
If q is the a-derivative as above, then Cp (a) ⊂ Cq (a), with obvious notation. The nicest result is perhaps the following. Let P be the polarized form of p, meaning that (ξ 1 , . . , ξ n ) → P (ξ 1 , . . , ξ n ) is a symmetric multilinear form, such that P (ξ, . . , ξ) = p(ξ) for every ξ ∈ R1+d (this is the generalization of the well-known polarization of a quadratic form). Then we have ξ 1 ∈ Cp (a), . . , ξ n ∈ Cp (a) =⇒ p(ξ 1 ) · · · p(ξ n ) ≤ P (ξ 1 , . . , ξ n )n . 34) is in the opposite sense.
A rather surprising byproduct is the concavity of the function6 H → (det H)1/n , H ∈ HPDn . This property is reminiscent of the Alexandrov–Fenchel inequality vol(K1 )vol(K2 ) ≤ V (K1 , K2 )2 for convex bodies, where V denotes the mixed volume. The van der Waerden inequality for the permanent of a doubly stochastic matrix can be rewritten in terms of an inequality for hyperbolic polynomials, applied to σn in n indeterminates. 1 Hyperbolicity of subsystems Let L = ∂t + α Aα ∂α be a hyperbolic n × n operator.
This may be rewritten as T u(t), L∗ φ(t) dt = 0. 28) 0 Let ψ be a slightly more general test function: ψ ∈ D(Rd × (−∞, T ))n . Choosing θ ∈ C ∞ (R) with θ(τ ) = 0 for τ < 1 and θ(τ ) = 1 for τ > 2, we define φ (x, t) = θ(t/ )ψ(x, t). 28) to φ , which gives T θ(t/ ) u(t), L∗ ψ(t) dt = 0 T 1 θ (t/ ) u(t), ψ(t) dt. 0 Using the continuity in time, we may pass to the limit as T → 0+ , and obtain u(t), L∗ ψ(t) dt = u(0), ψ(0) . 28) is valid for ψ as well, that is to test functions in D(Rd × (−∞, T ))n .