By F R Gantmakher
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13) γ= 1 (C1 + 2λ). 14) 1 1 d E(u(t)) ≤ 2γE(u(t)) + 2E(u(t)) 2 E(f (t)) 2 . 2. 13). 3) ∂t u ˆ + A(iξ)ˆ u = fˆ ˆ u ˆ|t=0 = h on [0, T ] × Rd , on Rd . It is natural to look for symmerizers that are defined on the Fourier side. 1. 4) m (1 + |ξ|2 )− 2 p ∈ L∞ (Rd ). 6) p(Dx )u m H s−m ≤ (1 + |ξ|2 )− 2 p L∞ u Hs . This extends immediately to vector valued functions and matrices p. 2. 4) is called a Fourier multiplier of order ≤ m and p(Dx ) is the operator of symbol p(ξ). 5) and Plancherel’s theorem immediately imply the following.
8). When r ∈ N and |α| = n ≥ r, there are α and α such that α = α + α and |α | = r. 19, we see that ∂xα Sk u L∞ ≤ C2n−r Sk ∂xα u ≤C2 (n−r) ∂xα u 59 L∞ L∞ ≤ C2k(n−r) u W r,∞ . 1). It is defined as soon as the multiplication by p acts from S to S . 1) p(x, Dx )u (x) = (2π)−d eix·ξ p(x, ξ)u(ξ)dξ , and to show that the properties above remain true, not in an exact sense but up to remainder terms which are smoother. 2 Operators with symbols in the Schwartz class As an introduction, we first study the case of operators defined by symbols in the Schwartz class.
This property depends on the behavior of the exponentials e−tA(iξ) when |ξ| → ∞. 8. There is a function C(t) bounded on all interval [0, T ], such that e−tA(iξ) ≤ C(t). 9. 18) u(t) Hs ≤ C(t) h Hs C(t − t ) f (t ) + 0 Hs dt . Proof. 18) implies that ˆ ˆ ≤ C(t) h(ξ) . e−tA(iξ) h(ξ) Thus, by Lebesgues’ dominated convergence theorem, if h ∈ L2 , the mapping ˆ t→u ˆ0 (t, ·) = u ˆ0 (t, ·) = e−tA(i·) h(·) is continuous from [0, +∞[ to L2 (Rd ). −1 0 2 Thus, u0 = F u ∈ C ([0, +∞[; L (Rd ). Moreover: u(t) L2 = 1 (2π)n u ˆ(t) L2 ≤ C(t) (2π)n ˆ h L2 ˆ = C(t) h L2 .