Notation xv in Chapter 2. From Chapter 3 on, we shall abbreviate “proper lower semi- continuous convex function” into just “convex function”. When X is a smooth Riemannian manifold, or a Banach space, and F is a continuous function on X, we shall denote by DF its differential map, and by DF (x) · v its first-order variation at some point x X, along some tangent vector v. When X is a smooth Riemannian manifold, we shall denote by TxX the tangent space at a point x, and by ·, ·x the scalar product on TxX defined by the Riemannian structure. We shall denote by D(X) the space of C∞ functions on X with compact support, and by D (X) the space of distribu- tions on X. We define the gradient operator on D(X) by the identity ∇F (x),v x = DF (x) · v so ∇F (x) belongs to TxX, while DF (x) lies in (TxX)∗. We shall denote by ∇· the divergence operator, which is the adjoint of on D(X). The gradient operator acts on real-valued functions, while the divergence operator acts on vector fields. We also define the Laplace operator Δ by the identity ΔF = · ∇F . By duality, all these operations are extended to D (X). Of course, if X = Rn, then ∇F = ∂F ∂x1 , . . . , ∂F ∂xn , · u = n i=1 ∂ui ∂xi , ΔF = n i=1 ∂2F ∂xi2 . We also denote by D2 the Hessian operator on X. Of course, if X = Rn, then D2F (x) can be identified with the Hessian matrix (∂2F (x)/∂xi ∂xj). The space of absolutely continuous (with respect to Lebesgue measure) probability measures on Rn will be denoted by Pac(Rn) it can be identified with a subspace of L1(Rn). The space of absolutely continuous probability measures with finite moments up to order 2 will be denoted by Pac,2(Rn). The Aleksandrov Hessian of a convex function ϕ on Rn will be denoted by DAϕ 2 it is only defined almost everywhere in the interior of the domain of ϕ. It should not be mistaken for the distributional Hessian of ϕ, denoted by DD 2 ϕ. The Hessian measure of ϕ will be denoted detH D2ϕ. All these notions will be explained within the text (see subsections 2.1.3 and 4.1.4). We shall use consistent notations for Laplace operators: the trace of DAϕ2 (resp. DD 2 ϕ) will be denoted by ΔAϕ (resp. ΔD ϕ). Whenever Ω is an open subset of Rn and k N, we denote by Ck(Ω) the space of functions u which are differentiable up to order k and, whenever α (0, 1), we denote by Ck,α(Ω) the space of functions u for which all partial derivatives at order k are older-continuous with exponent α.
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