By Luigi Ambrosio
In 2013, a college on Geometric degree concept and actual research, prepared via G. Alberti, C. De Lellis and myself, happened on the Centro De Giorgi in Pisa, with lectures via V. Bogachev, R. Monti, E. Spadaro and D. Vittone.
The e-book collects the notes of the classes. The classes supply a deep and recent perception on demanding mathematical difficulties and their contemporary advancements: infinite-dimensional research, minimum surfaces and isoperimetric difficulties within the Heisenberg team, regularity of sub-Riemannian geodesics and the regularity thought of minimum currents in any measurement and codimension.
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Extra info for Geometric Measure Theory and Real Analysis
Example text
A μ-measurable mapping F : X → E is stochastically Gˆateaux differentiable if there exists a measurable mapping D H F : X → H(H, E) such that for every h ∈ H we have F(x + th) − F(x) − DH F(x)(h) −−→ 0 in measure μ. t→0 t The derivative of the n th order D nH F is deVned inductively as D H (D n−1 H F). An alternative notation is ∇ Hn F. 3. Let 1 ≤ p < ∞. The space D p,1 (μ, E) is deVned as the class of all mappings f ∈ L p (μ, E) such that f is ray absolutely continuous, stochastically Gˆateaux differentiable and D H f ∈ L p μ, H(H, E) .
Var. Partial Diff. Equ. 45 (2012), n. 1-2, 101–124. [29] P. C ELADA and A.
The corresponding space of equivalence classes will be denoted by the same symbol. This is the exact analog of the class M H (μ) in Section 8. 1. The set M H (U, μ) is a Banach space with the norm f M := f L 1 (U,μ) + sup |h| H ≤1 fβhμ L 1 (U,μ) . Proof. Let us observe that the operator h → fβh from H to L 1 (U, μ) is linear and has a closed graph. Indeed, suppose that h n → h in H and fβh n → g in L 1 (U, μ). By the continuity of the embedding H → D(μ) we have βh n → βh in L 1 (μ), whence it follows that fβh n → fβh in measure on U , hence g = fβh .