A multidimensional Lyapunov type theorem

Alberto Bressan¹

¹

Studia Mathematica, Tome 106 (1993) no. 2, pp. 121-128

Résumé

Given functions $f_1,...,f_ν ∈ ℒ^1(ℝ^n;ℝ^m)$, weights $p_1,...,p_ν: ℝ^n → [0,1]$ with $∑ p_i ≡ 1$, and any finite set of vectors $v_1,...,v_k ∈ ℝ^n ∖ {0}$, we prove the existence of a partition ${A_1,...,A_ν}$ of $ℝ^n$ such that the two functions $f_p = ∑_{i=1}^ν p_i f_i, $f_A = ∑_{i=1}^ν χ_{A_i}f_i$ have the same integral not only over $ℝ^n$, but also over every single line $x' + ℝv_j$, for each j = 1,...,k and almost every x' in the orthogonal hyperplane $v_j^⊥$. Equivalently, the Fourier transforms of $f_p$, $f_A$ satisfy $f̂_p(y) = f̂_A(y)$ for every $y ∈ ⋃ v_j^⊥$.

Détail

DOI : 10.4064/sm-106-2-121-128