On discrete Borell–Brascamp–Lieb inequalities

• David Iglesias ; Jesús Yepes Nicolás ^[1]
  1. [1] Universidad de Murcia

     Universidad de Murcia

     Murcia, España

     
• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 36, Nº 3, 2020, págs. 711-722
• Idioma: inglés
• DOI: 10.4171/rmi/1145
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• Resumen

  • If f,g,h:Rn⟶R≥0 are non-negative measurable functions such that h(x+y) is greater than or equal to the p-sum of f(x) and g(y), where −1/n≤p≤∞, p≠0, then the Borell–Brascamp–Lieb inequality asserts that the integral of h is not smaller than the q-sum of the integrals of f and g, for q=p/(np+1).

    In this paper we obtain a discrete analog for the sum over finite subsets of the integer lattice Zn: under the same assumption as before, for A,B⊂Zn}, then ∑A+Bh≥[(∑rf(A)f)q+(∑Bg)q]1/q, where rf(A) is obtained by removing points from A in a particular way, and depending on f. We also prove that the classical Borell–Brascamp–Lieb inequality for Riemann integrable functions can be obtained as a consequence of this new discrete version.