Rearrangements of functions Article

Crowe, JA, Zweibel, JA, Rosenbloom, PC. (1986). Rearrangements of functions . JOURNAL OF FUNCTIONAL ANALYSIS, 66(3), 432-438. 10.1016/0022-1236(86)90067-4

cited authors

  • Crowe, JA; Zweibel, JA; Rosenbloom, PC

authors

abstract

  • Let f{hook}, g be measurable non-negative functions on R, and let \ ̄tf, ḡ be their equimeasurable symmetric decreasing rearrangements. Let F: R × R → R be continuous and suppose that the associated rectangle function defined by F(R) = F(a, c) + F(b, d) - F(a, d) - F(b, c) for R = [(x, y) ε{lunate} R2| a ≤ x ≤ b, c ≤ y ≤ d], is non-negative. Then ∝ F(f{hook}, g) dμ ≤ ∝ F( \ ̄tf, g ̄) dμ, where μ is Lebesgue measure. The concept of equimeasurable rearrangement is also defined for functions on a more general class of measure spaces, and the inequality holds in the general case. If F(x, y) = -θ{symbol}(x - y), where θ{symbol} is convex and θ{symbol}(0) = 0, then we obtain ∝ θ{symbol}( \ ̄tf - g ̄) dμ ≤ ∝ θ{symbol}(f{hook} - g) dμ. In particular, if θ{symbol}(x) = |x|p, 1 ≤ p ≤ +∞, then we find that the operator S:f{hook} → \ ̄tf is a contraction on Lp for 1 ≤ p ≤ +∞. © 1986.

publication date

  • January 1, 1986

published in

Digital Object Identifier (DOI)

start page

  • 432

end page

  • 438

volume

  • 66

issue

  • 3