Sharp estimates for integrals over small intervals for functions possessing some smoothness
by
V.I. Burenkov
Cardiff University

Theorem 1. Let 0 < p < \infty and let \lambda be a positive continuous function defined on (0, \infty). In order that, for some C₁ > 0, for all a > 0 and all f in L_p(0, a), the inequality

sup 0 < h <= a  \frac||f||Lp(0, h)\lambda(h) <= C1 æ è \frac||f||Lp(0, a)\lambda(a) + sup 0 < h <= a  \frac||f(x+h) - f(x)||Lp(0, a-h)\lambda(h) ö ø

be valid, it is necessary and sufficient that

1) for 0 < p <= 1, the function h^1/_p \lambda(h)^-1 be almost increasing on (0, \infty), i. e., for some C₂ >= 1, for all 0 < h < a < \infty

h1/p\lambda(h)-1 <= C2 a1/p \lambda(a)-1,

2) for 1 < p < \infty, for some 0 < \delta < 1, the function h^{\delta/ p}\lambda(h)^-1 be almost increasing on (0, \infty).

Theorem 2. Let 0 < p <= 1 and let \lambda, \mu and \nu be positive continuous functions defined on (0, \infty). In order that, for some C₃ > 0, for all h and a satisfying 0 < h <= a, and all f in L_p(0, a), the inequality

\frac||f||Lp(0, h)\lambda(h) <= C3 æ è \frac||f||Lp(0, a)\mu(a) + \frac||f(x+h) - f(x)||Lp(0, a-h)\nu(h) ö ø

be valid, it is necessary and sufficient that, for some C₄, C₅ > 0, for all h and a satisfying 0 < h <= a

\frach\frac1p\lambda(h) <= C4\fraca\frac1p\mu(a)

and

\frac1\lambda(h) <= C5 æ è \frac1\mu(a)+\frac1\nu(h) ö ø .

Date received: August 6, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cahz-36.