Boundedness of maximal operators on generalized local Morrey-type spaces
by
Huseyn Guliyev
University of Wales, Cardiff
Coauthors: Viktor I. Burenkov (University of Wales, Cardiff)

Let for x in Rⁿ and r > 0, B(x, r) denote an open ball centered at x of radius r.

Definition. Let 1 <= p < \infty, 0 < \theta < \infty and let \Phi > 0 be a measurable function. We denote by L_{p\theta, \Phi}, respectively by [L\tilde]_{p\theta, \Phi}, the generalised Morrey-type space, respectively the generalised local Morrey-type space, which are the spaces of all functions f locally integrable on Rⁿ with finite norms

|| f||Lp\theta, \Phi = sup x in Rn  æ è ó õ \infty 0  æ è  \Phi(r) |B(x, r)| ó õ B(x, r)  |f(y)|p dy ö ø \theta/p    dr r ö ø 1/\theta   ,

respectively

|| f||[L\tilde]p\theta, \Phi = æ è ó õ \infty 0  æ è  \Phi(r) |B(0, r)| ó õ B(0, r)  |f(y)|p dy ö ø \theta/p    dr r ö ø 1/\theta   .

If, for some r > 0, \int_r^\infty(\Phi(t)t^-n)^[(\theta)/p][ dt/t]=\infty, then L_{p\theta, \Phi}=[L\tilde]_{p\theta, \Phi}={0}.

Let M be the Hardy-Littlewood maximal operator:

Mf(x)= sup t > 0  |B(x, t)|-1 ó õ B(x, t)  |f(y)|dy,        x in Rn.

Theorem. Let 1 < p < \infty and 0 < \theta <= p. Then the operator M is bounded from [L\tilde]_{p\theta, \Phi} to [L\tilde]_{p\theta, \Phi} if, and only if, for some C > 0, for all r > 0

r-[(n\theta)/p] ó õ r 0  (\Phi(t))[(\theta)/p]  dt t <= C ó õ \infty r  t-[(n\theta)/p] (\Phi(t))[(\theta)/p]  dt t .

(1)

Corollary. If condition (1) is satisfied, then M is also bounded from L_{p\theta, \Phi} to L_{p\theta, \Phi}. (The case \Phi(t) = t^n-\lambda, 0 < \lambda < n, \theta = \infty was considered in [1].)

References

1. F. Chiarenza, M. Frasca, Morrey spaces and Hardy-Littlewood maximal function. Rend. Math. Duke Math., 7 (1987), p. 273-279.

Date received: July 4, 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 # cahv-53.