Recent progress in the study of the boundedness of the classical operators in Morrey-type spaces
by
V.I. Burenko
School of Mathematics, Cardiff University, UK

In my talks at the 3-d and 4-th ISAAC Congresses in Berlin and Toronto attention has been drawn to the importance of the study of the classical operators of real analysis in general Morrey-type spaces which are defined in the following way. Let 0 < p, q £ ¥ and let w be a non-negative measurable function on (0, ¥). We denote by \LMpf , \GMpf , the local Morrey-type spaces, the global Morrey-type spaces respectively, which are the spaces of all functions f Î \LpB for all r > 0 with finite quasinorms

|| f||\LMpf = || w(r) ||f||\LpB ||Lq(0, ¥),

|| f||\GMpf = sup x Î \Rn   || f(x+·)||\LMpf

respectively. (Here \LpBxr is the ball of radius r centered at the point x Î \Rn.) For w(r)=r^-[(l)/p], 0 < l < n the spaces \GMpf were introduced by C. Morrey in 1938 and appeared to be quite useful in various problems in the theory of partial differential equations.

The spaces \LMpf , \GMpf are mostly aimed at describing the behaviour of ||f||_\LpB, ||f||_\LpBxr respectively, for small r > 0. In the cases in which the main interest is in the behaviour of these quantities for large r it is useful to consider the dual local Morrey-type spaces \LMpfdual with the quasinorms

|| f||\LMpfdual = || w(r) ||f||\LpBdual ||Lq(0, ¥) .

It should be noted that if q = p, then the spaces \LMpf, \LMpfdual coincide with weighted L_p-spaces with radially monotonic weights, because \ - f\ - _LM_pp, w= \ - fV\ - _L_p(),     \ - f\ - __pp, w= \ - f\ - _L_p(), where for all x Î \Rn    V(x)=|| w||_{Lp(|x|, ¥)}, \Vdual(x)=|| w||_{Lp(0, |x|)}.

A survey will be given of recent results in which, for some values of the parameters, necessary and sufficient conditions are established for the boundedness of the maximal operator, fractional maximal operator and Riesz potential as operators from one general Morrey-type space to another one. Compared with the case of weighted L_p-spaces there are much more open problems which will also be under discussion.

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Date received: June 20, 2005

Copyright © 2005 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 # carf-61.