Necessary and sufficient conditions for boundedness of the maximal operator in the local Morrey-type spaces (original) (raw)

2004, Studia Mathematica, 163 (2) (2004), 157-176.

The problem of boundedness of the maximal operator in local and global Morrey-type spaces is reduced to the problem of boundedness of the Hardy operator in weighted L p-spaces on the cone of non-negative non-increasing functions. This allows obtaining sufficient conditions for boundedness for all admissible values of the parameters. Moreover, in case of local Morrey-type spaces, for some values of the parameters, these sufficient conditions coincide with the necessary ones. For x ∈ R n and r > 0, let B(x, r) denote the open ball centered at x of radius r. Definition 1. Let 0 < p, θ ≤ ∞ and let w be a non-negative measurable function on (0, ∞). We denote by LM pθ,w , GM pθ,w , the local Morrey-type spaces, the global Morrey-type spaces respectively, the spaces of all functions f ∈ L loc p (R n) with finite quasinorms f LM pθ,w ≡ f LM pθ,w (R n) = w(r) f Lp(B(0,r)) L θ (0,∞) , f GM pθ,w = sup x∈R n f (x + ·) LM pθ,w respectively. Lemma 1. Let 0 < p, θ ≤ ∞ and let w be a non-negative measurable function on (0, ∞). 0 2000 Mathematics Subject Classification: Primary 42B25 0 Supported by the grants of Russian Foundation for Basic Research (project 02-01-00602) and NATO (project PST.CLG.978412).