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KMID : 0380319870400000007
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Journal of Korean Research Institute for Better Living
1987 Volume.40 No. 0 p.7 ~ p.12
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A Weighted Norm Inequality for a Maximal Function with Respect to a One Parameter Monotonic Family of Intervals
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Kim Kyung-Hwa
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Abstract
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Let F be a one parameter monotonic family of intervals I_t 0＜t＜∞, with center 0 & G={J} be such that G={I+x｜I∈F, x∈R^n} & define
Mf(x)=sup_x∈J∈G 1/｜J｜∫_J｜f(x)｜dy
In this paper, we show the equivalence of the following, for p>1 and w∈^1_loc(R^n):
(1) ∫_Rn［Mf(y)］^pw(y)dy≤C∫_Rn｜f(y)｜^pw(y)dy, ∀f
(2) sup_J∈G［1/｜J｜∫_J w(y)dy］［1/｜J｜∫_J w(y)^(-1/p-1)dy］^p-1<∞
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KEYWORD
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