Abstract
Abstract The aim of this paper is to prove the weak type vector-valued inequality for the modified Hardy– Littlewood maximal operator for general Radon measure on ℝ n . Earlier, the strong type vector-valued inequality for the same operator and the weak type vector-valued inequality for the dyadic maximal operator were obtained. This paper will supplement these existing results by proving a weak type counterpart.
Highlights
The aim of this paper is to prove the weak type vector-valued inequality for the modi ed Hardy– Littlewood maximal operator for general Radon measure on Rn
The conclusion of this paper is that we can readily transplant the boundedness of the dyadic maximal operator to modi ed uncentered maximal operators
By a Radon measure, we mean a measure that is nite on all compact sets, outer regular on all Borel sets, and inner regular on open sets
Summary
The conclusion of this paper is that we can readily transplant the boundedness of the dyadic maximal operator to modi ed uncentered maximal operators. For a Borel measurable function f , de ne the modi ed uncentered maximal operator Mk,μ by. What is important here is that the constant C does not depend on μ. In this sense, Theorem 1.1 is a universal estimate. In [5, Theorem 1.7], the present author passed (1.2) to the vector-valued case: Proposition 1.3. Let < k, p, q < ∞ and μ be a Radon measure. A dyadic cube is a set of the form Qjk for some j ∈ Z, k = Kn) ∈ Zn. For each j ∈ Z, Dj(Rn) stands for the set of all dyadic cubes with volume −jn. D(Rn) stands for the set of all dyadic cubes
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