Abstract
Let ω be a A∞ Muckenhoupt weight. In this paper we get the estimate of rearrangement f*ω in homogeneous space that is . The similar estimate is obtained only on space of Rn .
Highlights
We first recall some basic notions about the homogeneous space and the weights we are going to use
Let μ be a positive measure on the σ -algebra of subsets of X generated by the d-bal= ls B ( x, r ) {y : d ( x, y) < r}, with x ∈ X and r > 0
We say that (X, d, μ) is a space of homogeneous type regular in measure if μ is regular, that is for every measurable set E, given ε > 0, there exists an open set G such that E ⊂ G and μ (G − E ) < ε
Summary
We first recall some basic notions about the homogeneous space and the weights we are going to use. A structure (X, d, μ), with d and μ as above, is called a space of homogeneous type. We say that (X, d, μ) is a space of homogeneous type regular in measure if μ is regular, that is for every measurable set E, given ε > 0 , there exists an open set G such that E ⊂ G and μ (G − E ) < ε. A non-negative locally integrable on homogeneous space X function ω ( x) is called a weight. (2014) Some Rearrangement Inequalities on Space of Homogeneous Type. Chen weight function we call the measure ω ( E ) = ∫E ω ( x) dx. Given a measurable function f on homogeneous space. A weight ω is in Muckenhoupt’s class A∞ respect to μ if there are positive constants C and ε such that the inequality:. The infimum of such C will be denoted by [ω] A∞
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