Multilinear Littlewood–Paley–Stein Operators on Non-homogeneous Spaces

Let m ≥ 2 , λ > 1 and define the multilinear Littlewood–Paley–Stein operators by g λ , μ ∗ ( f → ) ( x ) = ( ∬ R + n + 1 ( t t + | x - y | ) m λ | ∫ ( R n ) κ s t ( y , z → ) ∏ i = 1 κ f i ( z i ) d μ ( z 1 ) ⋯ d μ ( z κ ) | 2 d μ ( y ) d t t m + 1 ) 1 / 2 . In this paper, our main aim is to inve...

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Veröffentlicht in:The Journal of Geometric Analysis 2021-09, Vol.31 (9), p.9295-9337
Hauptverfasser: Cao, Mingming, Xue, Qingying
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description Let m ≥ 2 , λ > 1 and define the multilinear Littlewood–Paley–Stein operators by g λ , μ ∗ ( f → ) ( x ) = ( ∬ R + n + 1 ( t t + | x - y | ) m λ | ∫ ( R n ) κ s t ( y , z → ) ∏ i = 1 κ f i ( z i ) d μ ( z 1 ) ⋯ d μ ( z κ ) | 2 d μ ( y ) d t t m + 1 ) 1 / 2 . In this paper, our main aim is to investigate the boundedness of g λ , μ ∗ on non-homogeneous spaces. By means of probabilistic and dyadic techniques, together with non-homogeneous analysis, we show that g λ , μ ∗ is bounded from L p 1 ( μ ) × ⋯ × L p κ ( μ ) to L p ( μ ) under certain weak type assumptions. The multilinear non-convolution type kernels s t only need to satisfy some weaker conditions than the standard conditions of multilinear Calderón–Zygmund type kernels and the measures μ are only assumed to be upper doubling measures (non-doubling). The above results are new even under Lebesgue measures. This was done by considering first a sufficient condition for the strong type boundedness of g λ , μ ∗ based on an endpoint assumption, and then directly deduce the strong bound on a big piece from the weak type assumptions.
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In this paper, our main aim is to investigate the boundedness of g λ , μ ∗ on non-homogeneous spaces. By means of probabilistic and dyadic techniques, together with non-homogeneous analysis, we show that g λ , μ ∗ is bounded from L p 1 ( μ ) × ⋯ × L p κ ( μ ) to L p ( μ ) under certain weak type assumptions. The multilinear non-convolution type kernels s t only need to satisfy some weaker conditions than the standard conditions of multilinear Calderón–Zygmund type kernels and the measures μ are only assumed to be upper doubling measures (non-doubling). The above results are new even under Lebesgue measures. 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subjects Abstract Harmonic Analysis
Convex and Discrete Geometry
Differential Geometry
Dynamical Systems and Ergodic Theory
Fourier Analysis
Geometry
Global Analysis and Analysis on Manifolds
Kernels
Mathematics
Mathematics and Statistics
Operators
title Multilinear Littlewood–Paley–Stein Operators on Non-homogeneous Spaces
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