scholarly journals Carleson Measure and Tent Spaces on the Siegel Upper Half Space

2012 ◽  
Vol 2012 ◽  
pp. 1-23
Author(s):  
Kai Zhao
Keyword(s):  
Heisenberg Group ◽  
Half Space ◽  
Carleson Measure ◽  
Atomic Decomposition ◽  
Carleson Measures ◽  
Tent Spaces ◽  
Dual Spaces ◽  
Choquet Integrals ◽  
Hausdorff Capacity

The Hausdorff capacity on the Heisenberg group is introduced. The Choquet integrals with respect to the Hausdorff capacity on the Heisenberg group are defined. Then the fractional Carleson measures on the Siegel upper half space are discussed. Some characterized results and the dual of the fractional Carleson measures on the Siegel upper half space are studied. Therefore, the tent spaces on the Siegel upper half space in terms of the Choquet integrals are introduced and investigated. The atomic decomposition and the dual spaces of the tent spaces are obtained at the last.

scholarly journals Hardy-type inequalities on a half-space in the Heisenberg group

2013 ◽  
Vol 2013 (1) ◽  
Author(s):  
Heng-Xing Liu ◽  
Jing-Wen Luan
Keyword(s):  
Heisenberg Group ◽  
Half Space ◽  
Hardy Type Inequalities ◽  
Hardy Type

scholarly journals Geometric Hardy and Hardy–Sobolev inequalities on Heisenberg groups

2020 ◽  
Vol 10 (03) ◽  
pp. 2050016
Author(s):  
Michael Ruzhansky ◽  
Bolys Sabitbek ◽  
Durvudkhan Suragan
Keyword(s):  
Heisenberg Group ◽  
Half Space ◽  
Hardy Inequality ◽  
Euclidean Distance ◽  
Sobolev Inequalities ◽  
Sharp Constant ◽  
Heisenberg Groups ◽  
Hardy Inequalities ◽  
Angle Function ◽  
Distance To The Boundary

In this paper, we present geometric Hardy inequalities for the sub-Laplacian in half-spaces of stratified groups. As a consequence, we obtain the following geometric Hardy inequality in a half-space of the Heisenberg group with a sharp constant: [Formula: see text] which solves a conjecture in the paper [S. Larson, Geometric Hardy inequalities for the sub-elliptic Laplacian on convex domain in the Heisenberg group, Bull. Math. Sci. 6 (2016) 335–352]. Here, [Formula: see text] is the angle function. Also, we obtain a version of the Hardy–Sobolev inequality in a half-space of the Heisenberg group: [Formula: see text] where [Formula: see text] is the Euclidean distance to the boundary, [Formula: see text], and [Formula: see text]. For [Formula: see text], this gives the Hardy–Sobolev–Maz’ya inequality on the Heisenberg group.

scholarly journals Dual Spaces of Multiparameter Local Hardy Spaces

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Wei Ding ◽  
Feng Yu
Keyword(s):  
Hardy Spaces ◽  
Duality Theory ◽  
Carleson Measure ◽  
Dual Spaces ◽  
Local Hardy Spaces ◽  
The Relationship

In this paper, we study the duality theory of the multiparameter local Hardy spaces h p ℝ n 1 × ℝ n 2 , and we prove that h p ℝ n 1 × ℝ n 2 ∗ = cm o p ℝ n 1 × ℝ n 2 , where cm o p ℝ n 1 × ℝ n 2 are defined by discrete Carleson measure. Moreover, we discuss the relationship among cm o p ℝ n 1 × ℝ n 2 , Li p p ℝ n 1 × ℝ n 2 , and rectangle cm o rect p ℝ n 1 × ℝ n 2 .

The Boundedness of Maximal Functions in Orlicz–Campanato Spaces of Homogeneous Type

2008 ◽  
Vol 15 (2) ◽  
pp. 377-388
Author(s):  
Xiangxing Tao ◽  
Qinqin Chen
Keyword(s):  
Half Space ◽  
Maximal Operator ◽  
Carleson Measure ◽  
Normal Space ◽  
Space Of Homogeneous Type ◽  
Campanato Space ◽  
Homogeneous Type ◽  
Spaces Of Homogeneous Type ◽  
Doubling Property ◽  
Littlewood Maximal Operator

Abstract Let (𝑋,𝑑,μ) be a normal space of homogeneous type, 𝑋+ be the upper half-space equipped with a Carleson measure β, and let Φ be an N-function and φ a suitable function satisfying the doubling property. We prove that the generalized Hardy–Littlewood maximal operator 𝑀 is bounded from the Orlicz–Campanato space 𝐿Φ,φ (𝑋,μ) to 𝐿Φ,φ (𝑋+,β).

The Dirichlet Problem on the Heisenberg Group III: Harmonic Measure of a certain Half-Space

1986 ◽  
Vol 38 (3) ◽  
pp. 666-671 ◽  
Author(s):  
Bernard Gaveau ◽  
Jacques Vauthier
Keyword(s):  
Heisenberg Group ◽  
Half Space ◽  
Harmonic Measure ◽  
Short Note ◽  
Type Equation ◽  
Probabilistic Argument ◽  
Group Iii ◽  
3 Dimensional ◽  
Degenerate Hypergeometric Function ◽  
Made In

0. Introduction. In this short note we give an explicit computation of the harmonic measure of a half space x > 0 in the 3-dimensional Heisenberg group in terms of a degenerate hypergeometric function. A probabilistic argument reduces the whole problem to a Hermite-type equation on a half line, that we can solve in terms of the function G(l/4, 1/2; x2).A preliminary attempt to compute this kernel was done in [1] p. 107 and, cited by Huber [4]. Unfortunately a small mistake was made in [1] and the problem was still open until now. The first author is very grateful to Prof. Huber for having pointed out the weak argument of [1]. Since that time, other harmonic measures and even Green functions have been explicitly computed (see [2]).

Weighted Carleson Measure Spaces Associated with Different Homogeneities

2014 ◽  
Vol 66 (6) ◽  
pp. 1382-1412 ◽  
Author(s):  
Xinfeng Wu
Keyword(s):  
Hardy Spaces ◽  
Carleson Measure ◽  
Forthcoming Paper ◽  
Weighted Hardy Spaces ◽  
Dual Spaces ◽  
Measure Spaces

AbstractIn this paper, we introduce weighted Carleson measure spaces associated with different homogeneities and prove that these spaces are the dual spaces of weighted Hardy spaces studied in a forthcoming paper. As an application, we establish the boundedness of composition of two Calderón–Zygmund operators with different homogeneities on the weighted Carleson measure spaces; this, in particular, provides the weighted endpoint estimates for the operators studied by Phong–Stein.

scholarly journals A Hardy type inequality in the half-space on Rn and Heisenberg group

2008 ◽  
Vol 347 (2) ◽  
pp. 645-651 ◽  
Author(s):  
Jing-Wen Luan ◽  
Qiao-Hua Yang
Keyword(s):  
Heisenberg Group ◽  
Half Space ◽  
Type Inequality ◽  
Hardy Type Inequality ◽  
Hardy Type

scholarly journals Remarks on square functions in the Littlewood-Paley theory

1998 ◽  
Vol 58 (2) ◽  
pp. 199-211 ◽  
Author(s):  
Shuichi Sato
Keyword(s):  
Half Space ◽  
Carleson Measures ◽  
Square Function ◽  
Square Functions

We prove that certain square function operators in the Littlewood-Paley theory defined by the kernels without any regularity are bounded on , 1 < p < ∞, w ∈ Ap (the weights of Muckenhoupt). Then, we give some applications to the Carleson measures on the upper half space.

scholarly journals Duality for Outer $$L^p_\mu (\ell ^r)$$ Spaces and Relation to Tent Spaces

2021 ◽  
Vol 27 (4) ◽  
Author(s):  
Marco Fraccaroli
Keyword(s):  
Half Space ◽  
Full Range ◽  
Outer Measure ◽  
Strong Type ◽  
Finite Sets ◽  
Tent Spaces ◽  
Space Setting ◽  
Finite Set ◽  
Full Classification

AbstractWe study the outer $$L^p$$ L p spaces introduced by Do and Thiele on sets endowed with a measure and an outer measure. We prove that, in the case of finite sets, for $$1< p \leqslant \infty , 1 \leqslant r < \infty $$ 1 < p ⩽ ∞ , 1 ⩽ r < ∞ or $$p=r \in \{ 1, \infty \}$$ p = r ∈ { 1 , ∞ } , the outer $$L^p_\mu (\ell ^r)$$ L μ p ( ℓ r ) quasi-norms are equivalent to norms up to multiplicative constants uniformly in the cardinality of the set. This is obtained by showing the expected duality properties between the corresponding outer $$L^p_\mu (\ell ^r)$$ L μ p ( ℓ r ) spaces uniformly in the cardinality of the set. Moreover, for $$p=1, 1 < r \leqslant \infty $$ p = 1 , 1 < r ⩽ ∞ , we exhibit a counterexample to the uniformity in the cardinality of the finite set. We also show that in the upper half space setting the desired properties hold true in the full range $$1 \leqslant p,r \leqslant \infty $$ 1 ⩽ p , r ⩽ ∞ . These results are obtained via greedy decompositions of functions in the outer $$L^p_\mu (\ell ^r)$$ L μ p ( ℓ r ) spaces. As a consequence, we establish the equivalence between the classical tent spaces $$T^p_r$$ T r p and the outer $$L^p_\mu (\ell ^r)$$ L μ p ( ℓ r ) spaces in the upper half space. Finally, we give a full classification of weak and strong type estimates for a class of embedding maps to the upper half space with a fractional scale factor for functions on $$\mathbb {R}^d$$ R d .

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