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Title: Multipliers on bi-parameter Haar system Hardy spaces
Abstract Let$$(h_I)$$ ( h I ) denote the standard Haar system on [0, 1], indexed by$$I\in \mathcal {D}$$ I D , the set of dyadic intervals and$$h_I\otimes h_J$$ h I h J denote the tensor product$$(s,t)\mapsto h_I(s) h_J(t)$$ ( s , t ) h I ( s ) h J ( t ) ,$$I,J\in \mathcal {D}$$ I , J D . We consider a class of two-parameter function spaces which are completions of the linear span$$\mathcal {V}(\delta ^2)$$ V ( δ 2 ) of$$h_I\otimes h_J$$ h I h J ,$$I,J\in \mathcal {D}$$ I , J D . This class contains all the spaces of the formX(Y), whereXandYare either the Lebesgue spaces$$L^p[0,1]$$ L p [ 0 , 1 ] or the Hardy spaces$$H^p[0,1]$$ H p [ 0 , 1 ] ,$$1\le p < \infty $$ 1 p < . We say that$$D:X(Y)\rightarrow X(Y)$$ D : X ( Y ) X ( Y ) is a Haar multiplier if$$D(h_I\otimes h_J) = d_{I,J} h_I\otimes h_J$$ D ( h I h J ) = d I , J h I h J , where$$d_{I,J}\in \mathbb {R}$$ d I , J R , and ask which more elementary operators factor throughD. A decisive role is played by theCapon projection$$\mathcal {C}:\mathcal {V}(\delta ^2)\rightarrow \mathcal {V}(\delta ^2)$$ C : V ( δ 2 ) V ( δ 2 ) given by$$\mathcal {C} h_I\otimes h_J = h_I\otimes h_J$$ C h I h J = h I h J if$$|I|\le |J|$$ | I | | J | , and$$\mathcal {C} h_I\otimes h_J = 0$$ C h I h J = 0 if$$|I| > |J|$$ | I | > | J | , as our main result highlights: Given any bounded Haar multiplier$$D:X(Y)\rightarrow X(Y)$$ D : X ( Y ) X ( Y ) , there exist$$\lambda ,\mu \in \mathbb {R}$$ λ , μ R such that$$\begin{aligned} \lambda \mathcal {C} + \mu ({{\,\textrm{Id}\,}}-\mathcal {C})\text { approximately 1-projectionally factors through }D, \end{aligned}$$ λ C + μ ( Id - C ) approximately 1-projectionally factors through D , i.e., for all$$\eta > 0$$ η > 0 , there exist bounded operatorsA, Bso thatABis the identity operator$${{\,\textrm{Id}\,}}$$ Id ,$$\Vert A\Vert \cdot \Vert B\Vert = 1$$ A · B = 1 and$$\Vert \lambda \mathcal {C} + \mu ({{\,\textrm{Id}\,}}-\mathcal {C}) - ADB\Vert < \eta $$ λ C + μ ( Id - C ) - A D B < η . Additionally, if$$\mathcal {C}$$ C is unbounded onX(Y), then$$\lambda = \mu $$ λ = μ and then$${{\,\textrm{Id}\,}}$$ Id either factors throughDor$${{\,\textrm{Id}\,}}-D$$ Id - D more » « less
Award ID(s):
2349322
PAR ID:
10509669
Author(s) / Creator(s):
; ; ;
Publisher / Repository:
Springer Science + Business Media
Date Published:
Journal Name:
Mathematische Annalen
Volume:
390
Issue:
4
ISSN:
0025-5831
Format(s):
Medium: X Size: p. 5669-5752
Size(s):
p. 5669-5752
Sponsoring Org:
National Science Foundation
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