Multipliers on bi-parameter Haar system Hardy spaces

Lechner, R.; Motakis, P. (ORCID:0000000243953931); Müller, P_F_X; Schlumprecht, Th

doi:10.1007/s00208-024-02887-9

Abstract Let$$(h_I)$$

(h_{I})

denote the standard Haar system on [0, 1], indexed by$$I\in \mathcal {D}$$

I \in D

, the set of dyadic intervals and$$h_I\otimes h_J$$

h_{I} \otimes h_{J}

denote the tensor product$$(s,t)\mapsto h_I(s) h_J(t)$$

(s, t) \mapsto h_{I} (s) h_{J} (t)

,$$I,J\in \mathcal {D}$$

I, J \in 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} \otimes h_{J}

,$$I,J\in \mathcal {D}$$

I, J \in 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 \leq p < \infty

. We say that$$D:X(Y)\rightarrow X(Y)$$

D : X (Y) \to X (Y)

is a Haar multiplier if$$D(h_I\otimes h_J) = d_{I,J} h_I\otimes h_J$$

D (h_{I} \otimes h_{J}) = d_{I, J} h_{I} \otimes h_{J}

, where$$d_{I,J}\in \mathbb {R}$$

d_{I, J} \in 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}) \to V (δ^{2})

given by$$\mathcal {C} h_I\otimes h_J = h_I\otimes h_J$$

C h_{I} \otimes h_{J} = h_{I} \otimes h_{J}

if$$|I|\le |J|$$

| I | \leq | J |

, and$$\mathcal {C} h_I\otimes h_J = 0$$

C h_{I} \otimes 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) \to X (Y)

, there exist$$\lambda ,\mu \in \mathbb {R}$$

λ, μ \in R

such that$$\begin{aligned} \lambda \mathcal {C} + \mu ({{\,\textrm{Id}\,}}-\mathcal {C})\text { approximately 1-projectionally factors through }D, \end{aligned}$$

\begin{matrix} λ C + μ (Id - C) approximately 1-projectionally factors through D, \end{matrix}

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 ‖ \cdot ‖ 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 Like this