×

Dyadic structure theorems for multiparameter function spaces. (English) Zbl 1334.42050

In this paper, the authors prove that the multiparameter (product) space \(\mathrm{BMO}\) of functions of bounded mean oscillation is an intersection of finitely many dyadic product \(\mathrm{BMO}\) spaces with equivalent norms. They also show that the corresponding intersection results hold for the space VMO of functions of vanishing mean oscillation, for \(A_p\) weights, for reverse-Hölder weights and for doubling weights. The authors prove the equivalences of several definitions of VMO, in the continuous, dyadic, one-parameter and product cases. Moreover, for any product \(A_\infty\) weight \(\omega\), the authors show that the weighted product Hardy space \(H_\omega^1(\mathbb{R}\otimes \mathbb{R})\) is the sum of finitely many translates of dyadic weighted Hardy spaces, and the weighted strong maximal function \(M_{s,\omega}\) is pointwise comparable to the sum of finitely many dyadic weighted strong maximal functions, for each doubling weight \(\omega\). Their results hold in both the compact and non-compact cases.
To be precise, a function \(f\in L^1_{\mathrm{loc}}(\mathbb{R}\otimes \mathbb{R})\) is said to belong to the product BMO space \(\mathrm{BMO}(\mathbb{R}\otimes \mathbb{R})\) if there exists a positive constant \(C\) such that, for any open set \(\Omega\subset\mathbb{R}\otimes\mathbb{R}\) with finite measure, \[ \int\int_{T(\Omega)}|f\ast\psi_{y_1}\psi_{y_2}(t_1,\,t_2)|^2\, \frac{dt_1\,dy_1\,dt_2\,dy_2}{y_1y_2}\leq C|\Omega|, \] where \(T(\Omega):=\{(t_1,\,y_1,\,t_2,\,y_2):\;t_1,\,t_2\in\mathbb{R},\, y_1,\,y_2\in(0,\,\infty),\,(t_1-y_1,\,t_1+y_1)\times (t_2-y_2,\,t_2+y_2)\subset\Omega\}\) is the Carleson tent on \(\Omega\) and \(\psi_{y_1}\psi_{y_2}(t_1,\,t_2):=y_1^{-1}y_2^{-1}\psi(t_1/y_1)\psi(t_2/y_2)\) with \(\psi\in C_c^\infty(\mathbb{R})\) satisfying \(\int_{\mathbb{R}}\psi(t)\,dt=0\).
Let \(\mathcal{D}\) be the grid of dyadic integrals in \(\mathbb{R}\) and \(\mathcal{D}^\delta\) be the translate of \(\mathcal{D}\) for \(\delta\in\mathbb{R}\). A function \(f\in L^1_{\mathrm{loc}}(\mathbb{R}\otimes \mathbb{R})\) is said to belong to the dyadic product BMO space \(\mathrm{BMO}_{d,d}(\mathbb{R}\otimes \mathbb{R})\) if there exists a positive constant \(C\) such that, for any open set \(\Omega\subset\mathbb{R}\otimes\mathbb{R}\) with finite measure, \[ \sum_{R:=I\times J\in \mathcal{D}\times\mathcal{D},\,R\subset\Omega}(f,\,h_R)^2\leq C|\Omega|, \] where \(h_R:=h_I\times h_J\), and \(h_I\) and \(h_J\) are the Haar functions on the intervals \(I,\,J\in\mathcal{D}\). \(\mathrm{BMO}_{d,\delta}(\mathbb{R}\otimes \mathbb{R})\), \(\mathrm{BMO}_{\delta,d}(\mathbb{R}\otimes \mathbb{R})\), and \(\mathrm{BMO}_{\delta,\delta}(\mathbb{R}\otimes \mathbb{R})\) are defined similarly. They differ only in which of the dyadic grids \(\mathcal{D}\) and \(\mathcal{D}^\delta\) is used in each variable.
In this paper, the authors prove the following dyadic structure theorems: \[ \mathrm{BMO}(\mathbb{R}\otimes\mathbb{R}) =\mathrm{BMO}_{d,d}(\mathbb{R}\otimes\mathbb{R}) \cap \mathrm{BMO}_{d,\delta}(\mathbb{R}\otimes\mathbb{R}) \cap \mathrm{BMO}_{\delta,d}(\mathbb{R}\otimes\mathbb{R}) \cap \mathrm{BMO}_{\delta,\delta}(\mathbb{R}\otimes\mathbb{R}), \] with equivalent norms, for each \(\delta\in\mathbb{R}\) that is far from dyadic rationals.
If \(\omega\) is a product \(A_\infty\) weight, then it holds true that \[ H^1_{\omega}(\mathbb{R}\otimes\mathbb{R})=H^1_{d,d,\omega}(\mathbb{R}\otimes\mathbb{R}) +H^1_{d,\delta,\omega}(\mathbb{R}\otimes\mathbb{R}) +H^1_{\delta,d,\omega}(\mathbb{R}\otimes\mathbb{R}) +H^1_{\delta,\delta,\omega}(\mathbb{R}\otimes\mathbb{R}), \] with equivalent norms, for each \(\delta\in\mathbb{R}\) that is far from dyadic rationals.
If \(\omega\) is a product doubling weight, then it holds true that \[ M_{s,\omega}(f)\sim M_s^{d,d,\omega}(f)+M_s^{d,\delta,\omega}(f) +M_s^{\delta,d,\omega}(f)+M_s^{\delta,\delta,\omega}(f), \] with implicit constants independent of \(f\in L_{\mathrm{loc}}^1(\mathbb{R}\otimes\mathbb{R})\), for each \(\delta\in\mathbb{R}\) that is far from dyadic rationals.

MSC:

42B35 Function spaces arising in harmonic analysis
42B30 \(H^p\)-spaces
42B25 Maximal functions, Littlewood-Paley theory
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bernard, A.: Espaces H1 de martingales ‘a deux indices. Dualité avec les martin- gales de type “BMO” (French). Bull. Sci. Math. (2) 103 (1979), no. 3, 297-303. · Zbl 0403.60047
[2] Beznosova, O. and Reznikov, A.: Sharp estimates involving A\infty and L log L constants, and their applications to PDE. Algebra i Analiz 26 (2014), no. 1, 40-67; translation in St. Petersburg Math. J. 26 (2015), no. 1, 27-47. · Zbl 1325.42029 · doi:10.1090/S1061-0022-2014-01329-5
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.