The decomposition of product space \(H^1_L \times BMO_L\)
\textit{A. Bonami, T. Iwaniec, P. Jones} and \textit{Zinsmeister} [Ann. Inst. Fourier 57, No. 5, 1405--1439 (2007; Zbl 1132.42010)] proved the following: Let \(b \in BMO(\mathbb{R}^n)\) and \(h \in H^1(\mathbb{R}^n)\). Then the product \(b \cdot h\) can be given meaning as a Schwartz distribution and can be decomposed as \[ b \cdot h = \alpha + \beta, \quad \text{where} \;\alpha \in L^1(\mathbb{R}^n) \;\text{and} \;\beta \;\text{lies in a Hardy-Orlicz space}. \] The authors consider this theorem for Hardy space and BMO associated with Schrödinger operators. Let \(L = - \Delta + V\) be a Schrödinger operator on \(\mathbb{R}^n\), \(n \geq 3\), where \(V \neq 0\) is a nonnegative potential and \(V\) satisfies the reverse Hölder condition: \[ \left( \frac{1}{| B |} \int_B V^q (x) dx \right)^{1/q} \leq \frac{C}{| B |} \int_B V (x) dx \quad \text{for every ball} \quad B. \] Let \[ \frac{1}{m(x,V)} = \sup_{r>0} \left\{ r: \frac{1}{r^{n-2}}\int_{B(x,r)} V(y) dy \leq 1 \right\}. \] Let \(\{ T_t \}_{t>0}\) be the semigroup generated by the \(-L\) and \(T_t(x,y)\) be their kernel. A function \(f\) is said to belong to the Hardy space \(H_L^1\) if \(\| f \|_{H_L^1} = \| Mf \|_{L^1}\), where \(Mf(x) = \sup_{t>0} | T_t f(x)|\). The function space \(BMO_L\) is defined to be \[ BMO_{L}= \left\{ f \in BMO : \frac{1}{| B |} \int_B | f(x) | dx \leq C, \quad \text{for all} \quad B=B(x,r), r > \frac{1}{m(x,V)} \right\}. \] They prove the following: Let \(b \in BMO_L(\mathbb{R}^n)\) and \(h \in H_L^1(\mathbb{R}^n)\). Then there exist \(\alpha \in L^1 \) and \(\beta \in H_L^P\) (Hardy-Orlicz space associated with Schrödinger operator) such that \[ b \cdot h = \alpha + \beta. \]
scientific article; zbMATH DE number 5362174