Bilinear decompositions for the product space \(H^1_L\times \mathrm{BMO}_L\) (Q2922202)

This article studies the properties of pointwise products of functions in certain Hardy and BMO spaces on \(\mathbb{R}^d\).NEWLINENEWLINEBy the pioneering result of \textit{A. Bonami} et al. [Ann. Inst. Fourier 57, No. 5, 1405--1439 (2007; Zbl 1132.42010)], the product \(fg\) of functions \(f\in H^1\) and \(g\in \mathrm{BMO}\), while not necessarily integrable, belongs to the sum of \(L^1\) and a certain Hardy--Orlicz space. A recent improvement of this by \textit{A. Bonami} et al. [J. Math. Pures Appl. (9) 97, No. 3, 230--241 (2012; Zbl 1241.47028)] provided a bilinear decomposition \(fg = S(f,g) + T(f,g)\), where \(S(f,g)\) is in \(L^1\) and \(T(f,g)\) belongs to a new kind of Hardy space \(H^{\log}\), which is smaller than the previous Hardy--Orlicz space.NEWLINENEWLINEThe main result of this article extends the bilinear product decomposition to Hardy and BMO spaces associated with Schrödinger operators \(L=-\Delta + V\) whose nonnegative potentials \(V\) satisfy a reverse Hölder inequality. This class of potentials is by now standard and well-understood, guaranteeing, most importantly, suitable decay and Hölder continuity estimates for the associated heat kernel. More precisely, the main result establishes the product decomposition with continuous bilinear operators \(S_L : H^1_L \times \mathrm{BMO}_L \to L^1\) and \(T_L : H^1_L \times \mathrm{BMO}_L \to H^{\log}\), thus improving an earlier result of \textit{P. Li} and \textit{L. Peng} [J. Math. Anal. Appl. 349, No. 2, 484--492 (2009; Zbl 1161.42013)], which involved an \(L\)-adapted Hardy--Orlicz space. Note that the smaller space \(H^{\log}\) is independent of \(L\).NEWLINENEWLINEIn addition to this the author observes, in the spirit of the above mentioned works by Bonami et al., that the distributional interpretation of the product required for the rigorous treatment of the problem coincides, when restricted to test functions, with the standard pointwise product.NEWLINENEWLINEThe proofs rely on invoking an \(L\)-adapted partition of unity \(\{ \psi_{n,k} \}\) and expressing the function \(f\in H^1_L\) as the sum NEWLINE\[NEWLINE f = \sum_{n,k} (\psi_{n,k}f - \varphi_{2^{-n/2}} \ast (\psi_{n,k}f)) + \sum_{n,k} \varphi_{2^{-n/2}} \ast (\psi_{n,k}f), NEWLINE\]NEWLINE where the bump function \(\varphi_{2^{-n/2}}\) provides an averaging on the scale \(2^{-n/2}\). The first term on the right-hand side, belonging to the classical space \(H^1\), is analogous to earlier results by Bonami et al. [loc. cit.], while the second term is treated using local Hardy space techniques.