Local atomic decompositions for multidimensional Hardy spaces (Q2041886)

The maximal characterization of \(H^1(\mathbb{R}^N)\) says that \[f\in H^1(\mathbb{R}^N) \Leftrightarrow \sup_{t>0} |H_t*f| \in L^1(\mathbb{R}^N), \] where \(H_t(x)=\frac{1}{(4\pi t)^{n/2}} \exp(-\frac{|x|^2}{4t}) \) is the heat kernel in \(\mathbb{R}^N\). A result by \textit{D. Goldberg} [Duke Math. J. 46, 27--42 (1979; Zbl 0409.46060)], noticed that if the supremum in the maximal operator above is restricted to the range \(t\in (0,\tau^2)\), with \(\tau >0\), there is an atomic characterization but with atoms of the form \(a(x)=|B|^{-1} \chi_B (x)\), where \(\chi_B \) is the characteristic function of the ball \(B\) of radius \(\tau\). The paper under review deals with local atomic characterizations of the Hardy space \(H^1(L)\) consisting of the functions \(f\in L^1(X)\), \(X\subset \mathbb{R}^N\) such that \(\displaystyle{\sup_{t>0} |\exp(-tL) f|}\in L^1(X)\) where \(L\) is a non-negative self-adjoint operator on \(L^2(X)\). As an application, local atomic characterizations for multidimensional Bessel, Laguerre and Schrödinger operators are given.