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Duality of Hardy and BMO spaces associated with operators with heat kernel bounds. (English) Zbl 1078.42013
The main purpose of this paper is to prove a generalization of Fefferman and Stein’s result on the duality of \(H^1\) and BMO spaces. In a previous paper of one of the authors, a class of new Hardy spaces \(H^1_L\) were introduced where \(L\) is the infinitesimal generator of an analytic semigroup \(\{e^{-tL}\}_{t>0}\) on \(L^2\) with kernel \(p_t(x,y)\). Here they prove that if \(L\) has a bounded holomorphic functional calculus on \(L^2\) and the kernel \(p_t\) satisfies an upper bound of Poisson type, then the spaces \(\text{BMO}_{L^*}\) is the dual space of the Hardy space \(H^1_L\) where \(L^*\) denotes the adjoint operator of \(L\) and \(\text{BMO}_{L}\) is defined as the set of functions \(f\) such that \[ \sup_{B}{1\over | B| } \int_B | f(x)-P_{t_B}f(x)| dx <\infty \] and \[ P_{t_B}f(x)=\int_{\mathbb R^n} p_{t_B}(x,y) f(y) dy \] with \(t_B\) the radius of the ball \(B\). They also obtain a characterization of \(\text{BMO}_L\) in terms of Carleson measures and give a sufficient condition for the classical BMO space and \(\text{BMO}_L\) to coincide.

MSC:
42B30 \(H^p\)-spaces
42B35 Function spaces arising in harmonic analysis
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