On vector-valued tent spaces and Hardy spaces associated with non-negative self-adjoint operators (Q2816788)

Let \((M,d,\mu)\) be a complete doubling metric measure space, i.\,e. there exist positive constants \(C\) and \(n\) such that, for any \(\alpha\in[1,\infty)\) and any ball \(B\subset M\), NEWLINE\[NEWLINE\mu(\alpha B)\leq C\alpha^n\mu(B). NEWLINE\]NEWLINE Moreover, assume that the Banach space \(X\) has the UMD property, which is defined by a requirement for unconditionality of martingale differences.NEWLINENEWLINEIn this paper, the author developes the theory of vector-valued tent spaces and vector-valued Hardy spaces \(H^p_L(X)\) (\(p\in[1,\infty)\)) associated with a non-negative self-adjoint operator \(L\), which is defined on \(L^2(M)\) and satisfies an off-diagonal estimate. More precisely, the complex interpolation scales of vector-valued tent spaces and Hardy spaces are extended to the endpoint \(p=1\). Furthermore, it is also proved that \(L\) has a bounded \(H^\infty\)-functional calculus on the Hardy space \(H^1_L(X)\). Moreover, the atomic decompositions for functions in a dense subspace of \(H^1_L(X)\) are obtained in this paper.