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Part of the book series: Progress in Mathematics ((PM,volume 346))

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In order to extend the operator theory for L to Hardy spaces, we need to guarantee that certain operators f(L) preserve vanishing zeroth moments or have the conservation property f(L)c = c whenever c is a constant. In absence of integral kernels, the action of such operators on constants is explained via off-diagonal estimates. We discuss several conservation properties, in particular for resolvents and Poisson semigroups.

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Auscher, P., Egert, M. (2023). Conservation Properties. In: Boundary Value Problems and Hardy Spaces for Elliptic Systems with Block Structure. Progress in Mathematics, vol 346. Birkhäuser, Cham.

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