A Hörmander-type spectral multiplier theorem for operators without heat kernel

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Gaussian heat kernel upper bounds via the Phragmén–Lindelöf theorem (2006, unknown): “(b) In [8, 9, 10, 11, 12], Blunck and Kunstmann develop spectral multiplier theorems for operators which generate semigroups without heat kernel acting on spaces satisfying the doubling condition(4).”
Spectral multiplier theorems of H\"ormander type on Hardy and Lebesgue spaces (2012, methodology): “The approach in [15] makes no use of Hardy spaces, but uses the result of [9].”
Weighted norm inequalities, Gaussian bounds and sharp spectral multipliers (2010, background): “For more information about the background of this topic, the reader is referred to [1, 2, 4, 5, 13, 15, 18, 21, 29] and the references therein.”
Musielak–Orlicz–Hardy Spaces Associated with Operators and Their Applications (2012, background): “1) is bounded on various function spaces on X , which was extensively studied (see, for example, [2, 3, 10, 21, 33, 38, 78, 30] and their references).”