In this paper we prove spectral multiplier theorems for abstract self-adjoint operators on spaces of homogeneous type. We have two main objectives. The first one is to work outside the semigroup context. In contrast to previous works on this subject, we do not make any assumption on the semigroup. The second objective is to consider polynomial off-diagonal decay instead of exponential one. Our approach and results lead to new applications to several operators such as differential operators, pseudo-differential operators as well as Markov chains. In our general context we introduce a restriction type estimates à la Stein-Tomas. This allows us to obtain sharp spectral multiplier theorems and hence sharp Bochner-Riesz summability results in some situation. Finally, we consider the random walk on the integer lattice ${\mathbb{Z}}^n$ and prove sharp Bochner-Riesz summability results similar to those known for the standard Laplacian on ${\mathbb{R}}^n$.

在本文中，我们证明了齐次类型空间上抽象自伴算子的谱乘子定理。我们有两个主要目标。第一个是在半群上下文之外工作。与之前在该主题上的研究相比，我们对半群没有任何假设。第二个目标是考虑多项式非对角衰减，而不是指数式衰减。我们的方法和结果导致了对一些算子的新应用，例如微分算子，伪微分算子以及马尔可夫链。在一般情况下，我们引入一种限制类型估计值-Stein-Tomas。这使我们可以获得清晰的频谱乘子定理，因此在某些情况下可以得出清晰的Bochner-Riesz可加性结果。最后，我们考虑在整数晶格上的随机游动${\mathbb{ž}}^{ñ}$ 并证明了与标准Laplacian相似的清晰Bochner-Riesz可加性结果 ${\mathbb{[R}}^{ñ}$。