# Sharp spectral multipliers without semigroup framework and application to random walks

## TL;DR

This paper establishes sharp spectral multiplier theorems for self-adjoint operators on spaces of homogeneous type without relying on semigroup assumptions, using polynomial decay, and applies results to differential operators, pseudo-differential operators, Markov chains, and random walks.

## Contribution

It introduces a novel approach to spectral multipliers that avoids semigroup frameworks and handles polynomial off-diagonal decay, expanding applicability to various operators and random walks.

## Key findings

- Proved sharp spectral multiplier theorems without semigroup assumptions.
- Extended results to differential, pseudo-differential operators, and Markov chains.
- Established sharp Bochner-Riesz summability for random walks on rf6s lattice.

## Abstract

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 \`a la Stein-Tomas. This allows us to obtain sharp spectral multiplier theorems and hence sharp Bochner-Riesz summability results. Finally, we consider the random walk on the integer lattice $\mathbf{Z}^n$ and prove sharp Bochner-Riesz summability results similar to those known for the standard Laplacian on $\mathbb{R}^n$.