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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 Zn and prove sharp Bochner-Riesz summability results similar to those known for the standard Laplacian on Rn.
|Number of pages||30|
|Journal||Journal de Mathématiques Pures et Appliquées|
|Publication status||Published - Nov 2020|
- Spectral multipliers
- Polynomial off-diagonal decay kernels
- Space of homogeneous type
- Random walk
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Harmonic analysis of rough oscillations
Sikora, A., Portal, P., Hassell, A., Guillarmou, C. & van Neerven, J.
30/05/16 → …