Del Teso Méndez, FélixGómez-Castro, D.Vázquez, Juan Luis2023-06-162023-06-162021-08-231311-045410.1515/fca-2021-0042https://hdl.handle.net/20.500.14352/4992We introduce three representation formulas for the fractional p-Laplace operator in the whole range of parameters 0 < s < 1 and 1 < p < ∞. Note that for p ≠ 2 this a nonlinear operator. The first representation is based on a splitting procedure that combines a renormalized nonlinearity with the linear heat semigroup. The second adapts the nonlinearity to the Caffarelli-Silvestre linear extension technique. The third one is the corresponding nonlinear version of the Balakrishnan formula. We also discuss the correct choice of the constant of the fractional p-Laplace operator in order to have continuous dependence as p → 2 and s → 0+, 1−. A number of consequences and proposals are derived. Thus, we propose a natural spectral-type operator in domains, different from the standard restriction of the fractional p-Laplace operator acting on the whole space. We also propose numerical schemes, a new definition of the fractional p-Laplacian on manifolds, as well as alternative characterizations of the Ws, p(ℝn) seminorms.engjournal articlehttps://doi.org/10.1515/fca-2021-0042open access517517.9Fractional p-LaplacianBochner’s subordinationSemigroup formulaExtension problemBalakrishnan’s formulaSpectral formulationAnálisis matemáticoEcuaciones diferenciales1202 Análisis y Análisis Funcional1202.07 Ecuaciones en Diferencias