Resumen: In this paper we treat the following partial differential equation, the quasigeostrophic equation: d / dt + u · grad f = - s - A a f, 0 = a = 1, where (A, D (A)) is the infinitesimal generator of a convolution C 0 -semigroup of positive kernel on L p (R n), with 1 = p < 8. Firstly, we give remarkable pointwise and integral inequalities involving the fractional powers (- A) a for 0 = a = 1. We use these estimates to obtain L p -decayment of solutions of the above quasigeostrophic equation. These results extend the case of fractional derivatives (taking A = ¿, the Laplacian), which has been studied in the literature. Idioma: Inglés DOI: 10.1155/2019/4763450 Año: 2019 Publicado en: Journal of Function Spaces 2019, 4763450 (2019), [7 pp] ISSN: 2314-8896 Factor impacto JCR: 1.896 (2019) Categ. JCR: MATHEMATICS, APPLIED rank: 50 / 260 = 0.192 (2019) - Q1 - T1 Categ. JCR: MATHEMATICS rank: 24 / 323 = 0.074 (2019) - Q1 - T1 Factor impacto SCIMAGO: 0.455 - Analysis (Q3)