Sobolev algebras through heat kernel estimates

2015/05/06 by Bernicot, Frédéric, Coulhon, Thierry, Frey, Dorothee
#43A85 #46E35 #47D06 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1505.01442

Abstract

On a doubling metric measure space (M,d,μ) endowed with a "carré du champ", let L be the associated Markov generator and L^p_α(M,L,μ) the corresponding homogeneous Sobolev space of order 00 for the spaces L^p_α(M,L,μ) ∩ L^∞(M,μ) to be algebras for the pointwise product. Two approaches are developed, one using paraproducts (relying on extrapolation to prove their boundedness) and a second one through geometrical square functionals (relying on sharp estimates involving oscillations). A chain rule and a paralinearisation result are also given. In comparison with previous results ([29,11]), the main improvements consist in the fact that we neither require any Poincaré inequalities nor L^p-boundedness of Riesz transforms, but only L^p-boundedness of the gradient of the semigroup. As a consequence, in the range p∈(1,2], the Sobolev algebra property is shown under Gaussian upper estimates of the heat kernel only.

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