Sobolev algebras through heat kernel estimates

Abstract

Abstract. On a doubling metric measure space (M, d, µ) endowed with a "carré du champ", let L be the associated Markov generator andL p α (M, L, µ) the corresponding homogeneous Sobolev space of order 0 < α < 1 in L p , 1 < p < +∞, with norm L α/2 f p . We give sufficient conditions on the heat semigroup (e −tL ) t>0to 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 pboundedness 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.

Sobolev Algebras Through a ‘Carré Du Champ’ Identity

“…The algebra property. Following up on [4], we aim to prove that the (Besseltype) Sobolev spaces satisfy an algebra property under our assumptions. Such property is very well understood in the Euclidean space and goes back to initial works by Strichartz [13], Kato and Ponce [9], and then Coifman and Meyer [6,11] using the paraproduct decomopsition.…”

“…Then A(p, α) holds for every p ∈ (1, p 0 ) with α ∈ (0, 1), and for every p ∈ (p 0 , ∞) with 0 < α < p 0 p . The condition p 0 < ν is not relevant and not used, but for p 0 > ν the result was already obtained in [4] in a more general framework. That is why we restrict our attention here to the range 2 ≤ p 0 < ν.…”

“…We use a slightly different decomposition of the product than in [4]. Indeed in [4], the product of two functions was decomposed into two paraproducts. Here, we decompose it into three terms (two paraproducts and a 'resonant part').…”

“…We first observe that in the case of a self-adjoint and conservative operator L, then by integrating (1.4) implies exactly (1.5). So our current assumption is stronger than the one used in [4] and corresponds to a pointwise version; it is therefore natural that we are able to obtain a wider range of exponents. To be more precise, for p > p 0 we improve the range α ∈ (0, 1 − ν( 1 p 0 − 1 p )) (obtained in [4]) to α ∈ (0, p 0 p ).…”

“…So our current assumption is stronger than the one used in [4] and corresponds to a pointwise version; it is therefore natural that we are able to obtain a wider range of exponents. To be more precise, for p > p 0 we improve the range α ∈ (0, 1 − ν( 1 p 0 − 1 p )) (obtained in [4]) to α ∈ (0, p 0 p ). • Moreover, we only detail the proofs of [4] and of the current work in the setting where the semigroup is supposed to satisfy (UE), which corresponds to pointwise (or L 1 -L ∞ ) local estimates.…”