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Singular integral operators with non-smooth kernels on irregular domains.

Xuan Thinh Duong;  Alan McIntosh — 1999

Revista Matemática Iberoamericana

Let χ be a space of homogeneous type. The aims of this paper are as follows. i) Assuming that T is a bounded linear operator on L2(χ), we give a sufficient condition on the kernel of T such that T is of weak type (1,1), hence bounded on Lp(χ) for 1 < p ≤ 2; our condition is weaker then the usual Hörmander integral condition. ii) Assuming that T is a bounded linear operator on L2(Ω) where Ω is a measurable...

Littlewood-Paley-Stein functions on complete Riemannian manifolds for 1 ≤ p ≤ 2

Thierry Coulhon;  Xuan Thinh Duong;  Xiang Dong Li — 2003

Studia Mathematica

We study the weak type (1,1) and the L p -boundedness, 1 < p ≤ 2, of the so-called vertical (i.e. involving space derivatives) Littlewood-Paley-Stein functions and ℋ respectively associated with the Poisson semigroup and the heat semigroup on a complete Riemannian manifold M. Without any assumption on M, we observe that and ℋ are bounded in L p , 1 < p ≤ 2. We also consider modified Littlewood-Paley-Stein functions that take into account the positivity of the bottom of the spectrum. Assuming that...

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