Calderón reproducing formulas and new Besov spaces associated with operators

• Published in: Advances in mathematics (New York. 1965) 2012-03, Vol.229 (4), p.2449-2502
• Main Authors: Bui, Huy-Qui, Duong, Xuan Thinh, Yan, Lixin
• Format: Article
• Language: English
• Subjects: Analytic semigroup; Besov space; Calderón reproducing formula; Embedding theorem; Fractional integrals; Heat kernel; Schrödinger operator
• ISSN: 0001-8708; 1090-2082
• DOI: 10.1016/j.aim.2012.01.005

Let L be the generator of an analytic semigroup whose heat kernel satisfies an upper bound of Poisson type acting on L2(X) where X is a (possibly non-doubling) space of polynomial upper bound on volume growth. The aim of this paper is to introduce a new class of Besov spaces associated with the operator L so that when L is the Laplace operator −Δ or its square root −Δ acting on the Euclidean space Rn, the new Besov spaces are equivalent to the classical Besov spaces. Depending on the choice of L, the new Besov spaces are natural settings for generic estimates for certain singular integral operators such as the fractional powers Lα. Since our approach does not require the doubling volume property of the underlying space, it is applicable to any subset Ω of Euclidean spaces without any smoothness requirement on the boundary. We will also develop a number of Calderón reproducing formulas which play an important role in the theory of function spaces and are of independent interest. As an application, we study Besov spaces associated with Schrödinger operators with non-negative potentials satisfying reverse Hölder estimates.