Weighted Besov and Triebel–lizorkin Spaces Associated With Operators and Applications

Abstract

Let X be a space of homogeneous type and L be a nonnegative self-adjoint operator on L 2 (X) satisfying Gaussian upper bounds on its heat kernels. In this paper we develop the theory of weighted Besov spacesḂ α,L p,q,w (X) and weighted Triebel-Lizorkin spacesḞ α,L p,q,w (X) associated to the operator L for the full range 0 < p, q ≤ ∞, α ∈ R and w being in the Muckenhoupt weight class A∞. Similarly to the classical case in the Euclidean setting, we prove that our new spaces satisfy important features such as continuous charaterizations in terms of square functions, atomic decompositions and the identifications with some well known function spaces such as Hardy type spaces and Sobolev type spaces. Moreover, with extra assumptions on the operator L, we prove that the new function spaces associated to L coincide with the classical function spaces. Finally we apply our results to prove the boundedness of the fractional power of L and the spectral multiplier of L in our new function spaces.

Besov and Triebel–Lizorkin spaces on spaces of homogeneous type with applications to boundedness of Calderón–Zygmund operators

“…Using this decomposition, Liu et al [53,54] obtained the endpoint boundedness of commutators on a space of homogeneous type. Recently, Bui et al [10] introduced weighted Besov and Triebel-Lizorkin spaces associated with operators on a space of homogeneous type and showed that, if the operator has some good properties, these function spaces coincide with classical counterparts on a space of homogeneous type defined via atoms.…”

Boundedness of Some Paraproducts on Spaces of Homogeneous Type

“…Notice that the harmonic analysis on spaces of homogeneous type has a long history; see, for example, [27,28,35,36]. We refer the reader to [33,34,[37][38][39][40][41][42][43][44][45][46] for the real-variable theory of some function spaces and Calderón-Zygmund operators on RD-spaces. Furthermore, for some recent developments on the real-variable theory of function spaces and its applications on spaces of homogeneous type, please see [29,[47][48][49][50][51][52][53][54][55][56][57][58][59][60][61].…”

Real-Variable Characterizations of Hardy–Lorentz Spaces on Spaces of Homogeneous Type with Applications to Real Interpolation and Boundedness of Calderón–Zygmund Operators

“…We should also point out that the nite atomic characterization of Hardy spaces on X was applied to the bilinear decomposition of the product of Hardy spaces and their dual spaces, as well as the boundedness of operators; see, for instance, [39,[76][77][78]. On the other hand, the real-variable theory of function spaces associated with operators on X has also developed rapidly; see, for instance, [14][15][16][17][18].…”

“…14) where the implicit positive constant is independent of a and m.Let ϵ := {ϵm} m∈N with ϵm := δ m[ϵ+ω( − p )] for any m ∈ N. By the fact that δ ∈ ( , ) and p ∈ ( ω ω+ϵ , ], we nd that ∞ m= m(ϵm) p < ∞. (8.15) Thus, from (8.12), (8.13), (8.14), and (8.15), it follows that T(a) is a harmless constant multiple of a (p, , ϵ)molecule as in [54, De nition 5.4].…”