Triebel–Lizorkin spaces with non-doubling measures

Abstract

Abstract. Suppose that µ is a Radon measure on R d , which may be non-doubling. The only condition assumed on µ is a growth condition, namely, there is a constant C 0 > 0 such that for all x ∈ supp(µ) and r > 0,where 0 < n ≤ d. The authors provide a theory of Triebel-Lizorkin spacesḞ s pq (µ) for 1 < p < ∞, 1 ≤ q ≤ ∞ and |s| < θ, where θ > 0 is a real number which depends on the non-doubling measure µ, C 0 , n and d. The method does not use the vector-valued maximal function inequality of Fefferman and Stein and is new even for the classical case. As applications, the lifting properties of these spaces by using the Riesz potential operators and the dual spaces are given.

The Hardy Space H1 with Non-doubling Measures and Their Applications

“…/ or not. • In [48], Han and Yang established a theory of Triebel-Lizorkin spaces P F s pq . / for p 2 .1; 1/, q 2 OE1; 1 and s 2 .…”

“…Â; Â/, where  2 .0; 1/ depends on , C 0 , n and D as in (0.0.1). Moreover, in [48], the method, without using the vectorvalued maximal function inequality of Fefferman and Stein, is new even for the classical case. As applications, the lifting properties of these spaces by using the Riesz potential operators and the dual spaces were given.…”

Marcinkiewicz Integrals with Non-Doubling Measures

“…In recent years, many classical results concerning the theory of Calderón-Zygmund operators and function spaces have been proved still valid if the Lebesgue measure is substituted by a measure µ as in (1.1); see [14,15,21,22,23,16,17,24,9,10,7,12,3]. We mention that the analysis on non-homogeneous spaces played an essential role in solving the long-standing open Painlevé's problem by Tolsa in [25].…”

Limiting case of the boundedness of fractional integral operators on nonhomogeneous space

“…For example, Tolsa has defined the Hardy space H 1 (μ) [11]. Han and Yang have defined the Triebel-Lizorkin spaces [3].…”