Dual Properties of Triebel-Lizorkin-Type Spaces and their Applications

作者：Yang, Dachun^*; Yuan, Wen

来源：Zeitschrift fur Analysis und Ihre Anwendungen, 2011, 30(1): 29-58.

DOI：10.4171/ZAA/1422

• Hausdorff capacity
• Besov space
• Triebel-Lizorkin space
• tent space
• duality
• atom
• molecule
• embedding
• pseudo-differential operator
• trace

Let s is an element of R, p is an element of (1, infinity), tau is an element of [0, 1/p] and S-infinity(R-n) be the set of all Schwartz functions phi whose Fourier transforms (phi) over cap satisfy that partial derivative(gamma)(phi) over cap (0) = 0 for all gamma is an element of (N boolean OR {0})(n). Denote by (V)<(F)over dot>(s,tau)(p,p)(R-n) the closure of S infinity(R-n) in the Triebel-Lizorkin-type space (V)<(F)over dot>(s,tau)(p,p)(R-n). In this paper, the authors prove that the dual space of (V)<(F)over dot>(s,tau)(p,p)(R-n) is the Triebel-Lizorkin-Hausdorff space F<(H)over dot>(-s,tau)(p',p')(R-n) via their phi-transform characterizations together with the atomic decomposition characterization of the tent space F<(T)over dot>(-s,tau)(p',p')(R-Z(n+1)) where t' denotes the conjugate index of t is an element of [1, infinity]. This gives a generalization of the well-known duality that (CMO(R-n))* = H-1(R-n) by taking s = 0, p = 2 and tau = 1/2. As applications, the authors obtain the Sobolev-type embedding property, the smooth atomic and molecular decomposition characterizations, boundednesses of both pseudo-differential operators and the trace operators on F<(H)over dot>(s,tau)(p,p)(R-n); all of these results improve the existing conclusions.

• 出版日期2011
• 单位北京师范大学