Abstract :
Let s ∈ R, τ ∈ [0,∞), p ∈ (1,∞) and q ∈ (1,∞]. In this paper, we introduce a new class of function
spaces ˙ F
s,τ
p,q (Rn) which unify and generalize the Triebel–Lizorkin spaces with both p ∈ (1,∞) and p=∞
and Q spaces. By establishing the Carleson measure characterization of Q space, we then determine the
relationship between Triebel–Lizorkin spaces and Q spaces, which answers a question posed by Dafni and
Xiao in [G. Dafni, J. Xiao, Some new tent spaces and duality theorem for fractional Carleson measures
and Qα(Rn), J. Funct. Anal. 208 (2004) 377–422]. Moreover, via the Hausdorff capacity, we introduce
a new class of tent spaces F ˙ T
s,τ
p,q (Rn+1
+ ) and determine their dual spaces F ˙W
−s,τ/q
p
,q
(Rn), where s ∈ R,
p, q ∈ [1,∞), max{p, q} > 1, τ ∈ [0,
q
(max{p,q})
], and t
denotes the conjugate index of t ∈ (1,∞); as an
application of this, we further introduce certain Hardy–Hausdorff spaces F ˙H
s,τ
p,q (Rn) and prove that the
dual space of F ˙H
s,τ
p,q (Rn) is just ˙ F
−s,τ/q
p
,q
(Rn) when p, q ∈ (1,∞).
© 2008 Elsevier Inc. All rights reserved.Let s ∈ R, τ ∈ [0,∞), p ∈ (1,∞) and q ∈ (1,∞]. In this paper, we introduce a new class of function
spaces ˙ F
s,τ
p,q (Rn) which unify and generalize the Triebel–Lizorkin spaces with both p ∈ (1,∞) and p=∞
and Q spaces. By establishing the Carleson measure characterization of Q space, we then determine the
relationship between Triebel–Lizorkin spaces and Q spaces, which answers a question posed by Dafni and
Xiao in [G. Dafni, J. Xiao, Some new tent spaces and duality theorem for fractional Carleson measures
and Qα(Rn), J. Funct. Anal. 208 (2004) 377–422]. Moreover, via the Hausdorff capacity, we introduce
a new class of tent spaces F ˙ T
s,τ
p,q (Rn+1
+ ) and determine their dual spaces F ˙W
−s,τ/q
p
,q
(Rn), where s ∈ R,
p, q ∈ [1,∞), max{p, q} > 1, τ ∈ [0,
q
(max{p,q})
], and t
denotes the conjugate index of t ∈ (1,∞); as an
application of this, we further introduce certain Hardy–Hausdorff spaces F ˙H
s,τ
p,q (Rn) and prove that the
dual space of F ˙H
s,τ
p,q (Rn) is just ˙ F
−s,τ/q
p
,q
(Rn) when p, q ∈ (1,∞).
© 2008 Elsevier Inc. All rights reserved.
