Boundedness of Convolution-Type Operators on Certain Endpoint Triebel-Lizorkin Spaces
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Boundedness of Convolution-Type Operators on Certain Endpoint Triebel-Lizorkin Spaces Zhanying Yang
Received: 27 June 2010 / Accepted: 10 March 2011 / Published online: 22 March 2011 © Springer Science+Business Media B.V. 2011
Abstract In this paper, we are concerned with the boundedness of convolution-type Calderón-Zygmund operators on some endpoint Triebel-Lizorkin spaces. We establish the 0,q boundedness on F˙1 (2 < q < ∞) under a very weak pointwise regularity condition. The boundedness is established by the Daubechies wavelets and the atomic-molecular approach. Keywords Convolution-type Calderón-Zygmund operators · Endpoint Triebel-Lizorkin spaces · Daubechies wavelets · Atomic-molecular decomposition Mathematics Subject Classification (2000) 42B20 · 42C40
1 Introduction Let D = D(Rn ) denote the space of indefinitely differentiable functions with compact support and D the space of Schwartz distributions (the dual of D). Suppose that we have a linear continuous mapping T : D → D associated with a kernel K(x, y) (in the sense that Tf, g = g(x)K(x, y)f (y)dxdy (1.1) for test functions f and g with disjoint support). Assume that K(x, y) is continuous on = Rn × Rn \{x = y} and satisfies: |K(x, y)| ≤ C1 |x − y|−n ;
This research is supported by the Special Fund for Basic Scientific Research of Central Colleges, South-Central University for Nationalities (No. ZZQ10010) and Research Fund for the Doctoral Program of Higher Education (No. 20090141120010). Z. Yang () Department of Mathematics, South-Central University for Nationalities, Wuhan 430074, China e-mail: [email protected]
(1.2)
194
Z. Yang
if 2|x − x | ≤ |x − y|, then C2 |x − x |γ (1.3) |x − y|n+γ for 0 < γ ≤ 1. And for all f ∈ L2 (Rn ) with compact support, Tf (x) = K(x, y)f (y)dy holds for almost every x ∈ (suppf )c . Assume also that T extends to a bounded operator on L2 (Rn ). Then T is said to be a Calderón-Zygmund (C-Z) operator (see [2]), and written as T ∈ CZOγ . Ever since C-Z operators have been introduced by Coifman and Meyer, there has been significant progress on the study of their boundedness on various function spaces (see [1, 3, 6, 8]). The prototypical result is the famous T 1 theorem of David and Journé (see [3]), which states that under the conditions (1.2) and (1.3), T extends to a bounded operator on L2 if and only if it satisfies the T 1 condition: |K(x, y) − K(x , y)| + |K(y, x) − K(y, x )| ≤
T 1 ∈ BMO,
T ∗ 1 ∈ BMO,
(1.4)
and the weak boundedness condition: |Tf, g| ≤ C3 R n ( f ∞ + R ∇f ∞ )( g ∞ + R ∇g ∞ ), ∀R > 0, u ∈ Rn , f, g ∈ C01 (B(u, R)).
(1.5)
Since then, many mathematicians have been devoted to relaxing the regularity condition (1.3) (see [9,11]). But up to now, it is still unknown whether (1.3) can be replaced by the Hörmander condition: {|K(x, y) − K(x , y)| + |K(y, x) − K(y, x )|}dy < ∞. (1.6) sup x,x
|x−y|≥2|x−x |
In Deng-Yan-Yang [4], it is proved that the Hörmander condition can not ensure the boundedness on certain endpoint function spaces. To scale the extent of be
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