Hardy spaces associated with different homogeneities and boundedness of composition operators

Abstract

It is well known that standard Calderón-Zygmund singular integral operators with isotropic and nonisotropic homogeneities are bounded on the classical H p (R m ) and nonisotropic H p h (R m ), respectively. In this paper, we develop a new Hardy space theory and prove that the composition of two Calderón-Zygmund singular integral operators with different homogeneities is bounded on this new Hardy space. Such a Hardy space has a multiparameter structure associated with the underlying mixed homogeneities arising from the two singular integral operators under consideration. The Calderón-Zygmund decomposition and an interpolation theorem hold on these new Hardy spaces.Mathematics Subject Classification (2010): Primary 42B30; Secondary 42B20.

Multi-parameter Triebel–Lizorkin spaces associated with the composition of two singular integrals and their atomic decomposition

“…When 1 = 2 = 0, = 2 and 0 < ≤ 1, our spaceṡ , (ℝ ) = com (ℝ ). Hence, our Theorem 1.5 also veri es that the Hardy spaces com (ℝ ) given in [19] in the discrete form is actually equivalent to the one de ned in the continuous form.…”

“…Similarly, by comparing and 2 , we obtain | | ≤ 2, which is the aim to restrict the support of̂ (2) in the set {( ὔ , ) ∈ ℝ −1 × ℝ : 1 2 ≤ | | ℎ ≤ 2 1/2 }, smaller than the range in references [7,19]. Therefore applying Lemma 2.7 to each of the functions , * (2 − ∧ ὔ , 2 − ∧2…”

“…The following lemma is a variant of [19,Lemma 3.2]. The main di erence between them is that we restrict the sum in a collection.…”

“…However, the composition operator is bounded on neither the classical Hardy space nor the non-isotropic Hardy space. This motivates the authors of [19] to introduce a new Hardy space associated with the di erent homogeneities and establish the boundedness of composition singular integrals on such Hardy spaces. It is interesting to note that such Hardy spaces are of multi-parameter setting in nature.…”

“…As an application of our atomic decomposition, we obtain the necessary and su cient conditions for the boundedness of an operator on the multi-parameter Triebel-Lizorkin type spaces. In the special case of 1 = 2 = 0, = 2 and 0 < ≤ 1, our spaceṡ , (ℝ ) coincide with the Hardy spaces com associated with the composition of two di erent singular integrals (see [19]). Therefore, our results also give an atomic decomposition of com .…”