A Characterization of Besov Spaces of Para-Accretive Type and Its Application

There are two folds in this article. One fold is to characterize the Besov spaces of para-accretive type , which reduces to the classical Besov spaces when the para-accretive function is constant, by using a discrete Calderon-type reproducing formula and Plancherel-Polya-type inequality associated to a para-accretive function b in Rn. The other is to show that a generalized singular integral operator T with extends to be bounded from for and , where e is the regularity exponent of the kernel of T.


M TM WBP
∈ extends to be bounded

Introduction
Since Calderón and Zygmund developed the theory of singular integral operators in the fifties in last century, there have been lots of eagerness to generalize the theory in various ways. One kind of interest is to consider the boundedness of such operators on Hardy spaces, Triebel-Lizorkin spaces or Besov spaces (cf. [1]- [13]). The other interests include considering non-convolution operators such as the Calderón commutators (e.g. the 1 T and Tb theorems [14] [15]) or investigating operator-valued kernels (cf. [16] [17] [18]).
The remarkable 1 T theorem given by David and Journé [14] provides a general criterion for the 2 L -boundedness of these generalized singular integral operators. Frazier, Torres, and Weiss [4] considered the 1 T theorem on Triebel-Lizorkin spaces ,q p F α  , which include the classical p L spaces for 1 p < < ∞ and Hardy spaces p H for 0 1 p < ≤ , under the hypothesis * 0 Tx T x γ γ = = for a certain condition on γ . Afterward of of authors of the current paper extended the boundedness of singular integral operators acting on ,q p F α  to more relaxed restriction on Tx γ and * T x γ , see [12] [13] for details.
The Tb theorem for spaces of homogeneous type introduced by Coifman and Weiss was proved in [15]. If function 1 in the 1 T theorem is replaced by an accretive function, a bounded complex-valued function b satisfying ( ) 0 Reb x δ < ≤ almost everywhere, McIntosh and Meyer [10] showed the 2 L boundedness of the Cauchy integral on all Lipschitz curves. David, Journé, and Semmes [15] gave more general conditions on L ∞ functions, therefore one said para-accretive functions, and proved a new Tb theorem by substituting function 1 for para-accretive functions. It was also shown that if Tb theorem holds for a bounded function b , then b is necessarily para-accretive in [15].
In 2009, Lin and Wang [8] used a discrete Calderón-type reproducing formula and Plancherel-Pôlya-type inequality to characterize homogeneous Triebel-Lizorkin spaces of para-accretive type , , q b p F α  . A necessary and sufficient condition of singular integral operators which is bounded from with the regularity exponent ε of the kernel, is also derived in [8]. In this article, we study the Moreover, the operator T can be represented by for all We say that a singular integral operator is a Calderón-Zygmund operator if it can be extended to a bounded operator on ( ) 2 n L  . Coifman and Meyer [19] showed that every Calderón-Zygmund operator is bounded on p L for 1 p < < ∞ .
A locally integrable function defined on n  belongs to BMO if it satisfies ( ) where the supremum is taken over all cubes n Q ⊆  whose sides are parallel to the axes and . Note that these cubes need not be dyadic. For Definition 1.5. Suppose 1 b and 2 b are bounded complex-valued functions whose inverses are also bounded. A generalized singular integral operator is a continuous linear operator T Such an operator T is written as  M TM WBP ∈ for some , then T extends to a bounded linear operator from The main purpose and methods used in this paper is related to a Tb theorem in Besov spaces of para-accretive type 0, ,

which was introduced by
Han [20] for p , 1 q > , by Deng and Yang [21] for p , 1 q ≤ , denoted as Once one has an approximation to the identity, a Plancherel-Pôlya-type inequality follows immediately. For the terminology used in the rest of this section, see Section 2 for details.
another approximation to the identity with the same properties as the { } k k S ∈ .
Now it is ready to define a class of the homogeneous Besov spaces associated to para-accretive functions.
 be an approximation to the identity defined in Definition 2.1 and set , and 0 q < ≤ ∞ , the homogeneous Besov spaces of para-accretive type , , From Theorem 1.8, one can check that Definition 1.9 is independent of choices of approximations to the identity. As an application, one has the following.
Theorem 1.10. (Reduced Tb theorem for Besov spaces of para-accretive type) Assume that b is a para-accretive function. Let The proof of this main result is based on the discrete Calderón-type reproducing formula [5], a characterization of Besov spaces , , This paper is organized as follows. In Section 2, one gives some preliminaries.
Then one states and proves a Plancherel-Pôlya-type inequality in Section 3. Then one uses a Plancherel-Pôlya-type inequality to show norm equivalence between Besov space , , Finally one proves reduced Tb theorem for Besov spaces of para-accretive type in Section 5. Through the paper, one uses Q to denote a dyadic cube in n  , j k ∧ denotes the minimum of j and k and uses C to denote a positive constant independent of the main variables, which may vary from line to line.
means that there exist two positive constants 1 c and 2 c so that

Preliminaries
Recall the definition of approximation to the identity associated to a paraaccretive function and a related Calderón reproducing formula generated by such an approximation to the identity, and start with "test functions'' given by Han [20]. Fix two exponents 0 1 β < ≤ and 0 γ > . Suppose that b is a paraaccretive function. A function f defined on n  is said to be a test function Denote by We denote x d β γ β γ =   with equivalent norms. Thus, given In order to state the Calderón reproducing formula, one also needs an approximation to the identity (cf. [7] [15] [20]).  The following discrete Calderón reproducing formulae were given in [5].
such that, and ( ) where Q are all dyadic cubes with the side length 2 k N − − for some fixed positive large integer N and Q y is any fixed point in Q .

Plancherel-Pôlya-Type Inequalities
The classical Plancherel-Pôlya inequality has a long history and plays a central role in the theory of function spaces. Roughly speaking, if a tempered distribution f in n  , whose Fourier transform has compact support, then, by the Paley-Wiener theorem, it is an analytic function, or more precisely, entire analytic function of exponential type. The Plancherel-Pôlya inequality concludes is an appropriate set of points in n  , e.g., lattice points, where the length of the mesh is sufficiently small, then ( ) for all 0 p < ≤ ∞ with a modification if p = ∞ . The Fourier transform is the basic tool to prove such an inequality. See [22] for more details. For any cube Q and 0 λ > , one denotes by Q λ the cube concentric with Q whose each edge is λ times as long. A generalized Plancherel-Pôlya-type inequality for Triebel-Lizorkin spaces was given in [8]. In this section, one proves the following Plancherel-Pôlya-type inequalities in Besov sense.
Proof of Theorem 1.8. By Proposition 2.2, f can be written as where k Q y is any fixed point in k Q . To estimate using the inequality (see [7]) where ε ′ and ε ′′ are close enough to ε and satisfy α ε ε ε ′′ ′ < < < , one obtains Thus, For simplicity, let ( ) First one considers the case for 1 p ≤ . In this case, because we may choose ε ′ so that Note that the last inequality is followed from and If q p > , by Hölder's inequality, one has Next let us consider the case 1 p < < ∞ , by Hölder's inequality For q p > , by Hölder's inequality and (33) again, one obtains Since k Q y can be replaced by any point in k Q , it follows that (35) still holds for 1 q < < ∞ . With a modification for q = ∞ , (35) holds and therefore Conversely, if one interchanges the roles of S and V in the proof above, one immediately has ( ) ( ) and therefore the proof of part (a) is finished. The proof of part (b) is the same as the one of part (a).
First one considers the case for 1 p ≤ . In this case,