Since Calderón and Zygmund developed the theory of singular integral opera- tors 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.  -  ). The other interests include considering non-convolution operators such as the Calderón commutators (e.g. the and theorems   ) or investigating operator-valued kernels (cf.    ).
The remarkable theorem given by David and Journé  provides a general criterion for the -boundedness of these generalized singular integral operators. Frazier, Torres, and Weiss  considered the theorem on Triebel-Lizorkin spaces , which include the classical spaces for and Hardy spaces for , under the hypothesis for a certain condition on . Afterward of of authors of the current paper extended the boundedness of singular integral operators acting on to more relaxed restriction on and , see   for details.
The theorem for spaces of homogeneous type introduced by Coifman and Weiss was proved in  . If function 1 in the theorem is replaced by an accretive function, a bounded complex-valued function satisfying almost everywhere, McIntosh and Meyer  showed the boundedness of the Cauchy integral on all Lipschitz curves. David, Journé, and Semmes  gave more general conditions on functions, therefore one said para-accretive functions, and proved a new theorem by substituting function 1 for para-accretive functions. It was also shown that if theorem holds for a bounded function , then is necessarily para-accretive in  .
In 2009, Lin and Wang  used a discrete Calderón-type reproducing formula and Plancherel-Pôlya-type inequality to characterize homogeneous Triebel- Lizorkin spaces of para-accretive type . A necessary and sufficient condition of singular integral operators which is bounded from to ,
and with the regularity exponent of the kernel, is also derived in  . In this article, we study the boundedness of singular integral operators for wider ranges of and .
One begins by recalling some basic results about Calderón-Zygmund operator theory. As usual, denotes the set of functions with compact support and denotes the Schwartz class.
Definition 1.1. We say that is a singular integral operator, denoted by , if is a continuous linear operator from into its dual associated to a kernel , a continuous function defined on , satisfying the following conditions: there exist constants and such that
Moreover, the operator can be represented by
for all with .
We say that a singular integral operator is a Calderón-Zygmund operator if it can be extended to a bounded operator on . Coifman and Meyer  showed that every Calderón-Zygmund operator is bounded on for .
A locally integrable function defined on belongs to if it satisfies
where the supremum is taken over all cubes whose sides are parallel to the axes and . Note that these cubes need not be dyadic. For , and , let .
Definition 1.2. Let be a continuous linear operator. is called to have the weak boundedness property, denoted by , if for every bounded subset of , there is a constant such that
for all and in , , and .
David and Journé  gave a general criterion for the boundedness of singular integral operators as follows:
Proposition 1.3 ($T1$ theorem for L2) Suppose that for some and denotes its transpose. Then extends to be bounded on if and only if , , and .
Before stating the theorem of David, Journé and Semmes  , one recalls some definitions. Let denote the space of continuous functions with compact support such that
Definition 1.4. A bounded complex-valued function defined on is said to be para-accretive if there exist constants such that, for all cubes , there is a subcube with satisfying
Definition 1.5. Suppose and are bounded complex-valued functions whose inverses are also bounded. A generalized singular integral operator is a continuous linear operator from into , , for which the associated kernel satisfies inequalities (1)-(3) such that, for all , with ,
Such an operator is written as , where is the regularity exponent of in Definition 1.1.
Denote the multiplication operator by ; that is, . David, Journé and Semmes  proved the following theorem.
Proposition 1.6. ( theorem for ) Suppose that and are para- accretive functions and . Then extends to be bounded on if and only if (1) , (2) , and (3) .
Later on Lin and Wang gave the following result. For any , let denote the integer part of and . For , .
Proposition 1.7. (  ) Assume that is a para-accretive function. Let and for some . For , and , if , then extends to a bounded linear operator from to .
The main purpose and methods used in this paper is related to a theorem in Besov spaces of para-accretive type , which was introduced by Han  for , , by Deng and Yang  for , , 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.
Theorem 1.8 (Plancherel-Polya-type inequality) Let and . Suppose that is an approximation to the identity defined in Definition 2.1 and is another approximation to the identity with the same properties as the . Set and .
a) For all , if is finite then
b) For all , if is finite then
Now it is ready to define a class of the homogeneous Besov spaces associated to para-accretive functions.
Definition 1.9. Let be an approximation to the identity defined in Definition 2.1 and set for as before. For , , and , the homogeneous Besov spaces of para-accretive type is the collection of such that
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 follow- ing.
Theorem 1.10. (Reduced Tb theorem for Besov spaces of para-accretive type) Assume that is a para-accretive function. Let and for some . For , and , if then extended to a bounded linear operator from to .
The proof of this main result is based on the discrete Calderón-type reproduc- ing formula  , a characterization of Besov spaces , and a Plancherel- Pôlya-type inequality.
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 and its corresponding sequence space in Section 4. Finally one proves reduced theorem for Besov spaces of para-accretive type in Section 5. Through the paper, one uses to denote a dyadic cube in , denotes the minimum of and and uses to denote a positive constant independent of the main variables, which may vary from line to line. Also means that there exist two positive constants and so that .
Recall the definition of approximation to the identity associated to a para- accretive function and a related Calderón reproducing formula generated by such an approximation to the identity, and start with “test functions’’ given by Han  . Fix two exponents and . Suppose that is a para- accretive function. A function defined on is said to be a test function of type centered at with width if
Denote by the collection of all test functions of type centered at with width . For , the norm of in is defined by
We denote simply by .
It is clear that is a Banach space under the norm . Write
If and for , then the norm of is defined by . As usual, one uses and to denote the dual spaces of and , respectively. Use to denote the natural pairing of elements and
It is easy to check that for any and with equivalent norms. Thus, given , is well defined for all with any and .
In order to state the Calderón reproducing formula, one also needs an approximation to the identity (cf.    ).
Definition 2.1. Let be a para-accretive function. A sequence of linear operators is called an approximation to the identity associated to if the kernels of are functions from into such that there exist constant and some satisfying, for all and all , and ,
5) for all and ,
6) for all and .
The following discrete Calderón reproducing formulae were given in  .
Proposition 2.2. Suppose that is an approximation to the identity defined in Definition 2.1. Set . Then there exists a family of operators with kernel satisfying, for ,
where are all dyadic cubes with the side length for some fixed positive large integer and is any fixed point in .
Note that if and only if or equivalently, if and only if .
3. 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 in , 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 that if is an appropriate set of points in , e.g., lattice points, where the length of the mesh is sufficiently small, then
for all with a modification if . The Fourier transform is the basic tool to prove such an inequality. See  for more details.
For any cube and , one denotes by the cube concentric with whose each edge is times as long. A generalized Plancherel-Pôlya-type inequality for Triebel-Lizorkin spaces was given in  . In this section, one proves the following Plancherel-Pôlya-type inequalities in Besov sense.
Proof of Theorem 1.8. By Proposition 2.2, can be written as
where is any fixed point in . To estimate
using the inequality (see  )
where and are close enough to and satisfy , one obtains
For simplicity, let
First one considers the case for . In this case,
because we may choose so that
is finite. If , then
Note that the last inequality is followed from
If , by Hölder’s inequality, one has
Next let us consider the case , by Hölder’s inequality
For , one uses triangular inequality and (34) again to yield
For , by Hölder’s inequality and (33) again, one obtains
Since can be replaced by any point in , it follows that (35) still holds for . With a modification for , (35) holds and therefore
for , and .
Conversely, if one interchanges the roles of and 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).
4. Besov Spaces of Para-Accretive Type
Recall a definition and the duals of Besov sequence spaces introduced by Frazier and Jawerth   . For and , the space consists of all sequences satisfying
Proposition 4.1. (   ) Let , , . Then
with the pairing where and . As usual, when , interprets as .
Next one recalls the definition of almost diagonality and the boundedness of almost diagonal matrices acting on Besov sequence spaces.
Definition 4.2. For and , let . one says that a matrix is almost diagonal, denoted by , if there exist and such that, for all dyadic cubes and ,
Proposition 4.3. (   ) Let , . If , then is bounded on .
Theorem 4.4. Suppose that is an approximation to the identity defined in Definition 2.1 and set for . For , , and ,
In particular, the definition of is independent of the choice of approximations to the identity.
Proof. Let and be approximations to the identity defined in Definition 2.1. Set and . One wants to show that
By the Plancherel-Pôlya-type inequality, one has
Apply Proposition 4.3 and the Plancherel-Pôlya-type inequality again to yield
Conversely, if one interexchanges the roles and in the proof before, then one has
and the proof is completed.
Form the last theorem, the definition of homogeneous Besov spaces of para-accretive type is independent of the choice of approximations to the identity. For simplicity, one writes in stead of in the sequel.
Theorem 4.5. Suppose , and .
(a) If , then and .
(b) If , then and .
Furthermore, if , then the dual space of is and the dual space of is .
Proof. If , by Proposition 2.2,
By Theorem 4.4, one gets
The proof of case (b) is the same. To show the duality. By Propositions 2.2, if and , then
By the estimates for , it is routine to check that the matrix
is almost diagonal defined in Proposition 4.3. Thus it is bounded on , by Proposition 4.1. Thus
where the first inequality is followed from Propositions 4.1 and 4.3, and the second inequality is followed from Theorem 4.4. Therefore the duality follows immediately.
5. An Application
In this section one give a proof of reduced theorem for Besov case.
Proof of Theorem 1.10. For , by Theorem 4.4, one has
By the Calderón-reproducing formula,
Using the estimate given in  Lemma 3.13
where and are close enough to with .
First one considers the case for . In this case,
because one mays choose so that
is finite. When , one uses triangular inequality, and when , one uses Hölder’s inequality to yield
so is bounded from to for .
Now consider the case , by Hölder’s inequality
Similarly to this case, when , one uses triangular inequality, and when , one uses Hölder’s inequality. Thus extends to a bounded linear operator from to .
It is clear that , and hence is bounded from to if by Theorem 1.3 in  , where is a paraproduct operator defined by
for some fixed satisfying and . It is natural to ask what is the necessary and sufficient condition for the boundedness of paraproduct operators acting from to ?
Research by author was supported by Ministry of Science and Technology, R.O.C. under Grant #MOST 105-2115-M-259-002.
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 Frazier, M., Jawerth, B. and Weiss, G. (1991) Littlewood-Paley Theory and the Study of Function Spaces. CBMS Regional Conference Series, Vol. 79, A.M.S., Providence.