Research Article | Open Access
A Class of Unbounded Fourier Multipliers on the Unit Complex Ball
We introduce a class of Fourier multiplier operators on n-complex unit sphere, where the symbol . We obtained the Sobolev boundedness of . Our result implies that the operators take a role of fractional differential operators on .
In this paper, we introduce a class of unbounded holomorphic Fourier multipliers on complex unit sphere. We further study the boundedness of on Sobolev spaces. Our results generalize the theory of Fourier multipliers on Lipschitz curves in to complex unit sphere . We refer the reader to Gaudry et al. , McIntosh and Qian , and Qian [3, 4] for further information on multipliers on Lipschitz curves.
Our motivation originates from the following example on the unit sphere in . The explicit formula of the Cauchy-Szegö kernel is as follows: Let denote the orthonormal system in the space of holomorphic functions in . The following result is well-known: See Theorem 1 and (16) below for details. Formally, (2) can be seen as the special case of (4) below. Let be the sector defined as Assume that (1)is holomorphic on ;(2) is bounded near the origin;(3) for .
We consider the function: If , then (4) becomes (2). For , Cowling and Qian  introduced a class of bounded holomorphic multipliers on . In this paper, we consider the case . For this case, is unbounded on . We prove that if , then See Theorem 5.
In Section 4, we introduce a class of Fourier multipliers with . Unlike the ones of Cowling and Qian , our multipliers are unbounded on . Take . Plancherel's theorem implies that is not bounded on . Hence for such , we need to consider their boundedness on some function spaces with higher regularity. Let . We prove that if , is bounded from Sobolev space to Sobolev space , . Our result implies that the operators take a role of fractional differential operators on . See Theorem 11.
The rest of this paper is organized as follows. In Section 2, we state some basic preliminaries and notations which will be used in the sequel. In Section 3, we estimate the kernels generated by holomorphic multipliers . The Sobolev boundedness of the operators is given in Section 4.
Notations. represents that there is a constant such that whose right inequality is also written as . Similarly, one writes for .
2. Preliminaries and Notations
In this section, we state some preliminaries and notations and refer the reader to Gong , Hua , and Rudin  for further information. We use as a general element of ; that is, , , , . Denote . The notation is considered to be a row vector. Denote by the open unit ball , where . The unit sphere in is denoted by The open ball centered at with radius will be denoted by . A general element on is usually denoted by . The constant involved in the Cauchy-Szegö kernel is the surface area of and is equal to . For , we use the notation . The theory developed in this paper is relevant to the radial Dirac operator
Now we state some basis knowledge of basic functions in the space of holomorphic function in and some relevant function spaces on . We refer to Hua  for details. Let be a nonnegative integer. We consider the column vector with components The dimension of is
Let and be the Lebesgue volume element of and the Lebesgue area element of , respectively. Define It is easy to prove that and are positive definite Hermitian matrices of order . There exists a matrix such that where is a diagonal matrix and is the identity matrix. Set Denote by the components of the vectors . From (11), we can see that The following theorem is well known.
Theorem 1. The system of functions is a complete orthonormal system in the space of holomorphic functions in . The system is orthonormal, but it is not complete in the space of continuous functions on .
The explicit formula of the Cauchy-Szegö kernel on was first deduced in Hua  by using the system and the relation For , the nonisotropic distance is defined as It can be easily shown that is a metric on . For and , we define the ball corresponding to as The complement set of in is denoted by .
Set If , then where are the Fourier coefficients of : and for any positive integer , the series is uniformaly and absolutely convergent in any compact ball contained in in which is defined.
Denote by the unitary group of consisting of all unitary operators on the Hilbert space under the complex inner product . These are the linear operators that preserve inner products: Clearly, is a compact subset of . It is easy to verify that is invariant under . If , then is defined by its values on . In Section 3, we treat as identical to .
3. The Kernel Generated by Holomorphic Multipliers
The following function space is relevant.
Definition 2. Let . is defined as the set of all holomorphic functions in such that (a) is bounded for ;(b), , .
Remark 3. The classes are generalizations of which is introduced by McIntosh and his collaborators. We refer to Li et al. , McIntosh , McIntosh and Qian , Qian , and the reference therein for further information on .
Lemma 4. Let , . Then can be holomorphically extended to . Moreover, for and , where ; are the constants in Definition 2.
and is the ray , , where is chosen so that . Define
where is exponentially decaying as along . Then we get
which implies . Define
It is easy to see that is holomorphically and -periodically defined in the described region, and . Let
For , we write , where . Then . This implies that and . Therefore, (29) gives
Take the ball Applying Cauchy's integral formula, we obtain For any , we have . Then we have
Theorem 5. Let and Then is holomorphically defined for , such that , where is the function defined in Lemma 4, Moreover, for and , where , are the constant in the definition of the function space .
Proof. Recall that Then we have Therefore,
By [4, Theorem 3], we could obtain the following result.
Theorem 6. Let be a negative integer. If , then
Proof. The proof is similar to Theorem 5. we omit it.
4. Sobolev Spaces and Unbounded Fourier Multipliers
4.1. Integral Representation of Multipliers
Given , we define the Fourier multiplier operator by where are the Fourier coefficients of the test function .
For the above operator , the Plemelj type formula holds.
Theorem 7. Let . Take , where . Operator has a singular integral expression. For , where is a bounded function of and .
We can see that
which implies that . Then, we have
By integration by parts,
For any , we have
For , we have Now we consider . Let . For , write For such , . Without loss of generality, assume . We get which implies that The above estimate implies Since we obtain and then Denote Since and , we get . It is easy to see that is, . Because then we have , so Since , then that is Since , we have For , For , Then we get For , we have Then, we get Now we prove if , has a limit uniformly bounded for near . Integrating as before, we have Let . Then . We get By integration by parts, the inside integral with respect to the variable becomes We first estimate ,
Because , Since , we have So For , when , we obtain On the other hand, that implies
4.2. Sobolev Spaces on via Fourier Multipliers
Sobolev spaces on the -complex unit sphere are defined as follows. We define the fractional integral operator on as follows. Let For , the operator is defined by
For , we can see that the operators become the ordinary differential operators with higher orders.
Theorem 8. Let . on .
Proof. Without loss of generality, we assume that . Then where are the Fourier coefficients of : So
Definition 9. Let . The Sobolev norm on is defined as The Sobolev spaces on are defined as the closure of under the norm , that is, .
Remark 10. By the Plancherel theorem, if and only if
Now we consider the Sobolev boundedness of .
Theorem 11. Given and . The Fourier multiplier operator is bounded from to .
Conflict of Interests
The authors declare that they have no conflict of interests in this submitted paper.
This project is supported by NSFC Grant no. 11171203; New Teacher’s Fund for Doctor Stations, Ministry of Education, Grant no. 20114402120003; Guangdong Natural Science Foundation S2011040004131; Foundation for Distinguished Young Talents in Higher Education of Guangdong, China, LYM11063. Tao Qian is supported by MYRG116(Y1-L3)-FST13-QT; MYRG115(Y1-L4)-FST13-QT; FDCT 098/2012/A3.
- G. Gaudry, T. Qian, and S. Wang, “Boundedness of singular integral operators with holomorphic kernels on star-shaped closed Lipschitz curves,” Colloquium Mathematicum, vol. 70, no. 1, pp. 133–150, 1996.
- A. McIntosh and T. Qian, “Convolution singular integral operators on Lipschitz curves,” in Harmonic Analysis, vol. 1494 of Lecture Notes in Mathematics, pp. 142–162, Springer, Berlin, Germany, 1991.
- T. Qian, “Singular integrals with holomorphic kernels and -Fourier multipliers on star-shaped closed Lipschitz curves,” Studia Mathematica, vol. 123, no. 3, pp. 195–216, 1997.
- T. Qian, “A holomorphic extension result,” Complex Variables. Theory and Application, vol. 32, no. 1, pp. 59–77, 1997.
- M. Cowling and T. Qian, “A class of singular integrals on the -complex unit sphere,” Science in China. Series A, vol. 42, no. 12, pp. 1233–1245, 1999.
- S. Gong, Integrals of Cauchy Type on the Ball, vol. 1 of Monographs in Analysis, International Press, Cambridge, Mass, USA, 1993.
- L. K. Hua, Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains, American Mathematical Society, Providence, RI, USA, 1963.
- W. Rudin, Function theory in the unit ball of ℂn, Grundlehren der Mathematischen Wissenschaften, Springer, Berlin, Germany, 2008.
- C. Li, A. McIntosh, and S. Semmes, “Convolution singular integrals on Lipschitz surfaces,” Journal of the American Mathematical Society, vol. 5, no. 3, pp. 455–481, 1992.
- A. McIntosh, “Operators which have an functional calculus,” in Proceedings of the Mini Conference on Operator Theory and Partial Differential Equations, Center for Mathematical Analysis, vol. 14, ANU, Canberra, Australia.
- T. Qian, “Fourier analysis on starlike Lipschitz surfaces,” Journal of Functional Analysis, vol. 183, no. 2, pp. 370–412, 2001.
Copyright © 2014 Pengtao Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.