A Class of Unbounded Fourier Multipliers on the Unit Complex Ball

and Applied Analysis 3 If f ∈ A, then f (z) = ∞ ∑ k=0 Nk ∑ V=0 ckVp k V (z) , (20) where ckV are the Fourier coefficients of f: ckV = ∫ ∂Bn p V (ξ)f (ξ) dσ (ξ) , (21) and for any positive integer l, the series ∞ ∑ k=0 k l Nk ∑ V=0 ckVp k V (z) (22) is uniformaly and absolutely convergent in any compact ball contained in B(0, 1 + δ) in which f is defined. Denote by U the unitary group of C consisting of all unitary operators on the Hilbert spaceC under the complex inner product ⟨z, w⟩ = zw. These are the linear operators U that preserve inner products: ⟨Uz, Uw⟩ = ⟨z, w⟩ . (23) Clearly,U is a compact subset ofO(2n). It is easy to verify that A is invariant under U ∈ U. If f ∈ A, then f is defined by its values on ∂Bn. In Section 3, we treat ∂Bn as identical to f ∈ A. 3. The Kernel Generated by Holomorphic Multipliers Set Sω = {z ∈ Cz ̸ = 0 and 󵄨󵄨󵄨󵄨arg z 󵄨󵄨󵄨󵄨 < ω} , Sω (π) = {z ∈ Cz ̸ = 0, |Re (z)| ≤ π, 󵄨󵄨󵄨󵄨arg (±z) 󵄨󵄨󵄨󵄨 < ω} , Wω (π) = {z ∈ C | z ̸ = 0, |Re (z)| ≤ π, Im (z) > 0} ⋃Sω (π) , Hω = {z ∈ Cz = e iω , ω ∈ Wω (π)} . (24) The following function space is relevant. Definition 2. Let −1 < s < ∞. Hs(Sω) is defined as the set of all holomorphic functions in Sω such that (a) b is bounded for |z| ≤ 1; (b) |b(z)| ≤ Cμ|z| s , z ∈ Sμ, 0 < μ < ω. Remark 3. The classes Hs(Sω) are generalizations of H∞(Sω) which is introduced by McIntosh and his collaborators. We refer to Li et al. [9], McIntosh [10], McIntosh and Qian [2], Qian [11], and the reference therein for further information onH∞(Sω). Let φb (z) = ∞ ∑ k=1 b (k) z k . (25) Lemma 4. Let b ∈ Hs(Sω), −1 < s < ∞. Then φb can be holomorphically extended toHω. Moreover, for 0 < μ < μ < ω and l = 0, 1, 2, . . ., 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 (z d dz ) l φb (z) 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 ≲ Cμ󸀠 l! δ (μ, μ) |1 − z| l+1+s , z ∈ Hμ, (26) where δ(μ, μ) = min{1/2, tan(μ, μ}; Cμ󸀠 are the constants in Definition 2. Proof. Let Vω = {z ∈ C : Im (z) > 0}⋃Sω ⋃(−Sω) , Wω = Vω ⋂{z ∈ C : −π ≤ Re z ≤ π} (27) and ρθ is the ray r exp(iθ), 0 < r < ∞, where θ is chosen so that ρθ ⊆ Sω. Define Ψb (z) = 1 2π ∫ ρ(θ) exp (iξz) b (ξ) dξ, z ∈ Vω, (28) where exp(izξ) is exponentially decaying as ξ → ∞ along ρθ. Then we get 󵄨󵄨󵄨󵄨 |z| 1+s Ψb (z) 󵄨󵄨󵄨󵄨 = 󵄨󵄨󵄨󵄨󵄨󵄨󵄨 1 2π ∫ ρ(θ) exp (iξz) |z|b (ξ) dz 󵄨󵄨󵄨󵄨󵄨󵄨󵄨


Introduction
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 C to -complex unit sphere B  .We refer the reader to Gaudry et al. [1], McIntosh and Qian [2], and Qian [3,4] for further information on multipliers on Lipschitz curves.
Our motivation originates from the following example on the unit sphere in C  .The explicit formula of the Cauchy-Szegö kernel is as follows: Let { V  } denote the orthonormal system in the space of holomorphic functions in B  .The following result is wellknown: 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   = { ∈ C :  ̸ = 0,     arg      < } .
In Section 4, we introduce a class of Fourier multipliers   with  ∈   (  ),  ̸ = 0. Unlike the ones of Cowling and Qian [5], our multipliers  are unbounded on   .Take () =   .Plancherel's theorem implies that   is not bounded on  2 (B  ).Hence for such   , we need to consider their boundedness on some function spaces with higher regularity.
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.U ≈ V represents that there is a constant  > 0 such that  −1 V ≤ U ≤ V whose right inequality is also written as U ≲ V. Similarly, one writes V ≳ U for V ≥ U.

Preliminaries and Notations
In this section, we state some preliminaries and notations and refer the reader to Gong [6], Hua [7], and Rudin [8] The open ball centered at  with radius  will be denoted by (, ).A general element on B  is usually denoted by .The constant  2−1 involved in the Cauchy-Szegö kernel is the surface area of B  and is equal to 2  /Γ().For ,  ∈ C  , we use the notation   = ∑  =1     .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 B  and some relevant function spaces on B  .We refer to Hua [7] 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 C  and the Lebesgue area element of B  , respectively.Define It is easy to prove that   1 and   2 are positive definite Hermitian matrices of order   .There exists a matrix Γ such that where Λ = [  1 , . . .,    ] is a diagonal matrix and  is the identity matrix.Set .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 B  .The system {  ] } is orthonormal, but it is not complete in the space of continuous functions on B  .
The explicit formula of the Cauchy-Szegö kernel on B  was first deduced in Hua [7] by using the system {  V } and the relation For ,  ∈ B  ∪ B  , the nonisotropic distance (, ) is defined as It can be easily shown that (⋅, ⋅) is a metric on B  .For  ∈ B  and  > 0, we define the ball corresponding to (⋅, ⋅) as The complement set of (, ) in B  is denoted by   (, ).
for some  > 0} . ( where  V are the Fourier coefficients of : and for any positive integer , the series is uniformaly and absolutely convergent in any compact ball contained in (0, 1 + ) in which  is defined.Denote by U the unitary group of C  consisting of all unitary operators on the Hilbert space C  under the complex inner product ⟨, ⟩ =   .These are the linear operators  that preserve inner products: Clearly, U is a compact subset of (2).It is easy to verify that A is invariant under  ∈ U.If  ∈ A, then  is defined by its values on B  .In Section 3, we treat | B  as identical to  ∈ A.

The Kernel Generated by Holomorphic Multipliers
Set The following function space is relevant.Qian [11], and the reference therein for further information on  ∞ (  ). Let It is easy to see that   is holomorphically and 2-periodically defined in the described region, and Then where (,   ) = min{1/2, tan(  − )},    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.

Sobolev Spaces and Unbounded Fourier Multipliers
4.1.Integral Representation of Multipliers.Given  ∈   (  ), we define the Fourier multiplier operator   : A → A by where { V } are the Fourier coefficients of the test function  ∈ A.
For the above operator   , the Plemelj type formula holds.