Some properties of a T operator with B M kernel in the complex Clifford analysis

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R E S E A R C H

Open Access

Some properties of a T operator with B-M

kernel in the complex Clifford analysis

Zunfeng Li

1*

_{, Yuying Qiao}

2

_{and Nanbin Cao}

3

*_{Correspondence:}

[email protected]

1_{College of Science, Hebei}

University of Science and Technology, Shijiazhuang, P.R. China Full list of author information is available at the end of the article

Abstract

Teodorescu operator, or T-operator, plays an important role in Vekua equation systems and the generalized analytic function theory. It is a generalized solution to the nonhomogeneous Dirac equation. The properties of T operators play a key role in the study of boundary value problems and integral representation of the solutions. In this paper, we first define a Teodorescu operator with B-M kernel in the complex Clifford analysis and prove the boundedness of this operator. Then we give an inequality similar to the classical Hile lemma about real vector which plays a key role in the following proof. Finally, we prove the Hölder continuity and

γ

-integrability of this operator.

Keywords: Complex Cliﬀord analysis; Teodorescu operator; Boundedness; Hölder continuity;

γ

-integrability

1 Introduction

In some way, there are two branches of Clifford analysis. The first one is the real Clifford analysis introduced by Brack, Delanghe, and Sommen in [1] which studied function the-ory with values in a real Clifford algebra defined on a nonempty subset of the Euclidean spaceRn+1. Many important theoretic results, such as the Cauchy integral formula, the Cauchy theorem, the Taylor and the Laurent series expansion, the Liouville theorem, and the Morera theorem, have been obtained, and they are the extensions of the well-known classical theorems in one complex variable. Beyond these, a lot of scholars have studied many properties of function theory in the real Clifford analysis. Eriksson and Leutwiler [2–5] introduced the hypermonogenic function and studied some properties of it. Huang [6], Qiao [7–9], Xie [10–12], and Yang [13–15] obtained many results in Clifford analysis. The second one is the complex Clifford analysis. In the early 1990s, Ryan [16–19] in-troduced the definition of the complex regular function and obtained the Cauchy integral formula whose method is similar to the classical function with one complex variable. In recent years, Ku, Du [20,21] obtained some properties of complex regular functions using the isotonic function.

Based on the above theoretical study and practical background, we construct an ana-logue of Bochner–Martinelli kernel in several complex variables. We first define a Teodor-escu operator with B-M kernel in the complex Clifford analysis and prove the boundedness of this operator. Then we give an inequality similar to the classical Hile lemma about real

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vector which plays a key role in the following proof. Finally, we prove the Hölder continuity andγ-integrability of this operator.

2 Preliminaries

Let Cl0,n(C) be a complex Cliﬀord algebra over n+1-dimensional Euclidean space Cn+1.

Cl0,n(C) has the basis e0,e1,e2, . . . ,en;e1e2,e1e3, . . . ,e1en;e2e3, . . . ,e2en; . . . ;en–1en; . . . ;

e1· · ·en. Hence, an arbitrary element of the basis may be written aseA=eα1· · ·eαh, where A={α1, . . . ,αh} ⊆ {1, . . . ,n}, 1≤α1<α2<· · ·<αh≤nand whenA=∅,eA=e0= 1. So, the

complex Cliﬀord algebra is composed of elements having the typea=_AzAeA, wherezA

are complex numbers.

The basis in Cliﬀord algebra satisﬁes

e2_i = –1, i= 1, 2, . . . ,n, eiej= –ejei, 1≤i<j≤n, (i=j).

Deﬁne the norm of Cliﬀord numbers as follows:

A

zAeA

=(a,a) =

A |zA|2

1 2

.

Let ⊂Cn+1 _{be an open connected nonempty set. Then the function which is} de-ﬁned onand valued in Cl0,n(C) can be expressed asf(z) =

AfA(z)eA, wherefA(z) are

complex-valued functions. Let

F(r)=

ff :→Cl0,n(C),f(z) =

A

fA(z)eA,fA(z)∈Cr() .

Dirac operators are introduced as follows [6]:

Dlf=

n

i=0 ei

∂f

∂zi

; Dl_f ₌_e

0

∂f

∂z0–

n

i=1 ei

∂f

∂zi

;

Drf =

n

i=0

∂f

∂zi

ei; Drf =

∂f

∂z0 e0–

n

i=1

∂f

∂zi

ei.

Deﬁnition 2.1([16]) If⊂Cn+1_,_f_:_→_Cl

0,n(C) satisﬁes: (1) fA(z)is a holomorphic function for anyzj∈,

(2) Dlf(z) = 0,∀z∈,

thenf(z) is called a complex left regular function on.

Deﬁnition 2.2([16]) If⊂Cn+1,f:→Cl0,n(C) satisﬁes: (1) fA(z)is a holomorphic function for anyzj∈,

(2) Drf(z) = 0,∀z∈,

thenf(z) is called a complex right regular function on.

Lemma 2.1(Hadamard lemma [22]) Let⊂Rn+1_{be a bounded domain}_,_n_≥_2._If_α_,_β satisfy0 <α,β<n+ 1,andα+β>n+ 1,then for any x1,x2∈Rn+1_,_x1₌_x2_,_{we have}

|t–x1|–α|t–x2|–βdt≤J1|x1–x2|(n+1)–α–β,

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Lemma 2.2([22]) Let⊂Rn+1_{be a bounded domain}_,_when_α_<_n_{+ 1,}_{for any y}_∈_Rn+1_, we have

|x–y|–αdx≤M,

where M is a positive constant only related toαand the size of.

Lemma 2.3(Hölder inequality [23]) If fk∈Lpk(),k= 1, 2, . . . ,n,and

1

p=

1 p1

+ 1 p2

+· · ·+ 1 pn

≤1,

then f1f2· · ·fn∈Lp(),and

Lp(f1f2· · ·fn)≤Lp1(f1)Lp2(f2)· · ·Lpn(fn), p≥1.

Lemma 2.4(Minkowski inequality [23]) If f1,f2, . . . ,fn∈Lp(),then f1+f2+· · ·+fn∈Lp(),

and

Lp(f1+f2+· · ·+fn)≤Lp(f1) +Lp(f2) +· · ·+Lp(fn),p≥1.

Lemma 2.5([23]) Let Lp₍_{, Cl}

0,n(R))represent the set of all p order integrable functions

which are defined on the bounded domain⊂Rn+1_,_{and with values in the real Clifford} algebraCl0,n(R),define the norm ofϕas follows:

ϕ,p=

ϕ(x)pdVx

1

p

, p≥1,

when1≤r≤p,

Lp, Cl0,n(R)

⊂Lr, Cl0,n(R)

is true.

The notations used in this paper are as follows:

(1) ω2n+2represents the surface area of unit sphere in a2n+ 2-dimensional real

Euclidean space.

(2) Mi{i= 1, 2, 3},Ki{i= 1, . . . , 16}are constants only related tonand the size of domainin this paper.

(3) dVξ =dζ0∧dζ1∧ · · · ∧dζn∧dη0∧dη1∧ · · · ∧dηn,ζj∈R,ηj∈R, (j= 0, 1, . . . ,n),

ξj=ζj+iηj.

(4) dξ¯∧dξ=dξ¯0∧dξ¯1∧ · · ·dξ¯n∧dξ0∧dξ1· · · ∧dξn. (5) dξ¯∧dξ= (2i)n+1_dVξ_.

3 Some properties of a T operator with B-M kernel in the complex Clifford analysis

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Deﬁnition 3.1 Let⊂Cn+1_{be a bounded domain,}_ϕ_∈_Lp₍_{, Cl}

0,n(C)),z∈Cn+1, then

(Tϕ)(z)

= 1

ω2n+2(2i)n+1

ϕ(ξ)

n

k=0(ξk–zk)¯ek |ξ–z|2n+2 +

n

k=0(ξk–zk)e¯k |ξ–z|2n+2

dξ¯∧dξ

is called T operator with B-M kernel.

Theorem 3.1 Let⊂Cn+1_{be a bounded domain}_,_ϕ_∈_Lp₍_{, Cl}

0,n(C)),p>n+ 1,then T is

bounded on Lp₍_),_and

Tϕ,p≤M1ϕ,p. (1)

Proof Chooseq> 1 such that _p1+1_q = 1, whenp> 2(n+ 1), we have 1 <q< 2(₂n_n+1)₊₁, using Hölder’s inequality, we have

Tϕ(z)

= 1 2n+1_ω

2n+2 ϕ(ξ)

n

k=0(ξk–zk)¯ek |ξ–z|2n+2

dξ¯∧dξ

≤ K1

ω2n+2

ϕ(ξ)

n

k=0(ξk–zk)e¯k |ξ–z|2n+2

dVξ+

ϕ(ξ)

n

dVξ

≤K2

ϕ(ξ) 1

|ξ–z|2n+1dVξ+

ϕ(ξ) 1

|ξ–z|2n+1dVξ

≤K3

ϕ(ξ) 1

|ξ–z|2n+1dVξ

≤K4ϕ,p

|ξ–z|–(2n+1)qdVξ 1

q

.

Because 1 <q<2(₂n_n+1)₊₁, we have (2n+ 1)q< 2(n+ 1). Using Lemma2.2, for∀z∈, we have

|ξ–z|–(2n+1)qdVξ≤K5.

So we have

Tϕ(z)≤K4K5ϕ,p.

Hence,

Tϕ(z)pdVz

1

p

≤K4K5

ϕp_,_pdVz

1

p

.

LetM1=K4K5(dVz) 1

p_{, we have}

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Theorem 3.2 Let z=z0e0+z1e1+z2e2+· · ·+znen,ξ =ξ0e0+ξ1e1+ξ2e2+· · ·+ξnen∈

Cl0,n(C),z= 0,ξ= 0,and|z| =|ξ|,n(≥2),m(≥0)be integers,then for any i, 0≤i≤n,we

have zi

|z|m+2 –

ξi |ξ|m+2

≤|z–ξ|[Pm(z,ξ) +|z|

m

2|_ξ|m2_]

|z|m+1_|_ξ_|m+1 , (2)

where

Pm(z,ξ) = m

k=0

|z|m–k|ξ|k=|z|

m+1_–_|_ξ_|m+1

|z|–|ξ| .

Proof Suppose|z| ≤ |ξ|and insert the termzi|z|m+2_{in the following formula, then we have}

zi |z|m+2 –

ξi |ξ|m+2

=zi|ξ|

m+2_–_ξ_i|_z_|m+2

|z|m+2|_ξ|m+2

=zi|ξ|

m+2_–_z_i|_z_|m+2₊_z_i|_z_|m+2_–_ξ_i|_z_|m+2

|z|m+2_|_ξ_|m+2

≤|zi|||ξ|m+2–|z|m+2|+|zi–ξi||z|m+2 |z|m+2|_ξ|m+2

≤|z|||ξ|–|z||(|ξ|m+1+|ξ|m_||z|+· · ·+|z|m+1) +|z–ξ||z||ξ||z|m

z|m+2|_ξ|m+2

≤|z|||ξ|–|z||(|ξ|m+1+|ξ|m_||z|+· · ·+|ξ||z|m) +|z–ξ||z||ξ||z|m

z|m+2_|_ξ_|m+2

≤|z–ξ|[(|ξ|m+|ξ_||m–1|z|+· · ·+|z|m) +|z|m]

z|m+1|_ξ|m+1

≤|z–ξ|[Pm(z,ξ) +|z|

m

2|_ξ|m2_] |z|m+1_|_ξ_|m+1

.

When|ξ| ≤ |z|, insertξi|ξ|m+2_{in the above formula, we can prove the above inequality}

in a similar way.

Remark1 Because the original Hile lemma cannot be used directly in the complex Clif-ford analysis, we give the conclusion of Theorem3.2which is similar to the classical Hile lemma and plays an important role in proving the properties of T-operators and Cauchy operators. We insert the appropriate items according to the situation and prove that in-equality (2) holds. Inequality (2) is similar to the Hile lemma of the classical real vector and is complete symmetry with respect to the variableξ,z. It is a good tool to prove the Hölder continuity of the T operator with B-M kernel in the complex Cliﬀord analysis.

Theorem 3.3 Let⊂Cn+1be a bounded domain,ϕ∈Lp(),p> 2(n+ 1),then for any z1,z2∈,we have

(Tϕ)(z1) – (Tϕ)(z2)≤M2ϕ,p|z1–z2|α, (3)

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Proof Case 1. When|z1–z2| ≥1, using Theorem3.2we have

Tϕ(z1) –Tϕ(z2)≤2M1ϕ,p

≤2M1ϕ,p

1

|z1–z2|α

≤M2ϕ,p|z1–z2|α.

Case 2. When|z1–z2|< 1, we have

Tϕ(z1) –Tϕ(z2)

≤ 1

ω2n+22n+1

ϕ(ξ)

n

k=0(ξk–z1k)e¯k |ξ–z1|2n+2 –

n

k=0(ξk–z2k)e¯k |ξ–z2|2n+2

|dξ¯∧dξ|

+ 1

ω2n+22n+1

ϕ(ξ)

n

k=0(ξk–z1k)¯ek |ξ–z1|2n+2

–

n

k=0(ξk–z2k)e¯k |ξ–z2|2n+2

|dξ¯∧dξ|

≤ 1

ω2n+2

n

k=0

ϕ(ξ) (ξk–z1k)

|ξ–z1|2n+2 –

(ξk–z2k) |ξ–z2|2n+2

dVξ

+ 1

ω2n+2

n

k=0

|ξ–z1|2n+2 –

(ξk–z2k) |ξ–z2|2n+2

dVξ

= 2

ω2n+2

n

k=0

|ξ–z1|2n+2 –

(ξk–z2k) |ξ–z2|2n+2

dVξ. Let I=

|ξ–z1|2n+2 –

(ξk–z2k) |ξ–z2|2n+2

dVξ.

According to Theorem3.2, we can get

I≤

ϕ(ξ)|z1–z2|[P2n(ξ–z1,ξ–z2) +|ξ–z1|

n_|_ξ_–_z

2|n]

|ξ–z1|2n+1|_ξ_–_z2|2n+1

dVξ

=|z1–z2|

ϕ(ξ) P2n(ξ–z1,ξ–z2)

|ξ–z1|2n+1|_ξ_–_z2|2n+1 dVξ

+|z1–z2|

ϕ(ξ) |ξ–z1|

n_|_ξ_–_z2_|n |ξ–z1|2n+1|ξ–z2|2n+1

dVξ

=I1+I2.

ForI1, we have

I1=|z1–z2|

2n

k=0

|ξ–z1|–(k+1)|ξ–z2)|–(2n+1–k)|ϕ(ξ)|dVξ

=|z1–z2| 2n

k=0

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Using Hölder’s inequality we have

I1≤ |z1–z2|ϕ,p

2n

k=0

|ξ–z1|–(k+1)q|ξ–z2|–(2n+1–k)qdVξ 1

q

.

Becausep> 2n+ 2, 1

p+

1

q= 1, we can get

1 <q<2n+ 2 2n+ 1.

So

2n+ 1 < (2n+ 1)q< 2n+ 2,

0≤k≤2n, we get

(k+ 1)q≤2n+ 2,

(2n+ 1 –k)q≤2n+ 2,

and

(k+ 1)q+ (2n+ 1 –k)q> 2n+ 2.

By Hadamard’s lemma, we have

|ξ–z1|–(k+1)q|ξ–z2)|–(2n+1–k)qdVξ

≤K6|z1–z2|(2n+2)–(2n+1–k)q–(k+1)q

=K6|z1–z2|(2n+2)–(2n+2)q.

So we have

I1≤(2n+ 1)K6ϕ,p|z1–z2|1+

2n+2

q –(2n+2)

≤(2n+ 1)K6ϕ,p|z1–z2|1–

2n+2

p _.

As toI2, using Hölder’s inequality we have

I2 =|z1–z2|

ϕ(ξ) |ξ–z1|

n_|_ξ_–_z

2|n

|ξ–z1|2n+1|_ξ_–_z2|2n+1dVξ

≤ |z1–z2|ϕ,p

|ξ–z1|–(n+1)q|ξ–z2|–(n+1)qdVξ 1

q

.

Sincep> 2n+ 2 and1_p+1_q= 1, we get

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So

(n+ 1)q≤2n+ 2,

(2n+ 2)q≥2n+ 2.

From Hadamard’s lemma, we get

|ξ–z1|–(n+1)q|ξ–z2|–(n+1)qdVξ

≤K7|z1–z2|(2n+2)–(2n+2)q.

So

I2≤K7ϕ,p|z1–z2|1+

2n+2

q –(2n+2)

≤K7ϕ,p|z1–z2|1–

2n+2

p _.

Hence

I =I1+I2

≤(2n+ 1)(K6+K7)ϕ,p|z1–z2|1–

2n+2

p

=K8ϕ,p|z1–z2|1–

2n+2

p _.

Using Hölder’s inequality, we obtain

Tϕ(z1) –Tϕ(z2)≤ 2

ω2n+2

K8ϕ,p|z1–z2|1– 2n+2

p

≤K9ϕ,p|z1–z2|1–

2n+2

p

≤M2ϕ,p|z1–z2|α.

Remark2 In Case 2 of this theorem, we use the inequality of Theorem3.3, Hölder’s in-equality, and Hadamard’s lemma. This result enriches the theoretical system of the com-plex Cliﬀord analysis.

Theorem 3.4 Let⊂Cn+1 _{be a bounded domain}_, _ϕ_∈_Lp₍_{), 1 <}_p_{< 2}_n_{+ 2,}_γ _{is an}

arbitrary constant which satisﬁes1 <γ < ₍₂(2_nn_+2)–+2)p_p,then Tϕ isγ-integrable on,that is, Tϕ∈Lγ₍_),_{and the following inequality}

Tϕ,γ≤M3ϕ,p (4)

is true.

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Case 1. Whenp<γ<₍₂(2_nn_+2)–+2)p_p, 0 < p_γ < 1, thus 0 <p(1_p–_γ1) = 1 –p_γ < 1, again

p

γ +p

1 p– 1 γ = 1.

Chooseq> 0 such that1_p+1_q = 1, then we have

(2n+ 2) b 2– 1 γ

+ (2n+ 2) b 2– 1 q

= (2n+ 2)

b– 1

γ –

1 q

= (2n+ 2) 1 γ – 1 p+ 1 2n+ 2–

1

γ –

1 q

= (2n+ 2)

–1 + 1 2n+ 2

= –(2n+ 1).

Therefore,

Tϕ(z)

= 1 2n+1_ω

2n+2 ϕ(ξ)

n

dξ¯∧dξ

≤ K10

ω2n+2

ϕ(ξ) 1

|ξ–z|2n+1dVξ+

ϕ(ξ) 1

|ξ–z|2n+1dVξ

= 2K10

ω2n+2

ϕ(ξ) 1

|ξ–z|2n+1dVξ

= 2K10

ω2n+2

ϕ(ξ)

p

γ|_ξ_–_z|(2n+2)(b2–γ1)_ϕ₍_ξ₎p(

1

p–γ1)|_ξ_–_z|(2n+2)(b2–1q)_dVξ_.

Because 1 <p<γ, 1_γ + (1_p–1_γ) +1_q= 1, using Hölder’s inequality we get

Tϕ(z)

≤ 2K10

ω2n+2

ϕ(ξ)p|ξ–z|(2n+2)(γ2b–1)_dVξ

1

γ

ϕ(ξ)pdVξ 1

p–γ1

·

|ξ–z|(2n+2)(qb2–1)_dVξ

1

q

= 2K10

ω2n+2

ϕ(ξ)p|ξ–z|(2n+2)(γ2b–1)_dVξ

1

γ

ϕ1–

p

γ

,p

·

|ξ–z|(2n+2)(qb2–1)_dVξ

1

q

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Becauseb> 0, we have

(2n+ 2)

1 –γb 2

< 2n+ 2,

(2n+ 2)

1 –qb 2

< 2n+ 2.

From Lemma2.2we can know that two integrals are meaningful, we assume thatK11=

sup_ξ_∈|ξ–z|(2n+2)(qb2–1)_dVξ_.

Therefore we have

Tϕ(z)γ ≤

2

ω2n+2 γ

K

γ

q 11ϕ

γ–p

,p

ϕ(ξ)p|ξ–z|(2n+2)(γ2b–1)_dVξ

.

LetK12=supξ∈

|ξ–z|

(2n+2)(γ₂b–1)_dV

z, so we have

Tϕ(z)γ ≤K12ϕ_,γ–_ppϕp_,_p=K13ϕγ_,_p,

whereK13= (_ω2

2n+2)

γ_K

γ

q 11K12. Hence, we get

Tϕ,γ=

Tϕ(z)γdVξ 1

γ

≤K₁₃γϕ,p=K14ϕ,p,

whereK14=K γ 13.

(2) Whenp≥γ > 1, choosemsuch that 0 <₍₂(2_nn_+2)+γ+2)γ <m<γ, andmis an arbitrary pos-itive constant satisfyingm<γ <₍₂(2_nn₊₂₎₊+2)m_m. Becauseϕ∈Lp₍_),_m_<_p_{, we have}_ϕ_∈_Lm₍_).

Choose 1_p+1_q=_m1. Therefore, from the proof process of (1) and Lemma2.5, we get

Tϕ,p ≤K15ϕ,m

=K15

ϕ(ξ)·1m 1

m

≤K15

ϕ(ξ)·1p 1

p

ϕ(ξ)·1q 1

q

≤K15V

1

q ϕ,p ≤K16ϕ,p.

ThereforeTϕisγ integrable on. If we chooseM3=max{K14,K16}, then

Tϕ,γ≤M3ϕ,p.

Funding

(11)

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

ZFL has presented the main purpose of the article. All authors read and approved the ﬁnal manuscript.

Author details

1_{College of Science, Hebei University of Science and Technology, Shijiazhuang, P.R. China.}2_{College of Mathematics and}

Information Science, Hebei Normal University, Shijiazhuang, P.R. China.3_{Hebei GEO University, Shijiazhuang, P.R. China.}

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional aﬃliations.

Received: 6 July 2018 Accepted: 15 August 2018

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