Spherical 
                $$\Pi $$
                
                    
                                    
                        Π
                    
                
            -Type Operators in Clifford Analysis and Applications

Cheng, Wanqing; Ryan, John; Kähler, Uwe

doi:10.1007/s11785-017-0641-0

Cited by 1 publication

(7 citation statements)

References 10 publications

(6 reference statements)

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“…In the rest of this section, we will study the spectrum of the operators D RP n 1 and T; this helps us to show that the Π-operator defined in the next section also has an L 2 isometry property. Similar argument can be found in Cheng et al 7 Let H m denote the space of l n -valued harmonic polynomials with homogeneity of degree m on S n−1 . It is well known that L 2 (S n−1 ) = ∑ ∞ m=0 H 2m , see Axler et al 19 Now we consider a function f (x) defined on an open domain V ⊆ S n−1 , and it also satisfies that −x ∈ V for each x ∈ V and f (x) = f (− x).…”

Section: Dirac Operators On Real Projective Spacementioning

confidence: 56%

“…More details for these conformally flat manifolds can be found in Kraußhar and Ryan. 9,10 In the present paper, we will generalize the results in Euclidean space 4 and on the unit sphere 7 to the previous conformally flat manifolds through proper projection maps.…”

Section: Introductionmentioning

confidence: 88%

“…Recall that the generalized spherical Dirac operator D s and its conjugate on the n ‐dimensional unit sphere

𝕊^{n - 1}

are defined as follows:

D_{s} = x ((), {normalΓ}_{0} - \frac{n}{2}), \overset{D_{s}}{true} = \overset{x}{true} ((), \overset{{normalΓ}_{0}}{true} - \frac{n}{2})

, where

{normalΓ}_{0} = \sum_{j = 1}^{n} e_{0} e_{j} L_{0, j} - \sum_{i = 1, j > i}^{n} e_{i} e_{j} L_{i, j}

, and here the operators

L_{i, j} = x_{i} \partial_{x_{j}} - x_{j} \partial_{x_{i}}

are called the angular momentum operators. It is well known that the fundamental solution of Γ 0 is

G_{s} false(x, y false) = \frac{\overset{x - y}{true}}{false‖ x - y {false‖}^{n}}

, and the fundamental solution of

\overset{{normalΓ}_{0}}{true}

\overset{G_{s} false(x, y false)}{true} = \frac{x - y}{false‖ x - y {false‖}^{n}}

x, y \in 𝕊^{n - 1}

, see Cheng et al 7 for details.…”

Section: Preliminariesmentioning

confidence: 99%

“…Assume that Ω is a domain on

𝕊^{n - 1}

and

f \in C^{1} false(normalΩ, scriptC l_{n} false)

. One can define Cauchy transforms with respect to D s and

\overset{D_{s}}{true}

as below 7

T_{Ω} f (x) = \frac{1}{ω_{n}} \int_{Ω} G_{s} (x, y) f (y) d y = \int_{Ω} \frac{\overset{true‾}{x - y}}{‖ x - y ‖^{n}} f (y) d y, {\overline{T}}_{Ω} f (x) = \frac{1}{ω_{n}} \int_{Ω} \overset{true‾}{G_{s} (x, y)} f (y) d y = \int_{Ω} \frac{x - y}{‖ x - y ‖^{n}} f (y) d y .

Here, T Ω (

{\overset{T}{true}}_{normalΩ}

) is also a left and right inverse for D s (

\overset{D_{s}}{true}

) when considering functions with compact support, see Theorem 2 below.…”

Section: Preliminariesmentioning

confidence: 99%

“…In their study, 6 Blaya et al studied the Π‐operator in Clifford analysis by using two orthogonal bases of a Euclidean space, which allows to find the expression of the jump of the generalized Π‐operator across the boundary of the domain. The case of the Π‐operator and the Beltrami equation on the unit sphere has also been discussed in Cheng et al 7 with most useful properties inherited from the complex Π‐operator. The classical Ahlfors–Beurling inequality has also been generalized to higher dimensions by Martin in his study 8 …”

Section: Introductionmentioning

confidence: 99%

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The Π‐operator on some conformally flat manifolds and hyperbolic half space

Cheng

Ryan

2021

Math Methods in App Sciences

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The Π-operator, also known as Ahlfors-Beurling transform, plays an important role in solving the existence of locally quasiconformal solutions of Beltrami equations.In this paper, we first construct the Π-operator on a general Clifford-Hilbert module. This Π-operator is also an 2 isometry. Further, it can also be used for solving certain Beltrami equations when the Hilbert space is the 2 space of a measure space. Then, we show that this technique can be applied to construct the classical Π-operator in the complex plane and some other examples on some conformally flat manifolds, which are constructed by ∕Γ, where is a simply connected subdomain of either ℝ +1 or , and Γ is a Kleinian group acting discontinuously on . The Π-operators on those manifolds also preserve the isometry property in certain 2 spaces, and their norms are bounded by the norms of the Π-operators on ℝ +1 or , depending on where lies. The applications of the Π-operator to solutions of the Beltrami equations on those conformally flat manifolds are also discussed. At the end, we investigate the Π-operator theory in the upper-half space with the hyperbolic metric.

show abstract

Section: Dirac Operators On Real Projective Spacementioning

confidence: 56%

Section: Introductionmentioning

confidence: 88%

“…Recall that the generalized spherical Dirac operator D s and its conjugate on the n ‐dimensional unit sphere

𝕊^{n - 1}

are defined as follows:

D_{s} = x ((), {normalΓ}_{0} - \frac{n}{2}), \overset{D_{s}}{true} = \overset{x}{true} ((), \overset{{normalΓ}_{0}}{true} - \frac{n}{2})

, where

{normalΓ}_{0} = \sum_{j = 1}^{n} e_{0} e_{j} L_{0, j} - \sum_{i = 1, j > i}^{n} e_{i} e_{j} L_{i, j}

, and here the operators

L_{i, j} = x_{i} \partial_{x_{j}} - x_{j} \partial_{x_{i}}

are called the angular momentum operators. It is well known that the fundamental solution of Γ 0 is

G_{s} false(x, y false) = \frac{\overset{x - y}{true}}{false‖ x - y {false‖}^{n}}

, and the fundamental solution of

\overset{{normalΓ}_{0}}{true}

\overset{G_{s} false(x, y false)}{true} = \frac{x - y}{false‖ x - y {false‖}^{n}}

x, y \in 𝕊^{n - 1}

, see Cheng et al 7 for details.…”

Section: Preliminariesmentioning

confidence: 99%

“…Assume that Ω is a domain on

𝕊^{n - 1}

and

f \in C^{1} false(normalΩ, scriptC l_{n} false)

. One can define Cauchy transforms with respect to D s and

\overset{D_{s}}{true}

as below 7

T_{Ω} f (x) = \frac{1}{ω_{n}} \int_{Ω} G_{s} (x, y) f (y) d y = \int_{Ω} \frac{\overset{true‾}{x - y}}{‖ x - y ‖^{n}} f (y) d y, {\overline{T}}_{Ω} f (x) = \frac{1}{ω_{n}} \int_{Ω} \overset{true‾}{G_{s} (x, y)} f (y) d y = \int_{Ω} \frac{x - y}{‖ x - y ‖^{n}} f (y) d y .

Here, T Ω (

{\overset{T}{true}}_{normalΩ}

) is also a left and right inverse for D s (

\overset{D_{s}}{true}

) when considering functions with compact support, see Theorem 2 below.…”

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

The Π‐operator on some conformally flat manifolds and hyperbolic half space

Cheng

Ryan

2021

Math Methods in App Sciences

Self Cite

View full text Add to dashboard Cite

show abstract

Spherical $$\Pi $$ Π -Type Operators in Clifford Analysis and Applications

Cited by 1 publication

References 10 publications

The Π‐operator on some conformally flat manifolds and hyperbolic half space

The Π‐operator on some conformally flat manifolds and hyperbolic half space

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