| This thesis focuses on the κ-Cauchy-Fueter operator,κ-regular function on Hn and Penrose transformation, Radon-Penrose transformation on C4n, Radon transform over isotropic d-planes in R2n. We mainly discuss the expression of all solutions to the κ-Cauchy-Fueter equations, that is, the expression of all k-regular functions on Hn and the partial differential equations that the range of this Radon transform satisfies.In Chapter1, we give a comprehensive survey of the backgrounds and mod-ern developments of κ-Cauchy-Fueter operator, Penrose transformation, Radon-Penrose transformation and Radon transform. And then we introduce some concepts referring to this thesis and the main results of the subject.In Chapter2, we mainly discuss the κ-Cauchy-Fueter operator and Radon-Pcnrose transformation. We find an explicit Radon-Penrose type integral formula to realize the1-1correspondence:the first Dolbeault cohomology group of (?)-closed (0,1)-forms with coefficients in the (-κ-2)th power of the hyperplane section bundle H-k-2over the projective space P1and the solutions to the k-Cauchy-Fueter equations on the quaternionic space Hn.Chapter3is devoted to the study of κ-regular functions and Penrose trans-formation. A generalized Penrose integral formula gives the solutions to the holomorphic κ-Cauchy-Fueter equations on C4n, and conversely, all holomorphic solutions to thses equations are given by this formula. By restriction to the quaternionic space Hn∈C4n, we find all κ-regular functions. The integral for-mula also gives the series expansion of a κ-regular function by homogeneous κ-regular polynomials. In particular, the result holds for left regular functions, which are exactly1-regular.In the last chaper, we consider the Radon transform of rapidly decreasing functions on R2n over isotropic d-planes, d<n. For d=1, and d=2, we prove that the range of this Radon transform satisfies a system of partial differential equations of the second order with nonconstant coefficients, respectively. |
PDF Full Text Request