The Haseman Boundary Value Problem with Slowly Oscillating Coefficients and Shifts
The Haseman Boundary Value Problem with Slowly Oscillating Coefficients and Shifts
复制标题
DOI：
10.1007/978-3-319-49182-0_19
复制
发表时间：
2017
期刊：
影响因子：
--
通讯作者：
Y. Karlovich
中科院分区：
文献类型：
--
作者：
Y. Karlovich
本文致力于研究勒贝格空间中由对数螺线组成的星形卡尔森曲线 Г 上的哈斯曼边值问题 Φ+∘α=GΦ−+，其中 Φ± 是未知解析函数 Φ 在 Г、Gandgare 给定函数上的角边界值，α 是 Г 自身的保向同胚。这个问题被简化为勒贝格空间Lp(Γ)上具有shiftT=Vα++GP−的等效奇异积分算子，其中算子P±= 2−1(I±SΓ)与柯西奇异积分算子SΓ相关，并且移位算子Vα由Vαf=f∘α给出。应用具有有限平滑度的非正则符号的Mellin伪微分算子理论，本质上降低了平移α的平滑度，假设平移导数α’和系数G是在Γ上缓慢振荡的函数，建立了算子T的Fredholm判据和指数公式。
The paper is devoted to studying the Haseman boundary value problem Φ+∘α=GΦ−+gon a star-like Carleson curve Γ composed by logarithmic spirals in the setting of Lebesgue spaces, where Φ±are angular boundary values of an unknown analytic function Φ on Γ,Gandgare given functions, andαis an orientation-preserving homeomorphism of Γ onto itself. This problem is reduced to the equivalent singular integral operator with a shiftT=Vα++GP−on a Lebesgue spaceLp(Γ), where the operatorsP±= 2−1(I±SΓ) are related to the Cauchy singular integral operatorSΓ, and the shift operatorVαis given byVαf=f∘α. Applying the theory of Mellin pseudodifferential operators with non-regular symbols of limited smoothness and essentially decreasing the smoothness of the shiftα, we establish a Fredholm criterion and an index formula for the operatorTprovided that the shift derivativeα’ and the coefficientGare slowly oscillating functions on Γ.