Cauchy transform and the d-bar equation

[5/60%]

Show

LaTeX

Show LaTeX

Problem 19.13 from Бородин П.А., Савчук А.М., Шейпак И.А. "Задачи по функциональному анализу" [БСШ17]More problems in Section 19.2 "Fundamental solutions"

Find the fundamental solution for the Cauchy-Riemann operator $\frac{\partial }{\partial \bar{z}} = \frac{1}{2}\left(\frac{\partial }{\partial x}+i \frac{\partial }{\partial y}\right)$ .

Problem III.8.1 from Conway J.B. "A Course in Functional Analysis" [Con07]More problems in Section III.8 "An Application: Runge's Theorem"

Let $\mu$ be a compactly supported measure on $\mathbb {C}$ that is boundedly absolutely continuous with respect to area measure. Show that $\hat{\mu }$ is continuous on $\mathbb {C}_\infty$ .

Problem III.8.2 from Conway J.B. "A Course in Functional Analysis" [Con07]More problems in Section III.8 "An Application: Runge's Theorem"

Let $m =$ Lebesgue measure on $[0,1]$ . Show that $\hat{m}$ is not continuous at any point of $[0,1]$ .

Problem 5.3.28 from Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. "Problems and Solutions in Mathematics" [CJP+11]More problems in Section 5.3 "Complex Integration"

(a)(b)

(a)

Let $c$ be the unit circle in the complex plane, and let $f$ be a continuous $\mathbb {C}$ -valued function on $c$ . Show that

F_f(z) = \frac{1}{2\pi i} \int _c \frac{f(\zeta )}{\zeta - z}\, d\zeta

is a holomorphic function of $z$ in the interior of the unit disk.

(b)

Find a continuous $f$ on $c$ which is not identically zero, but so that the associated function $F_f$ is identically zero.

Problem VIII.4.11 from Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. "Избранные задачи по вещественному анализу" [МГЛ+04]More problems in Section VIII.4 "The Stieltjes Integral"

(a)(b)

Let $C$ be the Cantor set, $\sigma$ the Cantor function. Define the function $\varphi$ by the equality

\varphi (z)=\int _{0}^{1} \int _{0}^{1} \frac{d \sigma (\xi ) d \sigma (\eta )}{z-(\xi +i \eta )} \quad (z \in \mathbb {C})

Prove that

(a)

$\varphi \in C(\mathbb {C}), z \varphi (z) \rightarrow 1$ as $z \rightarrow \infty$ ;

(b)

the function $\varphi$ is holomorphic outside the set $C \times C$ (compare with Problem 3.46).

References

[БСШ17] Бородин П.А., Савчук А.М., Шейпак И.А. Задачи по функциональному анализу. МЦНМО, 2017.

[Con07] Conway J.B. A Course in Functional Analysis. 2nd ed., Springer, 2007.

[CJP+11] Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. Problems and Solutions in Mathematics. 2nd ed., World Scientific, 2011.

[МГЛ+04] Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. Избранные задачи по вещественному анализу. 2-е изд., Невский Диалект; БХВ-Петербург, 2004.