Understanding Vladimir Maz'ya's Problem 72 what is $|dy|$?

Question

In the article "Seventy Five (Thousand) Unsolved Problems in Analysis and Partial Differential Equations" by Vladimir Maz’ya

Problem 72 on page 36 is given as follows:

Let $C$ be the unit circle let $u(x)$ be a function on the unit circle. Define the operator $A$ as follows

$$ A(u)(x) = \int_{C} \frac{u(x)-u(y)}{|x-y|} |dy| $$

The spectrum of A is described in [79, Section 12.2.2]. Then the principal cauchy problem:

$$ \frac{d}{dt} + A, t> 0 $$

Which is reminiscent of the modified zeta function

$$ f(z) = \sum_{k=1}^{\infty} e^{-z \sum_{n=1}^{k} \frac{1}{n}} $$

Admits a meromorphic extension to the entire complex plane. Study the properties of this extension.

So I have a couple questions here.

1. What is $|dy|$? The only way I can try to make sense of this is that the sign of the integral should cancel out the sign of $dy$ for every point on the unit circle. I.E. this is the same as:

$$ A(u)(x) = \int_{y \in C} \frac{u(x)-u(y)}{y|x-y|} $$

But if that was true the author would've just gone ahead and said that. So I think I don't understand what $|dy|$ is supposed to be.

2. The author makes a mention that this is "reminiscent of the modified zeta function" but not equal to. I assume this means the modified zeta function's meromorphic continuation is known? Is there a reference for this and or an explicit construction?

The Reference [79] from the article is:

Maz’ya, V., Nazarov, S., Plamenevskij, B.: Asymptotic Theory of Elliptic Baundary Value Problems in Singularly Perturbed Domains, vol. I. Birkh¨auser, Basel (2000)

And I cannot get a hold of this article at this time.

Some More Notes:

Using approach0 I found this question where again a line integral on the unit circle is occuring and the symbol $|dz|$ indicates the line element here. So perhaps we need not know what $|dy|$ means at all to make sense of this (just consider it as a line integral over $C$). To be honest I'm still dissatisfied since usually with line elements $dl$ it is usually possible to expand $dl = a(x)dy + b(y)dy$. In this particular case I'm still not sure how to crack open $|dz| = ?dx + ?dy$

Answer 1

$|dy|$ is the surface measure on $C$. In coordinates $y(t) = (a(t), b(t))$ parameterizing $C$, it is given by $|dy| = |y'(t)|\,dt = \sqrt{a'(t)^2 + b'(t)^2}\,dt$. So $$\int_{C}f(y)|dy| = \int f(y(t))|y'(t)|\,dt.$$ For example, $\pi$ is defined as half the length of $C$: $$\pi := \frac{1}{2}\int_{C}1|dy|.$$