Questions tagged [singular-integral-equations]

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Construction of a regulariser for the boundary integral operator $\lambda\mathrm{Id} - K'$

$\newcommand\Id{\mathrm{Id}}$Assumptions and Notations : $\Omega$ is a bounded Lipschitz domain in $\mathbb R^2$, $\Gamma$ denotes its boundary and $n$ is the normal vector to the boundary $\Gamma$, ...
0 votes
0 answers
68 views

$ \int_{\mathbb{R}^n} f R_1 f=0$ if $f\in L^2$ Riesz transform

How can I see that? It seems that it has to do with the adjoint of the Riesz transform $R_1^*=-R_1$, but here we do not have the complex $L^2$ scalar product.
2 votes
0 answers
122 views

Sonin inversion formula, equivalence of two solutions of an integral equation

Let me first specify the problem I am facing, and then below explain where it arises. Given a function $f(x)$ on the interval $0<x<1$ and a real number $s\in(-1,1)$ I consider the integral ...
1 vote
0 answers
174 views

Dini continuity $\Rightarrow$ Hörmander condition

I'm trying to show that a Dini-continuous function $\Omega\in L^1(\mathbb{S}^{n-1})$ satisfies the Hörmander condition, i.e. $$\sup_{y\ne0}\int_{|x|\ge2|y|}|K_\Omega(x-y)-K_\Omega(x)|dx:=A_2<\infty,...
3 votes
0 answers
69 views

Analytic continuation of Fredholm integral equations

I am considering the equation of Fredholm type $$ f(x) = \lambda \int_\zeta \prod(y-l_i(x))^{\alpha_i} f(y) dy\label{1}\tag{1} $$ where $\zeta$ is an element of 1st homology of the complement to ...
4 votes
1 answer
260 views

Derivative of Cauchy PV is equivalent to Hadamard regularization?

Let $\mathcal C$ and $\mathcal H$ denote the Cauchy principal value and Hadamard finite part. According to the Wiki: $$ {\frac {\mathrm d}{\mathrm dx}}\left({\mathcal {C}}\int _{{a}}^{{b}}{\frac {...
4 votes
2 answers
263 views

Using $\delta$-method to "estimate" undefined moments of a random variable?

I posted this over on MSE without much luck. Not sure if posting here is considered cross-posting but I can remove it if it is. Let $X\sim\mathcal N(\sqrt 2,1/x^2)$. The expected value $\mathsf EX^{-1}...
2 votes
0 answers
58 views

How to prove this this integral equality which contain nonlocal operator, $(-\partial_{xx})^{1/2}$?

Suppose that $\theta(t,x)$ is even about $x$ and is smooth. $0\le \gamma<1/2$, $0<\delta<1-2\gamma$. $\Lambda=(-\partial_{xx})^{1/2}$ My Question: How to prove that $$ \int_0^{\infty} \frac{(\...
1 vote
1 answer
166 views

Unique solution for 2$\times$2 Fredholm integral equations system

Consider the following system of Fredholm integral equations with constant kernel matrix $$ f(x)=K(x)\int_{0}^{1}f(s)ds $$ where $K(x)\in C([0,1];M_{2\times 2}(% %TCIMACRO{\U{211d} }% %BeginExpansion \...