Singular integrals are integral operators with kernel $K:\mathbb{R}^n\times\mathbb{R}^n\to\mathbb{R}$ that is not defined on the diagonal $x=y.$ This tag is for questions related to singular integrals and their applications.

learn more… | top users | synonyms

0
votes
1answer
36 views

A general question about Cauchy operators

I want to familiarize myself more with the Cauchy operators. As soon as I say "operator" I have to specify on which space, Okay, that should be my first question: On which spaces the Cauchy operators ...
1
vote
1answer
30 views

Leibniz integral rule (singular)

Definte $I(\epsilon):=\int_{ \epsilon}^1\frac{\,\mathrm{d}x}{\sqrt{x-\epsilon}}$ for $\epsilon<0$ Want to show that ...
0
votes
1answer
23 views

The Riesz transform kernel satisfies Hörmander's condition

Define the kernel $$K_i(x) = \frac{x_i}{\lvert x\rvert^{d+1}}$$ as a function $\mathbb{R}^d \setminus \{ 0 \} \to \mathbb{C}$ for $d \geq 3$ (say). I want to prove that this kernel satisfies ...
1
vote
0answers
26 views

Analytic or numeric integration of Singular integral based on Bessel K0 and K1

thank you for reading this ! I need the following integral to be integrated from -1 to 1. It appears to have a singularity in -2/3. In string, Generated: ...
2
votes
1answer
85 views

Numeric integration of Greens Function over singularity

I'm currently using python to numerically evaluate the follow expression at various values of $r$ and $\theta$. \begin{equation*} f(r,\theta) = \int_{-\pi}^{\pi}\int_{0}^{R}\frac{\exp(ikS)}{2 \pi ...
1
vote
1answer
31 views

Riesz Projection as a Cauchy type integral

Let \begin{equation*} f(\zeta)=\sum_{k\in\mathbb{Z}}\widehat{f}(k)\zeta^k \end{equation*} be a complex-valued function on unit circle $\mathbb{T}=\{ \zeta\in\mathbb{C}:|\zeta|=1\},$ where ...
1
vote
0answers
28 views

“Transference” Argument

In the proof of the Iwaniec-Martin theorem (giving a bound in $L^p$ for the Riesz transform, $\|R_j\|_p=\cot(\frac{\pi}{2p^*})$ the proof of this equality is given by proving the inequalities $\leq$ ...
1
vote
0answers
60 views

Intuition Behind the Riesz Transform

Define the Riesz transform in singular integral form \begin{equation*} R_jf(x)=\lim_{\epsilon\to 0}\pi^{\frac{-(n+1)}{2}}\Gamma(\frac{n+1}{2})\int_{|y|>\epsilon}\frac{y_jf(x-y)}{|y|^{n+1}}dy. ...
1
vote
1answer
25 views

Is the integrand of this integral singular?

In a book I'm reading, integrals like the following appear and the author says the integrand is not singular (or perhaps, integrable) around the origin. $$ ...
10
votes
1answer
268 views

Showing a sequence of integrals converges to zero

Let $\varphi$ be a complex-valued function which is analytic on $\{z \in \mathbb C : |z| \leq 2\}$, let $\gamma$ be the unit circle in the complex plane, and define $$ F_n(z) = \int_\gamma ...
1
vote
1answer
94 views

integration concerning Fourier transform of homogeneous kernel(of degree 0)

Let $m\in C^\infty(\mathbb{R}^d-\{0\})$ be homogeneous of degree zero and has mean zero on the sphere$(S^{d-1})$. Then $m$ defines a tempered distribution and $\partial_j^dm\in S'(\mathbb{R}^d)$ is ...
10
votes
0answers
366 views

norm of a singular integral operator

My question is from Harmonic Analysis, about the study of singular kernels (in the Calderon Zygmund sense.) Suppose that a kernel $K$ is a singular kernel, extending to a bounded operator on $L^p$, ...
1
vote
0answers
35 views

Asymptotic behavior of oscillatory Hilbert transform

Does anyone know what is the leading term in the asymptotics of $$ P.V. \int\limits_{ -\infty }^{ +\infty } \frac{e^{i \lambda x^3 } f( x ) dx }{ x }, $$ as $ \lambda \to +\infty $? Assume $ f \in ...
1
vote
0answers
76 views

Numerically integrate an improper integral involving product of two bessel functions and a singularity

I am working on an elasticity problem which requires solving an integration with a rather complex kernel involving production of two Bessel's functions of the first kind (zeroth order) and a ...
1
vote
0answers
45 views

Is this function square-integrable? Able to be Fourier expanded?

I want to do a 3-dimensional Fourier series expansion on this function$$\frac{\cos (x) \cos (y) \cos (z)-\sin (x) \sin (y) \sin (z)}{\left[(a+\sin (y)+\cos (z))^2+(b+\cos (x)+\sin (z))^2+(c+\sin ...
0
votes
2answers
60 views

exact or numerical value of an improper integral

i am dealing with an improper integral which has been arised in my research. i will be greatful if you have any idea about the numeric value of this integral. $$ \int_{0}^{\frac{1}{4}} \frac{u^{4} ...
0
votes
0answers
51 views

How to make sense of the Fourier transform of this distribution

I want to compute the Fourier transform of this distribution: $$D(f)=\int_{\mathbb{R}} f(t,t^2) \frac{dt}{t}$$ ($f$ a Schwartz function on $\mathbb{R}^2$, the integral interpreted with a Cauchy ...
1
vote
1answer
53 views

Is the function $\frac{1}{\sqrt{|x_1|}}$integrable on the unit sphere $S^{n-1}\subset\mathbb{R}^n$?

Is the function $\frac{1}{\sqrt{|x_1|}}$integrable on the unit sphere $S^{n-1}\subset\mathbb{R}^n$? That is, is the integral $$\int_{S^{n-1}}\frac{1}{\sqrt{|x_1|}}d\sigma(x)$$ finite? Where $\sigma$ ...
2
votes
1answer
128 views

Integral involving convolution with Poisson kernel.

Suppose that $f \in L^2(\mathbb{R}^n)$ and let $P_y(x)$ ($x \in \mathbb{R}^n$, $y > 0$) be the dilation of the Poisson kernel: $$P_y(x) = \frac{C_n y}{(y^2 + |x|^2)^\frac{n+1}{2}},$$ where $C_n$ ...
2
votes
0answers
117 views

Fourier Transform of inverse powers of the absolute value

I don't think this question has been asked previously, so here goes. I need to evaluate the following integrals - $$ ...
2
votes
0answers
52 views

Singular Solutions of this Equation?

How would I find the singular solutions of this equation: y = $ce^{x^2}$ + $ce^{\sin x}$ (where $c$ is a constant). It should be $x^2$ if anyone gets confused by the first part of the equation. ...
4
votes
1answer
131 views

Calderón-Zygmund operators with positive kernel

Let $T$ be a Calderón-Zygmund operator. That is, $T$ maps $L^2(\mathbb{R}^d)$ to itself and satisfies the representation formula $$ Tf(x) = \int_{\mathbb{R}^d}K(x,y)f(y)\, dy $$ for all $f \in L^2$ ...
3
votes
2answers
142 views

What is a simple form of this integral?

This integral reminds me of something familiar but I cannot remember the rule to make it simple. $$\int_{-\infty}^{+\infty} \frac{\exp(i a \cdot v)}v \mathrm d v$$ where $a$ is a scalar for ...
0
votes
1answer
96 views

A singular integral along the arc and an interval

Is there a way to solve such and integral: $$\int_L\frac{\sin(\frac{1}{z})}{z(1-z)} \mathrm dz$$ $L$ can be: an arc of a semicircle with unit radius, centered at the origin and running clockwise ...
6
votes
2answers
211 views

Why are square functions important in analysis?

I have been reading through chapter 1 of E.M. Stein's textbook Harmonic Analysis: Real Variable Methods, Orthogonality, and Oscillatory Integrals. In chapter 1, Stein discusses the relationship ...
3
votes
0answers
176 views

Showing a Hölder continuous function acted on by a singular integral operator is Hölder continuous

Consider the following function defined by a singular integral \begin{equation} F(x)= \lim_{\epsilon \rightarrow 0} \int_{|x-y| \geq \epsilon} \partial_k \partial_j k_i(x-y) \left(Y_k(x)- Y_k(y) ...
4
votes
2answers
197 views

Fourier transform of $|x|^{-t}$

In $\mathbb{R}^d$, let $f(x)=|x|^{-t}$, its Fourier tranform $F(f)(ξ):=(2\pi)^{-\frac{d}{2}}∫_{\mathbb{R}^d} e^{ix\cdot ξ}f(x)dx$, is there any fast way to see that this integral converge at $ξ \neq0$ ...
4
votes
1answer
270 views

Radial limits of harmonic conjugate and Hilbert transform

Let $\mu$ be a real measure on the circle $\mathbf{T}$. Then the function $$f(z)=\int_\mathbf{T} \mathrm{Im}\left(\frac{\zeta+z}{\zeta-z}\right) d\mu(\zeta)$$ is harmonic on the unit disc and its ...