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.

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An inequality of first order partial derivatives.

Suppose $f:\mathbb R^2\to \mathbb C$ is $C^2$ with compact support. Show that $$\left\|\frac{\partial f}{\partial x_1}\right\|_p+\left\|\frac{\partial f}{\partial x_2}\right\|_p\le ...
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Singular integral operator

i got the following problem to solve. Let $0 < \alpha < 1$, $L \in L_\infty([0,1]^2)$, $D = \{(x,y) \in \mathbb{R}^2: x = y\}$ the diaagonal of $\mathbb{R}^2$ and $k:[0,1]^2 \setminus D \to ...
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Prove that $|Hf(x)|\le C|x|^{-\alpha}$ for $-1<x<0$ where $ Hf(x):=\frac1{\pi}\operatorname{p.v.}\int_{\Bbb R}\frac1sf(x-s)\,ds $

Let $f(x):=|x|^{-\alpha}\chi_{]-1,0[}(x)$, for some $\alpha\in]0,1[$; let's prove that $|Hf(x)|\le C|x|^{-\alpha}$ for $-1<x<0$, where the Hilbert transform of a measurable function $f:\Bbb ...
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Prove that $|k(x)|\le C|x|^{-n}$ under suitable hypothesis on $k\in\mathcal{C}^1(\Bbb R^n\setminus\{0\})$

DON'T BE AFRAID FROM THE +500 BOUNTY: it doesn't matter that I KNOW this problem is really hard, I put it only because I need to solve the problem really URGENTLY! Let $n\ge2$; given a kernel ...
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23 views

A Hilbert transform that takes several functions

While playing with some PDE I came across a singular integral that looks something like ...
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1answer
107 views

Riesz transform does not preserve continuity

I've read somewhere that the Riesz operator $R_j$ defined by $$R_j f(t) := c(n) \, \text{pv} \int_{\mathbb{R}^n} \frac{x_j}{|x|^{n+1}} f(t-x) \, dx$$doesn't preserve the continuity, but I can't ...
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48 views

Expansion of some singular kernel with the help of Bessel and Neumann spherical harmonic functions

With the following notations: $j_n$: spherical Bessel functions, $y_n$: spherical Neumann function, $P_n$: Legendre polynomial, $r$, $\rho$, $\theta$, $\lambda$ arbitrary complex, ...
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33 views

What happens to Chebyshev polynomials integration when n=1

The integration of Chebyshev polynomials of the first kind has the following value, $$\int T_{n}(x) \, dx = \frac{1}{2} \, \left( \frac{T_{n+1}(x)}{n+1} - \frac{T_{n-1}(x)}{n-1} \right)$$ what happens ...
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Commutator Characterization of $BMO(\mathbb{R})$

Let $a:\mathbb{R}\rightarrow\mathbb{C}$ be a locally integrable function, and let $H$ denote the Hilbert transform. Suppose that the commutator operator $[a,H]$ defined by $[a,H]f:=aH(f)-H(af)$ is ...
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34 views

Boundedness of singular integral operators on $L^{p}$ spaces

Let $\Omega \in L^{1}(S^{d-1})$ have mean zero. Prove that, if the operator $T_{\Omega}: L^{p} \rightarrow L^{q}$ given by $T_{\Omega}f(x) $:= p.v. $\int_{\mathbb{R}^{d}} \frac{\Omega ...
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34 views

Show that $K(x,y)=(2^{nk}\mathcal{F}^{-1}(2^kx))_{k\in\mathbb{Z}}$ is a singular kernel

I want to show that for a bump function $\psi$ with support in the annulus $\{\frac{1}{2}\leq\vert x\vert\leq2\}$ the kernel $K(x,y)=(2^{nk}\mathcal{F}^{-1}\psi(2^k(x-y)))_{k\in\mathbb{Z}}$ is a ...
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33 views

Fourier Transform of a kernel

Let $\omega \in \mathcal{S}(\mathbb{R}^{2})$ and define $u(x) = \int_{\mathbb{R}^{2}}\frac{(x-y)^{\perp}}{|x - y|^{2}}\omega(y)dy$, where $(x-y)^{\perp} = \begin{bmatrix}x_{2} - y_{2}\\-x_{1} + ...
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Continuous Littlewood-Paley Inequality

I am trying to prove Q13 from Terence Tao's Fourier Analysis notes (number 4): For every $t>0$, let $\psi_{t}:\mathbb{R}^{d}\rightarrow\mathbb{C}$ be a function obeying the estimates ...
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60 views

Is a bounded function always the Hilbert transform of some other function?

Given $f \in L^\infty(\mathbb R)$, does there always exist a $g$ (in some space) such that \begin{equation*} Hg=f, \end{equation*} where $Hg$ is the Hilbert transform of $g$ ? In other words, is the ...
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Example of a compactly supported Lipschitz function with non-Lipschitz Hilbert transform

Suppose $f$ is a compactly supported measurable function (say in the interval $[-1,1]$) which is Hölder continuous of order $\alpha\in (0,1)$. I have read that the Hilbert transform $Hf$ of $f$ is ...
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45 views

Calderon-Zygmund Operator Associated to Zero Kernel

We say that a Calderon-Zygmund operator (CZO) $T:L^{2}(\mathbb{R}^{n})\rightarrow L^{2}(\mathbb{R}^{n})$ (i.e. a bounded linear operator) is associated to a CZ kernel ...
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67 views

Are singular integral operators bounded on $L\log L$?

My question is regarding singular integrals of Calderon Zygmund type. It is known that the maximal function is bounded on $L\log L \mapsto L^1$ (but not on $L^1$) and satisfies the same operator ...
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The $p=\infty$ case for an $L^2$ convolution operator on $\mathbb{R}^n$

Let $T$ be a convolution operator on $L^2(\mathbb{R}^n)$, suppose $K$ is a tempered distribution in $\mathbb{R}^n$ that coincides with a locally integrable function on $\mathbb{R}^n\setminus \{0\}$. ...
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When the inequality $ |x|^{-k} * |f(x)|^2 \le C |f(x)|^2 $ holds?

When can I expect that the inequality $$ |x|^{-k} * |f(x)|^2 \le C |f(x)|^2 $$ for some positive constant $C$? I would like to know the range of $k>0$. Here $*$ means the convolution on $\mathbb ...
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53 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 ...
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57 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 ...
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66 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 ...
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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: ...
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52 views

Asymptotic value of a Cauchy Singular integral

Let, $\zeta(x,t) = A_0sin(k_0x)cos(\omega t) + \frac{2k_0A_0}{\pi} \{\int_{0}^{\infty}\frac{cos(kx)cos(\beta t)-cos(k_0x)cos(\omega t)}{k^2-k_0^2}dk\}$ Here $\beta ^2 = gktanh(kh)\ and\ \omega^2 = ...
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80 views

Evaluation of a cauchy Singular Integral

While evaluating the details of this paper by Taylor et al. I'm stuck at the following integral. In need to evaluate the following: Regard $x$ as spatial x-coordinate & $t$ as time. $I(x) = ...
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201 views

On the Hilbert Transform of a Bounded Function

Let $f: \mathbb R \rightarrow \mathbb R$ be a bounded function that is smooth and also in $L^1(\mathbb R) \cap L^2(\mathbb R)$. I want to prove that the Cauchy transform of this function $Kf$ is in ...
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88 views

$L^p$ boundedness of Riesz potential.

Why studying, I repeatedly see people use the following result. That is there exists $C > 0$ such that $$\|\nabla \Delta^{-1}\nabla \times u\|_p \le C \|u\|_p$$ for every $u \in ...
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163 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 ...
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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 ...
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“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 $\leq$ ...
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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. ...
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32 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. $$ ...
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276 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 ...
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What are the sequels to Rudin's Functional Analysis?

Briefly speaking my purpose, I'm looking for the sequels to Rudin's Functional Analysis. How about the following books by Stein? Are there any other nice ones? Harmonic Analysis: Real-Variable ...
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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 ...
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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 $L^p$, ...
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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 ...
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119 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 ...
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Cauchy singular integral operator

Help on proving the following equality: $$K(-\operatorname{sgn})=S$$ where $K$ is the operator defined by $K(f)=\mathcal{F}^{−1}f\mathcal{F}$ $\mathcal{F}$=Fourier transform, $f$=any function, sgn ...
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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 ...
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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} ...
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Question about the principal value of some integral

So here is my problem, is it possible that $$\int_{[0,1]}f(y)\cot(\pi(x-y))dy= p.v \int_{[0,1]}f(y)\cot(\pi(x-y))dy$$ I see that the left integral is singular for $x=0$ but since I never worked with ...
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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 ...
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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 $\sigma$ ...
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199 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 $C_n$ ...
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211 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 - $$ ...
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152 views

Does Hilbert Transform commute with Function Multiplication modulo Compact on $L^p(R)$?

Define Hilbert Transform (HT) as the convolution with the function $1/x$. E. Stein proves in his book Singular Integrals and Differentiability Properties of Functions that HT, when understood as a ...
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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. ...
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333 views

(Obvious?) Half-Space Poisson Kernel Estimate

In Stein's Singular Integrals and Differentiability Properties of Functions, Theorem 1 (a) of Chapter 8 (pg. 197) he makes the claim without proof that the Poisson Kernel for the half-space ...
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172 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 $f \in L^2$ ...