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Questions tagged [singular-integrals]

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.

A singular integral is an integral operator of the form

\begin{equation*} Tf=\int K(x,y)f(y)dy \end{equation*}

defined for "nice" (ie, Schwartz) functions and with kernel $K$ possessing a singularity at the origin. They can be of convolution type (the kernel must be locally integrable on $\mathbb{R}^n$ without the origin) or non-convolution type.

The most basic examples are the Hilbert transform, and its generalisation, the Riesz transform.

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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 $L^p$, for any $1\leq p<\infty$. On $L^2$ it is easy to…
michek
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What exactly do delta method estimates of moments for $1/\bar X_n$, $\bar X_n\sim\mathcal N(\mu,\sigma^2/n)$ approximate? (not as simple as you think)

Let me start with the excerpt out of Casella & Berger's Statistical Inference (2nd edition, pg. 470) that inspired this question. Definition 10.1.7 For an estimator $T_n$, if $\lim_{n\to\infty}k_n\mathrm{Var}T_n=\tau^2<\infty$, where $\{k_n\}$ is a…
Aaron Hendrickson
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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 $k\in\mathcal{C}^1(\Bbb R^n\setminus\{0\})$ such…
Joe
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Trying to compute limit of singular integrals : $L= \lim_{s\to 1}(1-s)\int_{\Omega}\frac{(u(x)-u(y))}{|x-y|^{d+2s}} d y.$

Let $\Omega\subset \Bbb R^d$ be a bounded $C^1$ domain. Let $u:\Bbb R^d\to \Bbb R$ be a function in $C^2_b(\Bbb R^d)$. I would like to compute the following limit: for $x\in \partial \Omega$ $$L= \lim_{s\to…
Guy Fsone
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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 \frac{1}{s-z} \int_{-2}^{2} e^{-nt^2}…
Antonio Vargas
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When is this integral a polynomial?

Consider the functions or integrals $f : \mathbb{R} \to \mathbb{R}$ of the form, $$ f(x) = \int_{a}^{b}K(|x-y|)u(y)dy $$ where $u$ is known and $K$ is a weakly singular kernel (or a continuous kernel for simplicity). How do we show that this…
Sam
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Trouble with "very simple" lemma from Kenig.

In Carlos Kenig's 1986 paper Elliptic Boundary Value Problems on Lipschitz Domains, he uses (but does not prove) the following result which he says is "very simple". Fix $n\geq 2$. Let $\lambda$ be the function satisfying $\lambda(0)=0$ and…
Milk
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An inequality equivalent to Hörmander's condition $\sup_{y\in\mathbb R^n}\int_{\{x: |x|>2|y|\}}|K(x-y)-K(x)|\,dx<\infty$

Let $K\in L_{\text{loc}}^1(\mathbb R^n\setminus\{0\})$. Prove that $$\sup_{y\in\mathbb R^n}\int_{\{x: |x|>2|y|\}}|K(x-y)-K(x)|\,dx<\infty\label{1}\tag{1}$$ if and only if $$\sup_{r>0}\frac1{r^n}\int_{B(0,r)}\int_{\{x:…
Feng
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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 between singular integrals, maximal functions, and…
Rob F.
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An estimate for singular integral.

This is an exercise in my course of harmonic analysis: assume that $f:\mathbb{R}^n\rightarrow \mathbb{R}$ for $n\geq 3$ satisfies $$\lim_{r\rightarrow 0}\sup_{x\in \mathbb{R}^n} \int_{|x-y|
Xin Fu
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Is the Cauchy principal value "invariant" under change of variables?

Let $f \in C^{\gamma}_c(\mathbb{R}) $. Let $K:\mathbb{R}^n \backslash \{\vec{0}\} \rightarrow \mathbb{R}^n$ be a singular integral kernel with the following properties: 1) K smooth everywhere except at $\vec{0}$ 2) K homogeneous of degree $-n$, in…
user78383
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What is a hypersingular integral kernel?

While reading literature about boundary element and finite element methods I have repeatedly seen that some integral kernels are singular and others are hypersingular. Could you explain what is the difference between singular and hypersingular and…
Džuris
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A question from Stein's book, Singular Integral.

A question from Stein's book, Singular Integral. Let $\left\{ f_{m}\right\} $ be a sequence of integrable function such that $$\int_{% \mathbb{R}^{d}}\left\vert f_{m}\left( y\right) \right\vert dy=1$$ and its support converge to the…
beginner
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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 $$|\nabla^{j}\psi_{t}(\xi)|\lesssim_{d}…
Matt Rosenzweig
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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 also Hölder continuous of order $\alpha$, but that…
Matt Rosenzweig
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