Mathematics

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…
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 that…
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…
C. A. Nastasi
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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 $$\lim_{r\rightarrow 0}\sup_{x\in \mathbb{R}^n} \int_{|x-y|<r} \frac{|f(y)|}{|x-y|^{n-2}}\,dy=0.$$ Prove that there…
Xin Fu
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Constructing $f$ so that $f(z) = f(z + 1)$, but only in a fixed region

Consider the two disjoint semicircles $A = \{-e^{i \phi} | -\pi/2 \leq \phi \leq \pi / 2\}$ and $B = \{1 + e^{i \phi} | -\pi/2 \leq \phi \leq \pi / 2\}$ in the complex plane: Let $f$ be defined and analytic on $\mathbb{C} \setminus A \setminus B$ .…
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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 origin:…
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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