# Tagged Questions

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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### Singular integral operator on a decay Hölder space

Denote $$E^{k,m}=\left\{f\in C^m(\mathbb{R}^3)\mid\sup_{x\in\mathbb{R}^3}(1+|x|)^{k+|\alpha|}|\partial^\alpha_xf(x)|<\infty\right\}.$$ Now given $f\in E^{2,m}$ for any fixed $m\in \mathbb{Z}^+$, ...
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### 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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### 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 ...