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A linear operator $P\colon C_c^\infty(X)\to \mathcal D'(X)$ with kernel $K\in\mathcal D'(X\times X)$ is local if and only if $\operatorname{supp}K\subseteq \Delta_X$, where $\Delta_X$ is the diagonal in $X\times X$. If $P$ happens to be a pseudodifferential operator, then its kernel $K$ is conormal with respect to $\Delta_X$. Having both properties the same ...

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The proof is not very complicated, but even a sketch needs more space than a comment. So here is a sketch. I want to prove that $$\int_{-a}^{a} dx \ e^{i y x} \ f(x) = \hat{f}(y)$$ with $f$ and $\hat{f}$ defined as in the question. First one observes that it suffices to prove the equality for $a=1$. Then because of $f(-x)=f(x)$ and the symmetric ...

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Jim Wright pointed out to me that Stein's spherical maximal function fails to be weak-type at the endpoint $L^{\frac{d}{d-1}}(\mathbb{R}^d)$ for $d \geq 3$. This can be seen in Stein's counterexample, provided in his paper http://www.ncbi.nlm.nih.gov/pmc/articles/PMC430482/pdf/pnas00037-0013.pdf. The example is $f(x) := |x|^{1-d} \cdot [\log (1/|x|)]^{-1}$ ...

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