Applications and main properties of hyperfunctions

I am trying to get familiar with hyperfunctions, and I do have some familiarity with the classical theory of distributions. I am wondering whether hyperfunctions have any advantages over distributions. Are there any applications of the former which cannot be obtained using the latter? Any important examples? I was told one general property of hyperfunctions … Read more

Distribution boundary value of analytic function and wave front sets

Assume f(z) is analytic in the tube domain Rn⊕iC, where C⊂Rn is a convex cone. Under the assumption |f(x+iy)|≤1/|y|k, we know by a Theorem of Martineau (see also Hormander, volume 1, Theorem 3.1.15) that the limit limy→0,y∈Cf(x+iy) exists as a tempered distribution f(x) on Rn, uniformly in proper cones y∈C′⊂C. The convergence is in the … Read more

Approximating compactly supported L2L^2 functions with Schwartz functions “from within”?

It is well known that the class of Schwartz functions S in dense in all Lp spaces therefore for each f∈L2 there exists a sequence of Schwartz functions (fk) such that ‖f−fk‖L2→0 as k→∞. If we suppose further that f has compact support can we find a sequence of Schwartz functions (fk) such that ‖f−fk‖L2→0 … Read more

Recognizing Schwartz regular distributions

Are there characterizations of Schwartz regular distributions other than being locally integrable (which does not lend itself to easy manipulations)? To be more detailed: if I want to show that some distribution is in fact a smooth function, I can try to use the concepts of singular support or wavefront, together with hypoellipticity. What can … Read more

Is the distribution $f\mapsto \int_{S} \frac{\partial^i }{\partial \nu^i}f\,\mathrm{dvol}$ in a Bessel potential space?

In order to finish a paper on ‘metric space magnitude’ I need to prove that a certain distribution on $\mathbb{R}^{2p+1}$ is in Mark Meckes’ weighting space (see Magnitude, Diversity, Capacities, and Dimensions of Metric Spaces). My question requires no knowledge of that background, however: for what I want to do, it suffices to show that … Read more

Literature on the product of two distributions satisfying the Hörmander condition

I am currently studying some basic questions concerning the product $uv\in \mathscr{D}'(\mathbb R^n)$ of two Schwartz distributions $u,v\in \mathscr{D}'(\mathbb R^n)$ satisfying the Hörmander wavefront set condition $(\mathrm{WF}(u)+\mathrm{WF}(v))\cap \mathbb R^n\times\{0\}=\emptyset$. I noticed that unfortunately most of the literature on products of distributions does not deal with this situation but rather with the (problematic) issue of defining … Read more

Extension of pseudodifferential operators

I’m very sorry if this is the wrong place to ask this question, but I’ve asked it on StackExchange and received no answers. ( https://math.stackexchange.com/questions/813063/convergence-to-a-schwartz-distribution ) Let M be a smooth manifold with countable atlas, and define the distributions D′(M) as the dual space to the smooth densities with compact support, and E′(M) as the … Read more

normal form of currents?

(this question did not get any answers on math.SE, so I am reposting it here) Let M be an n-dimensional manifold. Then the space of currents Dk(M) of degree k on M is the space of continuous linear functionals on the space of test n−k-forms. Two typical examples of currents are F(ω)=∫Γω where Γ is … Read more

How to show that this limit converges in the distributional sense to a dirac delta function

Let $$\begin{eqnarray}\nonumber f(y, t) &=& \frac{C}{\sigma ^2 t} \left[\frac{(1-\alpha) (b-y)}{\alpha t^{\alpha}} \, _1F_1\left[\frac{\alpha+1}{2 \alpha};\frac{3}{2};-\frac{ (b-y)^2}{2 \sigma^2 t^{2 \alpha}}\right]- \sqrt{2} \sigma \frac{\Gamma \left(\frac{3}{2}-\frac{1}{2 \alpha}\right)}{\Gamma \left(1-\frac{1}{2 \alpha}\right)} \, _1F_1\left[\frac{1}{2 \alpha};\frac{1}{2};-\frac{ (b-y)^2}{2 \sigma^2 t^{2 \alpha}}\right]\right] \end{eqnarray}$$ be defined in $f(y, t) \in (-\infty, b]$, where $0 < \alpha < 1$, $\sigma > 0$ and $t$ represents time and … Read more