# Questions tagged [schwartz-distributions]

A distribution is a continuous linear functional on the space $\mathcal{C}^{\infty}_c$ of smooth (indefinitely differentiable) functions with compact support. Though they appeared in formal computations in the physics and engineering literature in the late $19^{th}$ century, their formal setting ...

**1**

vote

**1**answer

125 views

### Convolution with Schwartz class function

Let $f, g\in \mathcal{S}(\mathbb R)$ (Schwartz class function), $\delta_0$ (dirac delta distribution).
Consider distribution as follows:
$$H(x, y)= f(x)g(x)\delta_0(y)-f(y)g(y)\delta_0(x), \ (x, y\...

**0**

votes

**0**answers

52 views

### is this explicit linear operator hypo-elliptic

Consider an operator of the form
$$L(\phi):=\Delta \phi + \gamma \phi_{rr}$$ here the $r$ denotes derivative with respect to the radial variable (we are in $ R^N$ say where $N \ge 3$).
I am ...

**4**

votes

**1**answer

77 views

### Is convolution jointly continuous on $\mathcal{E}'$?

Let $\mathcal{E}'(\mathbb{R})$ be equipped with its usual strong topology (being the dual space of $\mathcal{E}(\mathbb{R})$). Is convolution jointly continuous on $\mathcal{E}'(\mathbb{R})$?

**9**

votes

**2**answers

207 views

### Trace on $\mathcal{S}(\mathbb{R}^k) \mathbin{\hat{\otimes}_\pi} \mathcal{S}'(\mathbb{R}^k)$

I asked this question on Math StackExchange, but it did not receive an answer, despite my offering a bounty to attract attention. I am unsure whether it is appropriate for this venue, but I thought ...

**0**

votes

**1**answer

50 views

### Covergent net in $\mathcal{E}'(\mathbb{R})$ implies bounded?

Let $\mathcal{E}(\mathbb{R})$ be the space of all $C^\infty$ functions on $\mathbb{R}$ with its usual topology, and $\mathcal{E}'(\mathbb{R})$ be the dual space with the weak* topology.
Let $(T_i)_{...

**1**

vote

**0**answers

45 views

### Dense versus sequentially dense in $\mathcal{E}’$

Endow the dual space $\mathcal{E}’$ of smooth functions $C^\infty$ (with its metrizable topology described by uniform convergence on compacts for convergent sequences) with the weak* topology. Let $D$ ...

**8**

votes

**2**answers

188 views

### On the Fourier-Laplace transform of compactly supported distributions

Let $\mathcal{E}'(\mathbb{R})$ be the space of all compactly supported distributions on $\mathbb{R}$.
For $f\in \mathcal{E}'(\mathbb{R})$, let $\widehat{f}$ denote the entire extension of the ...

**1**

vote

**2**answers

153 views

### A question arising in the distribution theory of L. Schwartz

Let $R$ be the ring of distributions $T\in \mathcal{D}'(\mathbb{R})$ with support in $[0,\infty)$ and with the operations of pointwise addition and multiplication taken as convolution, and $I$ be the ...

**2**

votes

**0**answers

119 views

### Theory of distributions on various domains

The prototypical example of a distribution is the Dirac delta function, defined as a linear functional taking a well behaved test function $\phi:\mathbb{R} \to \mathbb{R}$ and returning its value at ...

**1**

vote

**1**answer

130 views

### Schwartz distributions, Colombeau algebra and applications

I have studied "enough" the theory of distributions , I would like to deepen some topic with applications. With some research I arrived at this book:
"Geometric Theory of Generalized Functions with ...

**2**

votes

**1**answer

213 views

### About the Fourier transform of the logarithm function

I want to calculate / simplify:
$$\mathcal{F} (\ln(|x|)\mathcal{F(f)}(x))=\mathcal{F} (\ln(|x|)) \star f$$
where $\mathcal{F}$ is the Fourier transform ($\mathcal[f](\xi)=\int_{\mathbb R}f(x)e^{ix\...

**5**

votes

**1**answer

126 views

### The division problem for tempered functions

It is well known (see for example S Łojasiewicz, Sur le problème de la division, Studia Math. 8 (1959), 87–136.) that any linear partial differential operator with constant coefficients is surjective ...

**-1**

votes

**2**answers

290 views

### Were there attempts to express derivatives of Delta function as polynomials of Delta function?

Is seems to me that it makes sense to presume some relations between derivatives of Dirac delta functions and its powers. I wonder, whether someone proposed a similar theory?
Particularly, it could ...

**5**

votes

**3**answers

724 views

### The “Spaces of Schwartz distributions are finite dimensional” challenge

The more I study Schwartz distributions and the corresponding spaces, the more the latter look "finite dimensional" to me. Of course they are not finite dimensional in the technical sense but they are ...

**16**

votes

**3**answers

1k views

### Research topics in distribution theory

The theory of distributions is very interesting, and I have noticed that it has many applications especially with regard to PDEs. But what are the research topics in this theory? also in terms of ...

**1**

vote

**0**answers

51 views

### Support of functions on Minkowski space and their Fourier transform

Suppose that a tempered distribution $f(x)$ on 4-dimensional Minkowski space with signature $+---$ vanishes for $x^2<0$ and its Fourier transform $\hat f(p)$ vanishes for $p^2<0$.
Are then $f$ ...

**3**

votes

**1**answer

269 views

### A Schwartz kernel type theorem for Sobolev spaces

The well-known Schwartz kernel theorem states that a continuous operator from smooth test functions to distributions, $T \colon C^\infty_c(\mathbb{R}^n) \to C^\infty_c(\mathbb{R}^n)'$ is continuous ...

**4**

votes

**1**answer

98 views

### Neumann DBAR problem with tempered distributions

It is well-known that the operator $$\frac{\partial}{\partial \overline{z}} : C^{\infty}(\mathbb{C}) \to C^{\infty}(\mathbb{C})$$ is surjective. (And it also works if we replace functions by Schwartz ...

**4**

votes

**2**answers

436 views

### Completion of $\mathcal{S}(\mathbb{R})$ for a given norm

Assume that $\lVert \cdot \rVert$ is a norm on the space of rapidly decaying functions $\mathcal{S}(\mathbb{R})$. Under which conditions on the norm can we say that the completion $\mathcal{X}$ for ...

**2**

votes

**0**answers

42 views

### Generalized Besov spaces with different integrability and smoothness in space and time?

Consider the family of Besov spaces $B_{p,q}^{s}(\mathbb{R})$ with $0<p,q \leq \infty$ and $s \in \mathbb{R}$.
Is there a natural way to define spaces of generalized functions $f(t,x) \in \mathcal{...

**1**

vote

**0**answers

39 views

### Is $\mathscr{S}_h'$ a complementary subspace for $\mathscr{S}'/\mathscr{P}$, the space of tempered distributions modulo polynomials?

Recall that in many Fourier Analysis texts, given a function $\Psi$ such that $\hat{\Psi}\in\mathcal{D}(\mathbb R^d)$, $\hat\Psi\ge0$ is supported in an annulus, and $\sum_{j\in\mathbb Z}\hat\Psi(2^j\...

**6**

votes

**1**answer

267 views

### Regularity of Fourier transforms of $L^p$ functions for $2<p\le\infty$

I was recently reading about the Mikhlin and Hörmander Multiplier Theorems, which give conditions for a measurable function $m:\mathbb R^d\to\mathbb C$ to be an $L^p$ multiplier, i.e. for there to ...

**2**

votes

**0**answers

75 views

### Extension of a generalized function to the plane

Let $\phi$ be a generalized function on $\mathbb{R}^2\backslash\{0\}$, and assume that its differential $d\phi$ extends to the whole plane $\mathbb{R}^2$.
Q. Does $\phi$ also extend to $\mathbb{R}^...

**5**

votes

**1**answer

174 views

### Cohomology of sheaf of Schwartz distributions with support in a submanifold

Let $M$ be a smooth manifold. Let $Z\subset M$ be a smooth submanifold which is a closed subset. Let $F$ denote the sheaf of generalized functions (equivalently, Schwartz distributions) on $M$, namely ...

**3**

votes

**1**answer

84 views

### A sufficient condition for a distribution to be temperate

Claim: Let $T$ be a distribution on $\mathbb R^n$ such that $\nabla T$ belongs to $L^p(\mathbb R^n)$ for some $p\in [1,+\infty]$. Then $T$ is a temperate distribution,
i.e. belongs to the topological ...

**4**

votes

**2**answers

211 views

### Composition of a smoothing operator with an $L^2$-bounded operator, non-compact Riemannian manifold

I'm trying to close in on a definitive answer to my own question BVPs for elliptic PDOs: When do Green functions ($L^2$ inverses) define pseudo-differential operators in the interior?, and think I ...

**5**

votes

**1**answer

278 views

### The elliptic regularity theorem for differential operators with variable coefficients

I'm following the book "Introduction to the theory of distributions" by Friedlander and Joshi. There is the following result p. 109
Theorem (8.6.1). Let $X \subset \mathbb{R}^n$ be an open set, and ...

**2**

votes

**2**answers

284 views

### Fourier transform inversion theorem for a function not in L1 or L2

For $\frac{1}{4}<a<1$ consider the following function:
$$f(x)=\frac{|x|^{\frac{1}{2}}}{(x^2+1)^{a+ib}}$$
If $1>a>\frac{1}{2}$ then $f(x) \in L^2$ and the Fourier inversion theorem can be ...

**3**

votes

**0**answers

85 views

### Question on de Rham complex with distributional coefficients

Let $X$ be a smooth manifold (usually assumed to be paracompact). Let us denote by $\underline{\Omega}^{p,-\infty}_X$ the sheaf of real valued $p$-forms with distributional coefficients in the ...

**11**

votes

**1**answer

545 views

### Is every continuous endomorphism of the Schwartz space a pseudo-differential operator?

Let $\mathcal{S}:= \mathcal{S}(\mathbb{R}^n)$ be the Schwartz space of smooth functions with rapid decay. The question is pretty simply stated in the title. Pseudo-differential act continuously on the ...

**5**

votes

**3**answers

348 views

### Exponential derivative of delta distribution?

This question is from here. I'm asking it here as well to increase the number of people who see it and might be able to help.
The question is, what is the result of the following integral for integer ...

**5**

votes

**0**answers

83 views

### Geometric characterization of Silva distributions

There is a well known geometric characterization of tempered distributions on $\mathbb{R}^n$.
A distribution $T\in \mathcal{D}'(\mathbb{R}^n)$ is an element of $\mathcal{S}'(\mathbb{R}^n)$ if and ...

**1**

vote

**0**answers

128 views

### Can we get rid of this test function?

I have a real-valued function $f$ defined on a ball $B$ of $\mathbb{R}^{N}$, $N\geq2$. I have found a constant $M>0$ such that for all $x\in B$ and $B(x,R)$ (ball of center $x$ and radius $R>...

**4**

votes

**1**answer

302 views

### 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 \...

**3**

votes

**0**answers

123 views

### (Non-)Existence of certain invariant distributions on a p-adic space

Following Bernstein-Zelevinski, an $\ell$-space is a Hausdorff, locally compact totally disconnected topological space. For an $\ell$-space $X$, denote $S(X)$ the space of Bruhat-Schwartz functions on ...

**0**

votes

**2**answers

176 views

### Is this equality between an integral and a series wrong?

In this paper (Maroun's PhD dissertation, 2013) at page 46 the following formula is given (apparently without a reference):
$$\int_0^{\infty } e^{i a x^s+i b x^p} \, dx=\sum _{n=0}^{\infty } \frac{\...

**1**

vote

**1**answer

213 views

### Fourier transform of delta function restricted to sphere [duplicate]

I want to compute $\mathcal{F}^{-1}\{\delta(|\cdot|-1)\}(x)$, which exactly means the following computation:
$$f(x) = (2\pi)^{-n/2} \int_{|\xi|=1}e^{ix\cdot\xi}\mathrm{d}\xi, \mbox{ where }~ \xi \in \...

**3**

votes

**1**answer

71 views

### Distributions on the mirabolic subgroup which are left invariant to the unipotent radical

I'm trying to find a reference or a proof to the following statement used by Matringe, N., 2012. Cuspidal representations of GL (n, F) distinguished by a maximal Levi subgroup, with F a non-...

**1**

vote

**0**answers

76 views

### Is a smooth function with compact support defined on adele groups of Schwartz class?

I'm reading Gelbart's Introduction to the Selberg Trace Formula https://arxiv.org/abs/math/0407288. In his paper he seems to have used the consequence that a smooth function with compact support is a ...

**5**

votes

**3**answers

227 views

### No kernel of the form $\lvert x - y\rvert^{-1}$ on tempered distributions?

Does there exist a continuous bilinear form $\mathcal{B}$ on $\mathcal{S}(\mathbb{R})\times \mathcal{S}(\mathbb{R})$ such that
\begin{equation}
\mathcal{B}(\varphi_1, \varphi_2) =\int_{\mathbb{R}\...

**3**

votes

**1**answer

198 views

### Wave front set of vector-valued Dirac delta distribution

Context: I am reading a physics paper Local Wick Polynomials and Time Ordered Products of Quantum Fields in Curved Spacetime which applies the notion of the wave front set to operator-valued ...

**3**

votes

**0**answers

137 views

### Extension of Paley-Wiener-Schwartz theorem to vector-valued distributions

Let $H_{j} := (H_{j}, \| \cdot \|_{H_{j}} ), j=0,1$ be a Hilbert space, and set
\begin{equation*}
{\mathscr S}'(\mathbb{R}^{n}, H_0; H_1) := {\mathscr L}( {\mathscr S}(\mathbb{R}^{n}, H_0), H_1)
\end{...

**6**

votes

**0**answers

176 views

### Uniform estimates of Fourier transform of tempered functions with parameters

Consider the following function in $\mathbb{R}^3$:
$$
f_t(x)=(1+|x|^2)^{-\alpha}e^{-g(x)t},\,\,\,\,\, \text{where}\,\, g(x)=\frac{x^2_1\cdot x^2_2}{1+|x|^2},
$$
where $\frac{1}{2}<\alpha<1$, and ...

**3**

votes

**2**answers

218 views

### limit of a sequence of distributions

Let $u\in \mathcal{D}'(\mathbb{R}^d)$, $f\in C_c^\infty(\mathbb{R}^d)$ and $f(x)=1$ if $|x|\leq 1$; $f(x)=0$ if $|x|>2$. Can we get the following conclusion: there exists $v\in \mathcal{D'}(\mathbb{...

**3**

votes

**1**answer

211 views

### Hörmander's hypoellipticity theorem for complex coefficients

Hörmander's theorem says that if $L = \sum _{i=1} ^r X_i ^2+ X_0 + f$ on some open subset $U \subseteq \Bbb R$ has the property that the Lie algebra generated by $\{X_0, \dots, X_r\}$ at every point ...

**0**

votes

**0**answers

27 views

### Positivity of continuous functions with values in distributions

Let $f\in C([0,T],\mathcal{S}')$ and assume $f(0)$ is a positive distribution. Does there exist $0<\epsilon\le T$ such that $f$ remains positive on $(0,\epsilon)$? If not, what further assumptions ...

**2**

votes

**1**answer

141 views

### Characterization of convex functions

Let $\Omega$ be a convex open subset of $\mathbb R^n$ and let $f:\Omega\rightarrow \mathbb R$ be a convex function. Since $f$ is continuous, it can be considered as a distribution on $\Omega$ and then ...

**3**

votes

**2**answers

261 views

### Completeness of an exponential family

The question is this: Does there exist an integrable function $f\colon\mathbb R\to\mathbb R$ such that $f$ differs from $0$ on a set of nonzero Lebesgue measure and
\begin{equation}
\int_{\mathbb R}...

**12**

votes

**1**answer

605 views

### On an Inequality of Lars Hörmander

Let $P(z)$ be a non-null complex polynomial in $\nu$ variables $z=(z_1,\dots,z_n)$ of degree $\mu$:
\begin{equation}
P(z)=\sum_{|\alpha| \leq \mu} c_{\alpha} z^{\alpha},
\end{equation}
where as usual ...

**2**

votes

**0**answers

121 views

### Does the reciprocal of a polynomial define a tempered distribution when it is locally integrable?

Consider a complex polynomial in $n$ variables $z=(z_1,\dots,z_n)$:
\begin{equation}
P(z)=\sum_{|\alpha| \leq N} c_{\alpha} z^{\alpha},
\end{equation}
where as usual for every $\alpha=(\alpha_1,\dots,\...