0
votes
0answers
15 views

Find $u:[0,T]\to H^2$ such that $u(0)=u_0\in H^2$ and $u_t(0)=u_1\in H^1$.

Let $u_0\in H^2$ and $u_1\in H^1$. If we define $$ \begin{align*}u:[0,T]&\longrightarrow L^2\\ t&\longmapsto u_0+\int_0^tu_1\;ds \end{align*}$$ then $u(0)=u_0$. Furthermore, the weak ...
6
votes
1answer
46 views

Which normed space have Fatou's property?

There is tool in mathematics, more specifically in Analysis, which has immense applications in handling delicates estimates, limiting arguments, etc... ; namely Fatou property. Let $E$ be a normed ...
2
votes
1answer
22 views

Is $C(\mathbb R) \subset H_{s}^{loc}$ (localized Sobolev space)?

We put, Sobolev space $$H_{s}=H_{s}(\mathbb R)=\{f\in L^{2}(\mathbb R):[\int_{\mathbb R} |\hat{f}(\xi)|^{2}(1+|\xi|^{2})^{s}d\xi]^{1/2}<\infty \}.$$ If $U$ is an open set in $\mathbb R,$ the ...
4
votes
1answer
73 views

Example of a function $u\in L^\infty(0,T,H^1)$ such that $u_t\notin L^\infty(0,T,H^1)$

Could someone give me an example of a function $u\in L^\infty(0,T,H^1)$ such that $u_t$ exists (in the distributional sense), $u_t\in L^\infty(0,T,L^2)$ and $u_t\notin L^\infty(0,T,H^1)$? Thanks. ...
1
vote
1answer
24 views

Is it true that, $x\rho(x/t)\in H^{s}$ for $\rho\in \mathcal{D}(\mathbb R)$ and $s>3/2$?

We put, $\mathcal{D}(\mathbb R)=$ The space of $C^{\infty}-$ functions on $\mathbb R$ with compact support; and the Sobolev space $$H^{s}(\mathbb R)=\{f\in L^{2}(\mathbb R):[\int_{\mathbb R} ...
1
vote
0answers
37 views

An exercise about Fourier transform and $H^s$ in Treves

In Treves, the Fourier transform is defined by $\hat{f}(\xi)=\int {e^{-i\langle\xi,\space x\rangle} f(x) dx}$. The following is the problem, where I have figured out almost all of the questions except ...
0
votes
1answer
22 views

Distribution: $f_a(x)=\frac{H(x+a)-H(x-a)}{2a}$. What is its derivative with respect to the parameter $a$ and the limit as $a\to 0$.

Consider the distribution $$ f_a(x)=\frac{H(x+a)-H(x-a)}{2a}$$ Determine the $a$-derivative of this distribution $$ \left < \frac{\partial f_a}{\partial a},\phi \right> = \lim_{h\to0} ...
4
votes
0answers
22 views

Distributions and primitives

I was wondering: if distributions are seen as a generalization of functions that "removes obstructions" to the operation of derivation, is there a generalization of functions that would remove any ...
1
vote
0answers
38 views

Can we characterize homogeneous Sobolev spaces by means of Sobolev embedding?

For $\alpha>0^{[1]}$, let $D^\alpha=(-\Delta)^{\frac{\alpha}{2}}$ be the operator defined on all Schwartz class functions $f$ as follows: $$ \widehat{D^\alpha f}(\xi)=\lvert \xi\rvert^\alpha ...
1
vote
0answers
20 views

Criteria to prove that a map is a tempered distribution

There is any simple sufficient condition to determine if a function is a tempered distribution? For example, given the map : $$ F \phi = \int_\epsilon^\infty \! \frac {\phi(x)}{\sqrt{x}} \, ...
3
votes
1answer
45 views

The Dirac Delta Distribution $\delta_0 : D \to \mathbb R$ is not regular

I have a question on the following proof, that $\delta_0$ is not a regular distribution. We define $\delta_0$ as the linear function on test function with $$ \delta_0(\varphi) = \varphi(0) $$ for ...
1
vote
1answer
65 views

Approximation of the Heaviside Function whose derivative has a compact support

I am looking for a smooth approximation $H_\delta$ of the Heaviside function, which has the property that $$ \lim_{\delta\rightarrow 0^+}H_\delta =H $$ in the distribution sense, and $$ ...
2
votes
1answer
102 views

Dirac delta distribution and measure?

Of course the Dirac delta is not a function. Despite, I think the concept of a measure is much easier than that of a distribution. Therefore, I was wondering: In what sense is the concept of a Dirac ...
3
votes
2answers
71 views

Topology of test functions $\mathcal{D}(\mathbb R)$

(My motivation for the following question is to understand the distribution theory) The space of test functions: $\mathcal{D}(\mathbb R)= \{\phi:\mathbb R \to \mathbb R : \phi \in C^{\infty}(\mathbb ...
1
vote
1answer
50 views

Is $\langle f,g\rangle$ defined for distributions $f,g$?

Consider a standard setting for the development of the theory of distributions. Let $D(\Omega)$ be the space of test functions and $D'(\Omega)$ be the space of distributions ("generalized functions"). ...
0
votes
1answer
20 views

Fourier transform of function similar to a Riesz kernel

I am trying to prove that the Fourier transform of $$\frac{x_1 x_2} {|x|^4}$$ in $\mathcal{R}^2$ (in the sense of distributions) is a bounded multiplier given by $\frac{\xi_1 \xi_2}{|\xi|^2}$ but am ...
4
votes
0answers
63 views

Limit of distribution

Let $T\in\mathcal{D}'(\mathbb{R})$ be a distribution on the set of smooth functions of compact support $\mathcal{D}(\mathbb{R})$ such that $$ \forall_{g\in\mathcal{D}(\mathbb{R})}~|\langle T, g ...
0
votes
1answer
54 views

p--laplacian problem

For $f \in W^{-1,p'}(\Omega)$, $f \geq 0$, we definr $v,$ $v_{\epsilon}$, $g_{\epsilon}$ by $$-\mathrm{div}(|Dv|^{p-2} Dv)=f \quad \mbox{in} \mathcal{D}'(\Omega), \quad v \in W^{1,p}_0(\Omega)$$ with ...
4
votes
1answer
56 views

Sobolev spaces in one-dimensional vs multidimensional

Here in Wikipedia, it is said that in the one-dimensional case, it is enough to assume that the $(k-1)$-th derivative of the function $f$, is differentiable almost everywhere and is equal almost ...
1
vote
0answers
35 views

Extension of a Pseudodifferential Operator

Let $M$ be a smooth manifold with countable atlas, and define the distributions $\mathscr{D}'(M)$ as the dual space to the smooth densities with compact support, and $\mathcal{E}'(M)$ as the dual ...
2
votes
1answer
31 views

Duals of embeddings in the space of distributions

If $ \Lambda \colon X \hookrightarrow \mathcal{D}'$ is a continuous embedding of a normed vector space $X$ into the space of distributions (for example $X=L^p$), is it true that the dual of ...
-2
votes
1answer
71 views

eigenvalues to Laplacian

Let $\Omega$ be a bounded subset of $\mathbb R^d$ and let $g\in L^2(\Omega )$. Let $(\lambda_n)_{n\in \mathbb{N}}$ be the eigenvalues of the Laplacian operator $\Delta$, and $(e_n)_n$ the ...
0
votes
0answers
36 views

When can we interchange summation with $L^2$ inner product?

(This question concerns a step in the solution given to Eignvalues of Laplacian operator and Sobolev spaces.) Why can we interchange the sum and the $L^2$ inner product in the following? $$(\sum_n ...
3
votes
0answers
45 views

Do we have $C^\infty \cap \mathcal{O}_C' = \mathcal{S}$ and/or $C^\infty \cap \mathcal{S}' = \mathcal{O}_M$?

We define the following traditional function spaces from distribution theory. $\mathcal{S}$ the space of rapidly decreasing smooth functions. $\mathcal{S}'$ the space of tempered distributions, dual ...
0
votes
0answers
46 views

Question with tried Eigenvalues of Laplacian operator and Sobolev spaces III.

Let $\Omega$ be a bounded subset of $\mathbb R^d$ and let $g\in L^2(\Omega )$. Let $(\lambda_n)_{n\in \mathbb{N}}$ be the eigenvalues of the Laplacian operator $\Delta$, and $(e_n)_n$ the ...
0
votes
0answers
21 views

eignvalues of laplacian operator and Sobolev spaces -II

Let $\Omega$ an open bounded in $\mathbb{R}^n$ and $I$ un interval in $\mathbb{R}$, let $t_0 \in I$, Let $F=(F_t) \in C^0(I,L^2(\Omega))$. Let $(\lambda_n)_{n\in \mathbb{N}}$ the eigen values od the ...
1
vote
2answers
200 views

Eignvalues of Laplacian operator and Sobolev spaces

Let $\Omega$ an open bounded in $\mathbb{R}^n$ and $I$ un interval in $\mathbb{R}$, let $t_0 \in I$, $f\in H^1_0(\Omega)$, $g\in L^2(\Omega)$. Let $(\lambda_n)_{n\in \mathbb{N}}$ the eigen values od ...
0
votes
0answers
33 views

Eigenvectors of Laplace operator on a particular functiona space

Exercise 8.15 in [1] is: "Show that $\Delta u=\lambda u$ has no solutions of polynomial growth if $\lambda > 0$, but does have such solutions if $\lambda < 0$." How should I make sense of this? ...
0
votes
0answers
41 views

Dilation and translation of the Dirac Delta distribution

Given $\delta(ax) = \frac{1}{|a|}\delta(x)$ and $\delta(ax-b) = \frac{1}{|a|}\delta(x-b/a)$ , it it true also that $\delta(a(x-b)) = \frac{1}{|a|}\delta(x-b)$ ?
1
vote
1answer
48 views

convolution -questions

I'm lost, can you help me please. How compute the product convolution between two distributions $T$ and $S$? (we suppose that $T * S$ exist)? How we compute the product convolution between an ...
2
votes
1answer
32 views

Convolution computing

How we can compute the convolution product $$\Big(\sum_{n=0}^{+\infty} \delta_n^{(n)}\Big) \star \Big(\sum_{n=0}^{+\infty} \delta_n\Big)$$ where $\delta$ is Dirac distribution? Thank's for the help
0
votes
1answer
42 views

Understanding a use of Leibniz Formula, functional analysis

I am working through the following text: http://homepage.ntlworld.com/ivan.wilde/notes/gf/gf.pdf Several times the author cites 'Leibniz Formula' which I understand as the product rule. I am not sure ...
1
vote
1answer
52 views

equation in Sobolev space

i have this exercice: Let $f\in L^2(\mathbb{R}^n)$. 1- Prouve that the equation $\Delta u - u = \dfrac{\partial f}{\partial x_i}$ admeit a unique solution $u \in H^1(\mathbb{R}^n)$? 2- Prouve the ...
1
vote
1answer
36 views

Examples of functions with values in distributions

What is an example of a function in $L^p((0,T);\mathcal{D}'(\mathcal{R}))$? I ask this because the Majda-Bertozzi book on Incompressible flow deals with vortex sheet initial data $\omega(t,\cdot)\in ...
1
vote
0answers
36 views

Calderón-Zygmund $\times$ Schwartz $=$ Calderón-Zygmund

I am in a functional analysis class, and we are being asked to show that if $\eta$ is a Schwartz function and $K$ is a Calderón-Zygmund distribution, then their product is also a Calderón-Zygmund ...
0
votes
1answer
19 views

Close fourier transforms implies close time domain functions?

What conditions do we need so that $A(f)\approx B(f) \Rightarrow \mathcal{F}^{-1}\left\{A\right\}(t)\approx\mathcal{F}^{-1}\left\{B\right\}(t)$ The Fourier transforms of my two things look alike. In ...
0
votes
1answer
33 views

$f, \hat {f} \in L^{p} \cap L^{\infty} \implies f\in A(\mathbb R)$?

$1\leq p \leq \infty $. We put, $$X_{p}= \{f\in L^{p}(\mathbb R)\cap L^{\infty}(\mathbb R) :\hat{f}\in L^{p}(\mathbb R)\cap L^{\infty}(\mathbb R)\};$$ and we consider the algebra of Fourier ...
3
votes
1answer
50 views

Limit of a functional

I'd like to find: $$ \lim_{\varepsilon\rightarrow 0}\frac{\varepsilon}{\varepsilon^2+x^2}\qquad \mbox{ in }\mathcal D'(\mathbb{R}) $$ And I started with the definition: $$ \left\langle ...
3
votes
1answer
41 views

“$L^\infty$ in the sense of distributions”

The following is Exercise 3 of Chapter 3 in Stein & Shakarchi Book 4: Show that a bounded function $f$ on $\mathbb{R}^d$ satisfies a Lipschitz condition $$|f(x)-f(y)| \leq C|x-y| ...
0
votes
0answers
52 views

Infinite solutions of Navier-Stokes equations

Is it a known fact that Navier-Stokes equations have exactly one (possibly infinite) solution in the space of distributions?
3
votes
1answer
164 views

Topologies of test functions and distributions

I'm wondering about some of the topological properties of $\mathcal D(\Omega)$ and $\mathcal D'(\Omega)$: I know $\mathcal D(\Omega)$ is not metrizable, so not first countable (right?). However, my ...
0
votes
0answers
31 views

Fourier transform of a function of characteristic function of a measure

Let $\mu$ be complex measure on $\mathbb{R}^2$ ($|\mu|$ is finite measure) and $\chi$ - its characteristic function $$ \chi(x_1,x_2) = \int_{\mathbb{R}^2} d\mu(p_1,p_2) \exp(i p_1 x_1+i p_2 x_2). $$ ...
3
votes
0answers
36 views

Exponential of the derivative operator on the Schwartz space?

We consider the derivative operator $\mathrm{D}$ on the space of smooth and rapidly decreasing function $\mathcal{S}$. We denote by $P_n = \frac{1}{0!} + \frac{X}{1!} + \frac{X^2}{2!} + \cdots + ...
1
vote
1answer
38 views

Is $L^2(0,T;H^{-1}(\Omega)) \subset \mathcal{D}^*((0,T)\times \Omega)$?

Let $\Omega \subset \mathbb{R}^n$ be a domain. Consider the space of test functions $\mathcal{D}((0,T)\times \Omega)$ and the space of distributions $\mathcal{D}^*((0,T)\times \Omega).$ Is it true ...
3
votes
1answer
111 views

How to build a compact support for a function

I was wondering if it is possible to build a distribution with compact support from a function. More precisely, consider a compact set $\mathbf{K}\subset\mathbb{R}^2\setminus\{0\}$, and a function ...
4
votes
0answers
76 views

Two possible definitions of “vector-valued distribution”

Let $X$ be a reflexive Banach space. Define $$\tag{1} \mathcal{D}^\star(0, T; X)=\left\{ u\colon \mathcal{D}(0, T)\to X\ \text{linear and continuous}\right\} $$ where the topology on the space of ...
2
votes
1answer
74 views

Why the tempered distribution is zero?

My question is derived from the proof of the equation $\Delta f=f$ which has no nonzero solution in $\mathscr{S}'(\mathbb{R}^n)$. The ideal to solve this equation is to use the Fourier transform. By ...
2
votes
0answers
115 views

Problem on the integral representation of a tempered distribution

Suppose $\mathscr{S}(\mathbb{R^n})$ is the space of Schwartz functions, in which the seminorms have the form $$\left \| \varphi \right \|_{m}=\underset{\underset{x\in \mathbb{R}^{n}}{|\alpha|\leq m ...
2
votes
2answers
27 views

Can we conclude that a distribution is a $L^2$ function by testing with $L^2$?

Let $T\colon \mathcal{D}\to\mathbb{R}$ be a distribution. Does $|T(f)|\leq\|f\|_2 \forall f\in\mathcal{D}$ imply $T=T_g$ for some $g\in L^2$? What if $T$ is tempered?
4
votes
1answer
115 views

Prove that the weak$^*$ topology on the space of tempered distributions is not 1st countable

Please, help me with a proof of this (apparently) known fact whose proof is out of my reach, even though I spent a considerable amount of time looking it up: The weak$^*$ topology on the space of ...