# Tagged Questions

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### Dirac delta distribution & integration against locally integrable function

I was reading the a lecture note online about distribution theory and it said: The Dirac delta distribution $\delta \in D'$ is defined as $\delta(\varphi)= \varphi(0)$, and there's no locally ...
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### Zigmund-Besov Spaces and Inverse Function Theorem, is the Inverse Zigmund?

Preliminary Definition Let $\Omega \subset \mathbb{R}^n$ be open. We define the Zygmund spaces $C^r_{*}(\Omega)$ with $r>0$, $r \in \mathbb{R}$ in the following way: (all the functions are allowed ...
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### Isomorphism between $C^\infty_0(B_1)$ and $\mathscr{S}(\mathbb{R}^n)$

Background: Related question I am trying to prove, that the countably-normed spaces $C^\infty_0(B_1)$ on the open unit ball (i.e. function and all derivatives vanish at the boundary) in ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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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 $$|f(x)-f(y)| \leq C|x-y| ... 0answers 55 views ### Infinite solutions of Navier-Stokes equations Is it a known fact that Navier-Stokes equations have exactly one (possibly infinite) solution in the space of distributions? 1answer 172 views ### Topologies of test functions and distributions I'm wondering about some of the topological properties of \mathcal D(\Omega) and \mathcal D'(\Omega): I know \mathcal D(\Omega) is not metrizable, so not first countable (right?). However, my ... 0answers 32 views ### Fourier transform of a function of characteristic function of a measure Let \mu be complex measure on \mathbb{R}^2 (|\mu| is finite measure) and \chi - its characteristic function$$ \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). $$... 0answers 38 views ### Exponential of the derivative operator on the Schwartz space? We consider the derivative operator \mathrm{D} on the space of smooth and rapidly decreasing function \mathcal{S}. We denote by P_n = \frac{1}{0!} + \frac{X}{1!} + \frac{X^2}{2!} + \cdots + ... 1answer 40 views ### Is L^2(0,T;H^{-1}(\Omega)) \subset \mathcal{D}^*((0,T)\times \Omega)? Let \Omega \subset \mathbb{R}^n be a domain. Consider the space of test functions \mathcal{D}((0,T)\times \Omega) and the space of distributions \mathcal{D}^*((0,T)\times \Omega). Is it true ... 1answer 112 views ### How to build a compact support for a function I was wondering if it is possible to build a distribution with compact support from a function. More precisely, consider a compact set \mathbf{K}\subset\mathbb{R}^2\setminus\{0\}, and a function ... 0answers 80 views ### Two possible definitions of “vector-valued distribution” Let X be a reflexive Banach space. Define$$\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 ...
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 ...