Articles of distribution theory

Is it true that $f\in W^{-1,p}(\mathbb{R}^n)$, then $\Gamma\star f\in W^{1,p}(\mathbb{R}^n)$?

I am trying to understand the following paper. In page 1191, in the beggining of the proof of Theorem 2.9. the authors consider the convolution $$v=\Gamma\star f$$ They claim that $v\in W^{1.p}(\mathbb{R}^n)$. Does anyone knows why this is true with $f$ being only in $W^{-1,p}(\mathbb{R}^n)$? Update: Maybe this can thrown some light upon the matter. […]

weak derivative of a nondifferentiable function

I am reading a book on Sobolev and having trouble understanding a notion of weak derivative. I consider a function $(x-1)^+=\max(x-1,0),x\in[0,2]$, I have a problem at $x=1$, so it is continuous and “almost everywhere differentiable” now, does it have a weak derivative everywhere? Can we “quantify” that? I know that the derivative in classical sense […]

How to show a function is a test function?

How to show that $$\psi =(x^2\phi)’$$where $\phi$ is a test function, is a test function if and only if $\int_{-\infty}^{\infty} \psi\, dx=\int_{0}^{\infty} \psi\, dx=\psi(0)=0$

Distributional derivative of a hölder function

Let $f$ be a $\alpha$-Hölder function in $\mathbb{R}^n$. Question : does it have distributional derivatives in a $L^p$ space ? (modulo a suitable relationship between $\alpha$ and $n$). I know the other way around : if we look at elements of Sobolev spaces, then we get a Hölder regularity depending on the dimension $n$ and […]

Why is a Harmonic distribution a function?

I want to know how one proves: “If for a distribution $T$ it holds that $\Delta T$=0, then $T$ is an Harmonic function (see Page in the book of Donoghue with the Theorem)”. In the proof in this book I do not understand the last sentence, can someone explain this: “It follows that $S$ is […]

How to pass from $L^2(0,T;V')$ to $\mathcal{D}'\big(\Omega\times (0,T)\big)$?

Let $\mathcal{V}=\{v\in\mathcal{D}(\Omega)\times\mathcal{D}(\Omega)\mid\operatorname{div} v=0\}$, where $\Omega$ is a bounded open subset of $\mathbb{R}^2$ with smooth boundary. Set $V={\overline{\mathcal{V}}}^{H^1(\Omega)\times H^1(\Omega)}$. It is possible to show that $$V=\{v\in H_0^1(\Omega)\times H_0^1(\Omega)\mid\operatorname{div} v=0\}.$$ Assume that $f\in L^2(0,T;V’)$ and define $$F(t)=\int_0^tf(s)\;ds.\tag{1}$$ Question: Why $F$ belongs to $\mathcal{D}'(\Omega\times (0,T))$? And why the partial distributional derivative $\frac{\partial F}{\partial t}$ equals $f$ in $\mathcal{D}'(\Omega\times […]

Why is the topology of compactly supported smooth function in $\mathbb R^d$ not first countable?

In other words, given a countable sequence of neighborhoods of $f(x)=0$, how to construct another open neighborhood that doesn’t contain any of these neighborhoods? Thanks.

Integral of Dirac delta function/distribution $\delta(x)$ with upper boundary equal to zero

I would like to find the value of $$\int_{a<0}^0 \delta(x) dx$$ In particular, I would like to know if I can break down the integral $$\int_a^b \delta(x)f(x) dx=\int_a^0 \delta(x)f(x) dx + \int_0^b \delta(x)f(x) dx $$ with $a<0$ and $b>0$ and $f(x)$ a well-behaved function. Is it wrong to break down the integral like this, doing […]

Continuous inclusions in locally convex spaces

Let $(X, \left \| \cdot \right \|_X )$, $(X, \left \| \cdot \right \|_Y)$ two normed vector spaces with $X \subset Y$, by definition we have $X \hookrightarrow Y$ if $\left \| x \right \|_Y \leq C \left \| x \right \|_X$ for some $C \geq 0$ (general definition of continuous inclusions in normed spaces) […]

Does zero distributional derivative imply constant function?

If a real function $f\colon[a,b]\to\mathbb{R}$ is differentiable and its derivative $f’$ is zero, then $f$ is constant. Does this result still hold when $f$ has a weak derivative? Explicitly, suppose $f\colon[a,b]\to\mathbb{R}$ is an integrable function such that its distributional derivative $Df$ is zero. Does this mean that $f$ is constant?