0
votes
1answer
60 views

If $u\in L^p$, is $u\in L^q$ for some $q>p$?

(Motivation is below) Let $\Omega$ be an open bounded subset of $\mathbb{R}^n$. Let $p\in [1,\infty)$ and $u\in L^p(\Omega)$. Is there any $q>p$ such that $u\in L^q(\Omega)$? I already know that ...
3
votes
1answer
92 views

Gauss–Ostrogradsky formula for Distributions

Let $\Omega\subset\mathbb{R}^N$ be a bounded domain and $u\in W^{1,p}(\Omega)$, $v\in L^{p'}(\Omega)^N$ with $p\in (1,\infty)$. Let $\operatorname{div}(v)$ be the divergence of $v$ in the sense of ...
2
votes
1answer
83 views

Regularizing a solenoidal vector field $u\in L^p(\Omega)^N$.

Let $\Omega\subset\mathbb{R}^N$ be a bounded smooth domain and suppose that $u\in L^p(\Omega)^N$, $p\in (1,\infty)$. Assume that in the sense of distributions, $\operatorname{div}u=0$ where ...
0
votes
1answer
85 views

A basic question about $\operatorname{supp}f$ (support of f).

Is it true that $\operatorname{supp}f$ is the complement of the biggest open set where $f=0 $? Here $\operatorname{supp}f=$ {$x\in \Bbb R^n ; f(x)\not=0$} and $f\in C$ (collection of continuous maps ...
8
votes
2answers
118 views

Various kinds of derivatives

Let $f\colon \mathbb{R}\to \mathbb{R}$ be a measurable function. Let us introduce the following notions of "derivative" of $f$. Classical derivative. The unique function $f'_c$ defined pointwise by ...
2
votes
1answer
187 views

About a property of the Dirac delta function

How can I show that there is no $u$ satisfying both (i) and (ii):$$(i) \; u \in L^p (\Bbb R^n )$$ and $$(ii) \int_{\Bbb R^n} \delta (x) \phi(x) dx= \int_{\Bbb R^n} u (x) \phi(x)dx\; ( \forall \phi ...
1
vote
1answer
93 views

Convergence of distributions in $L^p$

If I understand correctly, distributions $F_n \in C^\infty_c(\mathbb{R})^*$ are defined based on how they act on test functions $\phi \in C^\infty_c(\mathbb{R})$. What does it mean then to say $F_n ...
2
votes
2answers
64 views

Convergence of a sequence in $L^1(\mathbb{R}^3)$

All function spaces are over $\mathbb{R}^3$. Let $u_n \in C^\infty_0$, $u_n\rightarrow u$ in $L^1$. Let $v\in L^1_\text{loc}$ be such that $uv \in L^1$. Does $u_n v \rightarrow uv$ in $L^1$? What ...