Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 $\omega(t,\cdot)\in \mathcal{M}\cap H^{-1}$ but the time dependence is assumed to be Lipschitz. I am wondering about the more general case, for instance how you can have a delta distribution in $x$ that is integrable in $t$ behaving like $\delta(x)f(t,x)$ for some $f\in L^1$, since multiplication with a distribution is only defined for $C^\infty$ functions. Thanks!

share|cite|improve this question

To answer your first question, suppose you have a map $$f:(0,T)\to D'(\Omega),\quad t\to f(t)$$such that $\forall \phi\in D(\Omega)$ the function $$g_\phi:(0,T)\to\Bbb R,\quad g_\phi(t) = \langle f(t),\phi\rangle,$$ belongs to $L^p(0,T)$. Then we can say that $$f\in L^p((0,T);D'(\Omega)).$$

Like this we can give sense to expessions like, for example, $f(t,x)=\sin t \cdot\delta_0(x)$.

share|cite|improve this answer
By the way: This approach is called "weak" or Gelfand-Pettis-integration. These keywords may help a search for relevant material. – Johannes Hahn May 12 '14 at 13:29
@JohannesHahn Hm, thank you for this reference, I didn't know it had a special name. I just made an analogy with the definition of function spaces like $L^p((O,T);L^q(\Omega))$. – TZakrevskiy May 12 '14 at 15:59

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.