1

Let $\sigma \in {W}^{1,\gamma}(0,\infty)$ $(\gamma\geq 1$) and $\phi \in {W}^{1,p}(0,\infty)$ $(p\geq 1$).

I am wondering if $\sigma\phi \in {W}^{1,p}(0,\infty)$ $(1 \leq p <\gamma$)?

Thanks for helps.

user895874
  • 55
  • 5
  • 1
    This is true for $p = \gamma = 2$ on $\mathbb{R}$. It can be proven using the Fourier transform: https://math.stackexchange.com/questions/314820/sobolev-space-hs-mathbbrn-is-an-algebra-with-2sn – Mason Jul 19 '22 at 19:48

1 Answers1

0

You need a Theorem to claim that the function $\sigma\phi$ is at least integrable.

The Hölder's inequality is very useful for this general functions. If $\gamma,p\in[1,\infty]$ and $1/\gamma+1/p=1$, then $\sigma\phi\in L^1(0,\infty)$.

With general functions $\sigma$ and $\phi$ you can't claim that $\sigma\phi\in L^p(0,\infty)$ and this is neccessary for $\sigma\phi\in W^{1,p}(0,\infty)$.

So if $\gamma$ and $p$ verify $1/\gamma+1/p=1$, you can affirm that $\sigma\phi\in W^{1,1}(0,\infty)$.

We assume $1/\gamma+1/p=1$, let's see that the derivate $(\sigma\phi)'$ is integrable. $$ (\sigma\phi)'=\sigma '\phi+\sigma\phi' $$ Because $\sigma\in W^{1,\gamma}$ and $\phi\in W^{1,p}$, then $\sigma,\sigma'\in L^\gamma$ and $\phi,\phi'\in L^p$. Using the Hölder inequality: $\sigma '\phi\in L^1$ and $\sigma\phi'\in L^1$. Because the sum of two functions in $L^1$ is in $L^1$ then $(\sigma\phi)'\in L^1$.

So you can assure that $\sigma\phi\in W^{1,p}(0,\infty)$ when $1/\gamma+1/p=1$. In a general situation you can say nothing.

user_1234
  • 51
  • Hi. In which space can you write that $(σϕ)′=σ′ϕ+σϕ′ $ ? – user895874 Jul 14 '22 at 14:38
  • I think in $L^1(0,\infty)$, because the derivates of $\sigma$ and $\phi$ are weaks. Is my point that the derivate of the product $\sigma\phi$ is in $L^1(0,\infty)$. – user_1234 Jul 14 '22 at 14:38
  • 1
    I hope my answer and my comment help you. At least is what I know about the $L$ spaces and their properties. Maybe you can say something else for the general case – user_1234 Jul 14 '22 at 14:39
  • 1
    I proved that if $\gamma=\infty$ then $\sigma\phi \in {W}^{1,p}(0,\infty)$ $(p\geq1$). – user895874 Jul 14 '22 at 14:46
  • If $\sigma\in W^{1,\gamma}$ then $\sigma\in L^\infty$ whence $\int \vert \sigma\phi\vert^p \leq |\sigma|_\infty\int\vert\phi|^p<\infty$ – user895874 Jul 15 '22 at 04:50
  • Do you have find the reference ? – user895874 Jul 16 '22 at 07:25