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I have a question concerning Young's inequality stated as follows:

$||a∗b||_{ℓ_q}≤||a||_{\ell_1}||b||_{ℓ_q},~~~~ 1≤q≤∞$.

Here you can find something on $\ell_q\big(\mathbb{Z}\big)$: Young's inequality for discrete convolution

Edit: From discussion in comments it is true for $\ell_q\big(\mathbb{N}\big)$ from the fact that it's true for $\ell_q\big(\mathbb{Z}\big)$. Does anyone can give straightforward proof for space $\ell_q\big(\mathbb{N}\big)$ without using $\ell_q\big(\mathbb{Z}\big)$ ?

Community
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Bart
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1 Answers1

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You have that if $a \in l_q\big(\Bbb N\big)$ then $a \in l_q\big(\Bbb Z\big)$ since a sequence of natural numbers is a sequence of integers.

You also have if $a$ and $b$ are two sequences of natural numbers then so is $ a*b$, since the convolution only involves multiplications and additions.

So the equality you wrote holds for sequences of natural numbers too, not just sequences of integers.

Andrea
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  • $\mathbb{Z}$ and $\mathbb{N}$ are supports of $\ell_p$ hence domains not ranges. We are talking about sequences of real ( or even complex) numbers. – Bart Jun 30 '15 at 09:32
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    Nevertheless your comment showed me the right way. We can treat sequences indexed with natural numbers as sequences defined on integers with $a_n=0$ when $n<0$. Then from the inequality for Z we get inequality for N. Correct me if I'm not right :) – Bart Jun 30 '15 at 10:14
  • But it is not enough in my case. If someone could provide a straightforward proof for $\mathbb{N}$ without using the case of integers. Maybe some Minkowski or Holder for series? – Bart Jun 30 '15 at 10:20
  • Why is it not enough? If you have a property of a vector space, $l_q\big(\Bbb Z\big)$, then it holds for any subspace. If you can show that another vector space, $l_q\big(\Bbb N \big)$, can be isometrically embedded in it as a subspace, then you showed that the property holds for the second space. – Andrea Jun 30 '15 at 10:58
  • Although I am not sure about how you define the convolution explicitly on $l_q\big( \Bbb N\big)$. – Andrea Jun 30 '15 at 11:00
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    Yes, my question is answered but I want to prove Young's inequality for some symmetric sequence spaces other than $\ell_p$ and then I do not have such an embedding. Hence I have to found some straightforward proof for $\ell_p(\mathbb{N})$, to get some tools. – Bart Jun 30 '15 at 11:03