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Young's inequality for convolution of functions states that for $f\in L^p(\mathbb{R}^d)$ and $g\in L^q(\mathbb{R}^d)$ we have

$$\|f\star g\|_r\le\|f\|_p\|g\|_q$$

for $p$, $q$, $r$ satisfying

$$\frac{1}{p}+\frac{1}{q}=\frac{1}{r}+1.$$

Does this inequality hold for sequences? That is, can we replace $L^n(\mathbb{R}^d)$ with $\ell_n$, $n=p,q$ respectively, where convolution of sequences is the discrete convolution?

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    Yes, Young's inequality is true for convolution on locally compact groups (not necessarily abelian), in particular $\mathbb{Z}$. See e.g. [Theorem 20.18 on page 296](http://books.google.com/books?id=uf11K1wXEYUC&pg=PA296) of Hewitt-Ross, *Abstract Harmonic Analysis, I* but that's serious overkill. [The slick argument given by robjohn here](http://math.stackexchange.com/questions/69921/convolution-of-an-l-p-mathbbt-function-f-with-a-term-of-a-summability-k/69945#69945) should carry over without any pain.2011-10-05
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    Thanks! I wonder can I cite this theorem in the article, or should I prove this result myself with the hints given. After I posted the question I found the result proven in [Bogachev's book](http://books.google.com/books?id=CoSIe7h5mTsC&lpg=PR1&dq=bogachev%20measure%20theory&hl=fr&pg=PR1#v=onepage&q&f=false), his argument is similar to robjohn.2011-10-05
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    I would say that Young's inequality is standard and straightforward enough and thus very likely to fall victim to a referee in the publication process anyway, so I wouldn't waste time to write up the proof in an article I'm writing for publication unless I was really unable to find it in the desired form in the literature. You seem to have two references already, so the statement with a reference should be amply sufficient.2011-10-05
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    There is Minkowski's inequality: $$\|a\ast b\|_{\ell^q}\le \|a\|_{\ell^1}\|b\|_{\ell^q},\ 1\le q\le \infty.$$2013-04-26

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Yes, Young's inequality can be shown to hold for arbitrary locally compact groups — under suitable integrability assumptions on $f$ and $g$, see Hewitt–Ross, Abstract Harmonic Analysis, I, Theorem (20.18) on page 296 for the precise statement.

If $G$ happens to be abelian, compact, discrete (or, more generally, unimodular) then these assumptions translate to: If $f \in L^{p}$, $g \in L^q$ and $\frac{1}{p} + \frac{1}{q} = 1 + \frac{1}{r}$ for $1 \leq p, q, r \leq \infty$ then $f \ast g \in L^r$, and

$$\|f \ast g\|_r \leq \|f\|_p\,\|g\|_q.$$

Replacing integrals by sums robjohn's argument here carries over painlessly to $\mathbb{Z}$ or $\mathbb{Z}^d$.

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    Over a year late, but I just came across this post. Summation is just integration over a discrete set :-)2012-10-31
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    Sorry for fishing an old thread. But doesnt that mean, that $l^P=l^1$ (settheoretic) whenever $p\in[1,2]$? Since $l^1\subset l^p$ we would get by convolution with $e^0$, where $e^0_z=1$ for $z=0$ and $e^0_z=0$ for $z\neq 0$ $$e^0\ast f(z)=f(z)~\text{for all}~z$$ and so $$\|f\|_1=\|e^0\ast f\|_1\leq\|e^0\|_q\|f\|_p=\|f\|_p.$$ Thanks for reply.2012-10-31
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    @Burn: Are you talking about $\ell^1$ and $\ell^p$? In that case, $\ell^1\subset\ell^p$ for $p\ge1$, but the reverse containment is not true.2012-10-31
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    @robjohn: yeah, right :) Thanks, hi and bye, Theo.2012-11-29
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    @robjohn in that case what is wrong in the argument of @user47666?2016-10-31
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    I see it sorry, $q<1$ is the problem.2016-10-31