# Questions tagged [functional-inequalities]

For questions about proving and manipulating functional inequalities.

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### Doubt obtaining the inequality (3) via (1)

I am reading a proof from the 2009-paper Travelling waves for the Gross-Pitevskii equation II (Béthuel, Gravejat, Saut) and I am really stuck in one step. Please, help me with this: In the paper, ...
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### Does there exist any solution for this inequality?

Let $\Omega,\Omega^*$ be disks in $\mathbb{R}^2$, such that $\Omega^*\subsetneq\Omega$ and their boundaries meet at one point (so they are tangent at that point; consider $N((1,0),1)$ and $N((2,0),2)$ ...
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### Injectivity of integral operators

Let $K:L^2[0,1]^{d_1}\to L^2[0,1]^{d_2}$ be integral operator $$(Kf)(y) = \int f(x)k(x,y)d x.$$ If $d_1>d_2$ is it possible for $K$ to be injective?, e.g. let's take $d_1=2,d_2=1$. More generally, ...
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### Product of distributions satisfying log-sobolev inequality

Let $f,g\in C^\infty(\mathbb{R})$ be two smooth positive functions satisfying $\int f = \int g = 1$. Suppose that both $f$ and $g$ satisfy the log-Sobolev inequality (LSI) with constant $C$, so that ...
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### Proving complicated transcendental inequality

Suppose we have a function $f$ of four posirive real numbers $a,b,c$ and $d$ in a domain that, for a given real number $0<r<1$ they satisfy $$rc<b<a,$$ $$rc<rd<a.$$ We then have ...