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I'm trying to check that certain examples of Young functions in the harmonic analysis literature are actually Young functions, and in doing so need to prove the following convexity-like inequality for ...

If we have a product of functions $fg$ with $f\in L^r$ and $g\in L^s$ for some $s,r>1$ satisfying $1/r+1/s=1$, then we know that $fg\in L^1$. But if $g$ is a little bit more than $L^s$, say $L^s \...

When a sequence $f \in \ell_1$, there is a very simple bound on its $\ell_q$-norms given by $\|f\|_q^q \leq \|f\|_1 \cdot \|f\|_\infty^{q-1}$. This inequality is a special (or rather limit) case of ...

I have been reading from Weak Convergence and Empirical Processes, and came across the following: Let $a_1,\ldots,a_n$ be constants and $\epsilon_1,\ldots,\epsilon_n\sim$Rademacher. Then $\mathbb{P}...