$L^p$ spaces 2-side inequality

Question

I am currently studying $L^p$ spaces. Let's consider probability space in the following. Also consider measurable $f$'s.

We know that (for example, Proposition 1.4 here) when $1\leq p< q$, we have $||f||_p \leq ||f||_q$.

Is there a known inequality for the reverse as well? That is, for $1\leq p< q$, can we get $$||f||_q \leq C ||f||_p$$ for some "nice" $C$? We can assume $f$ is bounded: $|f(x)|\leq B$.

This thread provides $||f||_q^q \leq C ||f||_p^p$ with $C=B^{1 - (p/q)}$. This is close to what I am looking for, but can we get a version without the exponents just like in my request above?

Thanks in advance for all inputs!

Answer 1

In general, a reverse inequality of the type $\| f\|_q\leq C\| f\|_p$, valid for all $f\in L^p$ with $p<q$ and $C$ independent of $f$ will fail. For example $f(x)=1/\sqrt{x}$, and $p<q=2$ on $L^p[0,1]$.

1. One case where such an inequality is true is when your underlying sigma algebra is finite, i.e. $L^p=\ell^p_n$ for some $n\in \mathbb N$ (see here for the little $\ell$ notation, the subindex just means only the first $n$ entries are allowed to be nonzero). In this case $\| f\|_{\ell^q_n}\leq \| f\|_{\ell^p_n}\leq n^{1/p-1/q}\| f\|_{\ell^q_n}$ for any $p<q$.

2. If you restrict the inequality to hold, say for all $f\in X\subset L^p$ a linear subspace (and $C$ still independent of $f$ of course), things are a bit more interesting: If $L^p=L^p[0,1]$ and $(r_j)_j$ is the Rademacher system, $X=\text{Span}_{L^2}((r_j)_j)$, then Khintchine's inequality says $$ c_r\left( \sum_j |a_j|^2\right)^{1/2} \leq \lVert \sum_j a_jr_j\rVert_{L^r} \leq C_r\left( \sum_j|a_j|^2\right)^{1/2}. $$ This means that on $X$ we have $\| f\|_{L^r}\sim \| f\|_{L^2}$ for any $r>0$. In fact, one way to interpret Khintchine's inequality is as saying there's a copy of $\ell^2$ (viewed as the $L^2[0,1]$ closure of the span of the $r_j$) embedded in $L^r[0,1]$.

3. In your example, where $X=X_B:=\{ f\in L^\infty: \|f\|_{L^\infty}\leq B\}$, you're treating a variant of the above type of problem (i.e. item 2), where $X$ is no longer a (linear) subspace. You saw that we can get $\| f\|_{L^q}\leq C_{B,p,q} \| f\|_{L^p}^a$, for some constant $0<a<1$. The constant $a$ can't be removed in general, as can be seen by considering $f_n(x)=x^n$ for $n\in \mathbb N$ in $[0,1]$.