Trying to find $K$ such that for $1\leq p<q\leq\infty$ we have $||f||_{L^p}\leq K||f||_{L^q}$

Question

Let $1\leq p<q\leq\infty$

I want to find a $K=K(a,b,p,q)$ such that $||f||_{L^p}\leq K||f||_{L^q}$

In other words $L^q(a,b)\subset L^p(a,b)$

For now I just have that $|||f|^{q\over p}||_{L^p}=(||f||_{L^q})^{q\over p}$ and don't know how to use that...

Answer 1

The following theorem can be found is almaost all books on measure theory:

Theorem: If $1\leq p<q\leq\infty$ and if $A$ is a Borel-set in $\mathbb R^n$ with $\lambda_n(A)< \infty$, then we have $ L^q(A)\subset L^p(A)$ and

$$||f||_{L^p}\leq \lambda_n(A)^{1/p-1/q}||f||_{L^q}$$

for all $f \in L^q(A)$. If $n=1$ and $A=(a,b)$, then $\lambda_n(A)^{1/p-1/q}=(b-a)^{1/p-1/q}$.

Notation: $\lambda_n$ is Lebesgue measure on $\mathbb R^n$.

Answer 2

Write $\left\lVert f\right\rVert^p=\int_{(a,b)}\left\lvert f(x)\right\rvert^p\mathbf 1_{(a,b)}(x)\mathrm dx$, apply Hölder's inequality $\int gh\leqslant \left\lVert g\right\rVert_r\left\lVert h\right\rVert_r$ (where $r,r'\geqslant 1$ and $1/r+1/r'=1)$ to $g\colon x\mapsto \left\lvert f(x)\right\rvert^p$, $h\colon x\mapsto \mathbf 1_{(a,b)}(x)$, $r=q/p$.

The constant you get can be seen to be optimal by choosing $f\colon x\mapsto \mathbf 1_{(a,b)}(x)$.