Proving $L^q(\mathbb{T}) \subseteq L^p(\mathbb{T})$ for $1 \leq p < q \leq \infty$

Question

First, I'd like to address that this question is similar to " $L^p$ and $L^q$ space inclusion" or "Proving that $L^p \subset L^q$ when $1 \le q \le p$". But I was wondering if we can prove it without using Jensen's inequality or using $$||f||_p\leq ||f||_q\lambda(\mathbb{T})^{1/q-1/p}$$ which can be obtained by applying Holder's inequality on $|f|^p\cdot 1$. As for the latter case, if I was using it, what would then $\lambda(\mathbb{T})$ be? (Here $\lambda$ is the Lebesgue measure). How do I know that $\lambda(\mathbb{T})$ is finite if that's the case?

Let $\mathbb{T}$ be a set $\mathbb{R}\backslash 2\pi\mathbb{Z} = \{x + 2\pi\mathbb{Z} : x \in [-\pi, \pi)\}$. Let $f \in L^q(\mathbb{T})$ and show that $L^q(\mathbb{T}) \subseteq L^p(\mathbb{T})$. Then $$||f||_q^q=\int_{\mathbb{T}}|f|^q\ dx<\infty.$$ As a hint, I apparently should use the Holder's inequality. But I don't quite see how to use this inequality.

Answer 1

$\Bbb T$ is the circle parametrized by $e^{2\pi ix}$ with $x\in[0,2\pi]$. So you can view any function defined on $\Bbb T$ as a function on the interval $I=[0,2\pi]$ and $$\int_{\Bbb T}f=\int_If(x)dx\;.$$

Now take $f\in L^q$, that is, the integral $\int_I|f|^q$ is finite. Partition $I$ into subset $A$ where $|f(x)|\le 1$ and subset $B$ where $|f(x)|>1$. To show $f\in L^p$, we have to show the integral $\int_I|f|^p$ is finite. That last integral breaks into the sum of $\int_A|f|^p+\int_B|f|^p$.

The first integral is finite because $$\int_A|f(x)|^p dx\le\int_A1 dx\le\int_0^{2\pi}1dx=2\pi\;.$$

The second integral is finite because on $B$ we have $|f(x)|^p\le|f(x)|^q$ (since $|f(x)|>1$ and $p\le q$) and we assumed the corresponding integral with exponent $q$ was finite.