I understand that if a function $f$ is in $L^\infty([0, 1]^2)$ then $f$ can be pictured as some function, possibility discontinuous at some points, that fits inside the cube $[0,1] \times [0,1] \times [-M, M]$ for some $M \in \mathbb{R}$.
But what about the other $L^p$ spaces, that is $L^1, L^2, L^3, \dots,$ defined on $[0,1]^2$? How should I picture functions in those spaces (or not in those spaces)? As the index $p$ increases do the functions start to look more like functions that are in $L^\infty$?
Some intuition: You may think of a function $f$ with say $\|f\|_p=1$, $1\leq p<+\infty$ as a mostly bounded function but possibly with some 'spikes'. With growing $p$ the spikes have to be more 'narrow' (take up less and less ground volume). More precisely, if $\mu$ is Lebesgue measure on the square, you may define the 'ground' measure of what exceeds your $M$-box:
$$ A(M) = \mu( |f|>M ) \in [0,1]$$ in terms of which you have: $$ 1 = \left( \|f\|_p \right)^p = p \int_0^\infty A(M) \ M^{p-1}\ d M$$ This 'constraint' shows that $A(M)$ has to go zero as $M$ increases and the faster the larger the value of $p$.