Intuition of $L^{p}$ spaces

Question

Let $\Omega$ a unbounded domain of integracion of $\mathbb{R}^{n}$. What is the relation between a function belongs to a $L^{p}(\Omega)$ space ( $1\leq p < \infty$) and its decay at infinity? What is the improvement that a function that decay faster to infinity has?

Answer 1

As a commenter pointed out, I assume you mean to ask about unbounded domains of integration otherwise the question doesn't have much substance.

In general, there is no real connection between a function $f\in L^p(\Bbb R^n)$ and its decay at infinity. For instance, just consider $L^p(\Bbb R)$ (so $n=1$). Then we can find functions that do not decay at all, for instance $$ f(x) = \sum_{k=1}^\infty k\,\chi_{I_k}(x) $$ where $I_k$ is an interval of width $2^{-k}k$ centered at $k$. Notice that $\limsup_{x\to\infty} f(x) = \infty$, so there is no decay at infinity. The function $f$ can obviously be modified to be continuous, too. However, by the monotone convergence theorem, $\int f^p = \sum \big(\frac{k}{2^kk}\big)^p = \sum 2^{-kp} < \infty$, so $f$ is in $L^p(\Bbb R)$.

If you impose something like uniform continuity however, then suddenly we do get that $f\in L^p(\Bbb R^n)$ satisfies $\limsup_{x\to\infty}f(x) = 0$.