Suppose there exists a Lebesgue Space, $L_1$ and functions functions $\phi$, $\phi'$, $f$, and $f'$ functions where $$\phi, \phi' \in L_1$$
By rule of integration by parts, $$uv|_a^b = \int_a^b udv + \int_a^b vdu$$
Let $$ u = \phi, du= \phi'$$ $$ v = f, dv = f'$$
Are there any properties of Lebesgue functions that allow
$$ uv|_a^b = 0$$
Are $\phi$ and $\phi'$ convergent as integrals? Do the unbounded limits of $\phi$ and $\phi'$ converge?
On
It is not true in general that $\phi \in L_1(\mathbb{R})$ implies $\lim_{x \to \infty} \phi (x) = 0$. The limit may not exist. For example, let $\phi (x) = 1$ for $x \in [n, n+2^{-n})$ for each natural number $n$ (including zero), and let $\phi \equiv 0$ otherwise. Then, we have
\begin{equation*} \int_{-\infty}^\infty |\phi| = \sum_{n=0}^\infty 2^{-n} = 2 < \infty, \end{equation*} but $\lim_{x \to \infty} \phi (x)$ does not exist. In fact, we can modify this example to construct an unbounded function which has no limit at infinity but belongs to $L^p$ for every finite $p$.
It is not clear to me what your question is. Typically the situations in which boundary terms for integration by parts vanish are when we know the functions are periodic (e.g. they are solutions to a PDE on which we impose periodic boundary conditions) or when one of the functions is compactly supported on the domain of integration.
By the definition of Lebesgue spaces,
If $\phi \in L_1$ then $\int_{-\infty}^{\infty} \phi(x)\, dx < \infty$
If $\lim_{x \to \infty} \phi(x) > 0$ or $\lim_{x \to \infty} \phi(x) < 0$ then $\int_{-\infty}^\infty |\phi(x)| dx = \infty$
The contrapositive of the preceding statement is $ \int_{-\infty}^\infty | \phi(x) | dx \ne \infty$ then $\lim_{x \to \infty} \phi(x) = 0$
Integration by parts is usually applied as $a = -\infty$ and $b = \infty$ $uv|_{\infty}^{\infty}$ will evaluate to 0 because the unbounded integral of $\phi(x)$ converges.