  We develop the framework of $L_p$ operations for functions by introducing two
primary new types $L_{p,s}$ summations for $p>0$: the $L_{p,s}$ convolution sum
and the $L_{p,s}$ Asplund sum for functions. The first type is defined as the
linear summations of functions in terms of the $L_p$ coefficients
($C_{p,\lambda,t}$, $D_{p,\lambda,t}$), the so-called the $L_{p,s}$
supremal-convolution when $p\geq1$ and the $L_{p,s}$ inf-sup-convolution when
$0<p<1$, respectively. The second type $L_{p,s}$ summation is created by the
$L_p$ averages of bases for $s$-concave functions. We show that they are
equivalent in the case $s=0$ (log-concave functions) and $p\geq1$. For the
former type $L_{p,s}$ summation, we establish the corresponding
$L_p$-Borell-Brascamp-Lieb inequalities for all $s\in[-\infty,\infty]$ and
$p\geq1$. Furthermore, in summarizing the conditions for these new types of
$L_p$-Borell-Brascamp-Lieb inequalities, we define a series of the $L_{p,s}$
concavity definitions for functions and measures. On the other hand, for the
latter type $L_{p,s}$ Asplund summation, we discover the integral formula for
$L_{p,s}$ mixed quermassintegral for functions via tackling the variation
formula of quermassintegral of functions for $p\geq 1$.
