On the framework of $L_{p}$ summations for functions

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 $p\geq 1$.