New inequalities related to sums of $L^p$ functions in connection with Carbery's problems
Asadollah Aghajani, Juha Kinnunen
TL;DR
This paper advances Minkowski-type inequalities for sums of nonnegative functions in $L^p$ by deriving new two-function bounds valid for all $p\in(1,\infty)$, expressed via the explicit functions $\underline{J}_p$ and $\overline{J}_p$ and depending on cross-terms such as $\|fg^{p-1}\|_1$ and $\|fg^{1/(p-1)}\|_{p-1}$. It then extends these ideas to sums of $n$ nonnegative functions, establishing a sharp $C_p$-dependent inequality for $p\in[2,\infty)$ and, for $p\in[1,2]$, a corresponding reverse inequality, yielding necessary and sufficient conditions for $\sum_j f_j$ to belong to $L^p$ in both regimes. The proofs employ a differential-inequality approach to $F(t)=\int (f+tg)^p$ together with a key lemma bounding $\|fg\|_1$, as well as a fundamental convexity-based inequality for sums of nonnegative terms. The results refine and, in certain cases, improve upon prior bounds in the Carbery–Carlen–Lieb–Mooney framework and resolve regimes of $p$ not previously covered, with clear equality criteria and implications for sequences of $f_j$.
Abstract
Carbery (2006) proposed novel estimates for the $L^p$ norm of a sum of two nonnegative measurable functions. Subsequently, Carlen, Frank, Ivanisvili and Lieb (2018) provided stronger bounds, which Ivanisvili and Mooney (2020) further refined to achieve estimates that are, in a certain sense, optimal. Continuing this line of research, the present work establishes new upper and lower bounds for the range \(p\in(1,\infty)\). Carbery also asked under what conditions on a sequence \((f_j)\) of nonnegative measurable functions the inequality \(\sum \|f_j\|_p^p < \infty\) implies that \(\sum f_j \in L^p\). Ivanisvili and Mooney (2020) resolved this question for \(p\in[1,2]\), and the present work proposes an answer for \(p\in[2,\infty)\).