Motivated by the inequality
∥∥f+g∥22≤∥f∥22+2∥fg∥1+∥g∥∥22,
Carbery (2009) raised the question of what is the “right” analogue of this estimate in
Lp for
p≠2.
Carlen, Frank, Ivanisvili and Lieb (2018) recently obtained an
Lp
version of this inequality by providing upper bounds for
∥∥f+g∥∥pp in terms of
the quantities
∥∥f∥∥pp,
∥∥g∥∥pp and
∥∥fg∥∥p∕2p∕2 when
p∈(0,1]∪[2,∞), and lower
bounds when
p∈(−∞,0)∪(1,2),
thereby proving (and improving) the suggested possible inequalities of
Carbery. We continue investigation in this direction by refining the
estimates of Carlen, Frank, Ivanisvili and Lieb. We obtain upper bounds for
∥∥f+g∥∥pp also when
p∈(−∞,0)∪(1,2) and lower
bounds when
p∈(0,1]∪[2,∞).
For
p∈[1,2]
we extend our upper bounds to any finite number of functions.
In addition, we show that all our upper and lower bounds of
∥∥f+g∥∥pp for
p∈R,
p≠0, are the best possible in
terms of the quantities
∥∥f∥∥pp,
∥∥g∥∥pp and
∥∥fg∥∥p∕2p∕2, and
we characterize the equality cases.
Keywords
triangle inequality, Lp spaces, concave envelopes,
Bellman function
