Inequalities for $$L^p$$-Norms that Sharpen the Triangle Inequality and Complement Hanner’s Inequality

Abstract

In 2006 Carbery raised a question about an improvement on the naïve norm inequality f + g p p ≤ 2 p−1 (f p p + g p p) for two functions f and g in L p of any measure space. When f = g this is an equality, but when the supports of f and g are disjoint the factor 2 p−1 is not needed. Carbery's question concerns a proposed interpolation between the two situations for p > 2 with the interpolation parameter measuring the overlap being f g p/2. Carbery proved that his proposed inequality holds in a special case. Here, we prove the inequality for all functions and, in fact, we prove an inequality of this type that is stronger than the one Carbery proposed. Moreover, our stronger inequalities are valid for all real p = 0.

Inequalities that sharpen the triangle inequality for sums of $N$ functions in $L^p$

“…The proof of Theorem 1.1, for all N , is actually relatively simple compared to the proof of the slightly more incisive result for N =2 that is proved in [4].…”

“…The simplest way to demonstrate that things do go wrong for r=r(N, p), N >2 is to compute K N,p (0): K N,p (0)≥1 if and only if The sufficient value r (3,4) is quite close to the nececessary value specified in (3.9). Numerical experiments show that with r given by the right side of (3.8), the inequality K N,p (a)≥1 is likely to be valid, but of course, the inequality K N,p (a)≥1 is only a case of the inequality (1.10) for a very special choice of the functions f 1 , ..., f N .…”

“…These trial functions were chosen to make N j=1 f p j and i<j f i f j constant. A convexity argument was used in [4] to reduce to this case, but the convexity on which this reduction relied fails already for N =3 and p=4. Thus, while it is possible that the exponent r in Theorem 1.1 could be improved slightly for N >2, it cannot be improved by much.…”

“…However, as we shall show here, N ≥3 is significantly different from N =2. The inequality (1.10) would reduce to the inequality found in [4] for N =2 were it possible to replace r(N, p) by (1.12)r(N, p)…”

“…The case N =2 is special. This case was considered by us in [4] where (1.10) was proved with 2/p in place of r (2, p). Note that r(2, p)= 4 4+3(p−2) < 2 p .…”