Abstract
In 2006 Carbery raised a question about an improvement on the naïve norm inequality Vert f+gVert _p^p le 2^{p-1}(Vert fVert _p^p + Vert gVert _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 Vert fgVert _{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 pne 0.
Highlights
Introduction and Main Theorem1.1 Main ResultSince |z|p is a convex function of z for p ≥ 1, for any measure space, the L p unit ball, { f : | f |p ≤ 1}, is convex
(1.1) is equivalent to Minkowski’s inequality, it becomes an equality in fewer circumstances
There is another well-known refinement of Minkowski’s inequality for 1 < p < ∞, namely Hanner’s inequality, [2,6,9] which gives the exact modulus of convexity of the unit ball in L p, Bp := { f : | f |p ≤ 1}
Summary
Since |z|p is a convex function of z for p ≥ 1, for any measure space, the L p unit ball, { f : | f |p ≤ 1}, is convex. We note that Carbery’s proposed inequality (1.2) involves three kinds of quantities on the right side (namely, f g p/2, f g p p and f p g p), while our inequality (1.4) involves only two (namely, f g p/2 and f g pp) This both strengthens the result and simplifies the proof. Carbery proved that his proposed inequality (1.2) is valid when f and g are characteristic functions. Another important special case is when f and g are proportional to each other Even in this special case, inequality (1.4) is quite nontrivial and, constitutes the core of the proof of Theorem 1.1.
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