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

Abstract

AbstractIn 2006 Carbery raised a question about an improvement on the naïve norm inequality $$\Vert f+g\Vert _p^p \le 2^{p-1}(\Vert f\Vert _p^p + \Vert g\Vert _p^p)$$ ‖ f + g ‖ p p ≤ 2 p - 1 ( ‖ f ‖ p p + ‖ g ‖ p p ) for two functions f and g in $$L^p$$ L p of any measure space. When $$f=g$$ f = g this is an equality, but when the supports of f and g are disjoint the factor $$2^{p-1}$$ 2 p - 1 is not needed. Carbery’s question concerns a proposed interpolation between the two situations for $$p>2$$ p > 2 with the interpolation parameter measuring the overlap being $$\Vert fg\Vert _{p/2}$$ ‖ 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 $$p\ne 0$$ p ≠ 0 .