Sharpening Hölder's inequality

Abstract

We strengthen Hölder's inequality. The new family of sharp inequalities we obtain might be thought of as an analog of Pythagorean theorem for the L p -spaces. Our treatment of the subject matter is based on Bellman functions of four variables.Here, P e ⊥ denotes the orthogonal projection onto the orthogonal complement of a nontrivial vector e. At this point, we note that since P e ⊥ f 0, the identity (1.1) implies the Cauchy-Schwarz inequality | f, e | f e , e, f ∈ H.We also note that (1.1) leads to Bessel's inequality:N n=1 | f, e n | 2 f 2 , f ∈ H, for an orthonormal system e 1 , . . . , e N in H. We may think of (1.1) as of an expression of the precise loss in the Cauchy-Schwarz inequality. Our aim in this paper is to find an analogous improvement for the well-known Hölder inequality for L p norms. Before we turn to the analysis of L p spaces, we need to replace the norm of the projection, P e ⊥ f , by an expression which does not rely on the Hilbert space structure. It is well known that( 1.2) where α ranges over all scalars (real or complex).

A contractive Hardy–Littlewood inequality

“…In [1, Lemma 2.2], this is proved for and in [7] this is proved for (in [7], this lemma is proved even for , but with as ). Nevertheless we present here a simple uniform proof of this lemma.…”

A contractive Hardy-Littlewood inequality

“…In [8,Lemma 2.2] this is proved for p ≤ 1 and in [9] this is proved for 1 < p ≤ 2 (in [9] this lemma is proved even for f ∈ L p , but with γ p → ∞ as p → 1). Nevertheless we present here a simple uniform proof of this lemma.…”