We determine the set of all triples 1 ≤ p,q,r ≤ ∞ for which the so-called Marcinkiewicz-Zygmund inequality is satisfied: There exists a constant c≥ 0 such that for each bounded linear operator $T:L_q\left(\mu \right)\to L_p\left(\nu \right)$, each n ∈ ℕ and functions $f_1,...,f_n\in L_q\left(\mu \right)$, $\left(ʃ\right({\sum }_{k=1}^n|T{f}_k{|}^r{)}^{p/r}{d\nu )}^{1/p}\le c\parallel T\parallel \left(ʃ\right({\sum }_{k=1}^n|f_k{|}^r{{)}^{q/r}d\mu )}^{1/q}$. This type of inequality includes as special cases well-known inequalities of Paley, Marcinkiewicz, Zygmund, Grothendieck, and Kwapień. If such a Marcinkiewicz-Zygmund inequality holds for a given triple (p,q,r), then we calculate the best constant c ≥ 0 (with the only exception:...

We answer a question of Aharoni by showing that every separable metric space can be Lipschitz 2-embedded into c₀ and this result is sharp; this improves earlier estimates of Aharoni, Assouad and Pelant. We use our methods to examine the best constant for Lipschitz embeddings of the classical ${\ell }_p$-spaces into c₀ and give other applications. We prove that if a Banach space embeds almost isometrically into c₀, then it embeds linearly almost isometrically into c₀. We also study Lipschitz embeddings into...

We determine the norm in $L^p\left(ℝ₊\right)$, 1 < p < ∞, of the operator $I-{ℱ}_s{ℱ}_c$, where ${ℱ}_c$ and ${ℱ}_s$ are respectively the cosine and sine Fourier transforms on the positive real axis, and I is the identity operator. This solves a problem posed in 1984 by M. S. Birman [Bir] which originated in scattering theory for unbounded obstacles in the plane. We also obtain the $L^p$-norms of the operators aI + bH, where H is the Hilbert transform (conjugate function operator) on the circle or real line, for arbitrary real a,b. Best...

We investigate best constants for inequalities between the Orlicz norm and Luxemburg norm in Orlicz spaces.

We prove an almost isometric reverse Hölder inequality for the euclidean norm on an isotropic generalized Orlicz ball which interpolates Paouris concentration inequality and variance conjecture. We study in this direction the case of isotropic convex bodies with an unconditional basis and the case of general convex bodies.

We present the classical Paley-Wiener-Schwartz theorem [1] on the Laplace transform of a compactly supported distribution in a new framework which arises naturally in the study of the Mellin transformation. In particular, sufficient conditions for a function to be the Mellin (Laplace) transform of a compactly supported distribution are given in the form resembling the Bochner tube theorem [2].

Let 0 < p ≤ 1, let ω: ℤ → [1,∞) be a weight on ℤ and let f be a nowhere vanishing continuous function on the unit circle Γ whose Fourier series satisfies ${\sum }_{n\in \mathbb{Z}}{\left|f̂\left(n\right)\right|}^p\omega \left(n\right)<\infty$. Then there exists a weight ν on ℤ such that ${\sum }_{n\in \mathbb{Z}}{\left|\widehat{(1/f)}\left(n\right)\right|}^p\nu \left(n\right)<\infty$. Further, ν is non-constant if and only if ω is non-constant; and ν = ω if ω is non-quasianalytic. This includes the classical Wiener theorem (p = 1, ω = 1), Domar theorem (p = 1, ω is non-quasianalytic), Żelazko theorem (ω = 1) and a recent result of Bhatt and Dedania (p = 1). An analogue of...

Given a locally compact abelian group G with a measurable weight ω, it is shown that the Beurling algebra L¹(G,ω) admits either exactly one uniform norm or infinitely many uniform norms, and that L¹(G,ω) admits exactly one uniform norm iff it admits a minimum uniform norm.

For a locally compact group G and p ∈ (1,∞), we define and study the Beurling-Figà-Talamanca-Herz algebras $A_p(G,\omega )$. For p = 2 and abelian G, these are precisely the Beurling algebras on the dual group Ĝ. For p = 2 and compact G, our approach subsumes an earlier one by H. H. Lee and E. Samei. The key to our approach is not to define Beurling algebras through weights, i.e., possibly unbounded continuous functions, but rather through their inverses, which are bounded continuous functions. We prove that...