Nikolskii inequality and functional classes on compact Lie groups

In this note we study Besov, Triebel-Lizorkin, Wiener, and Beurling function spaces on compact Lie groups. A major role in the analysis is played by the Nikolskii inequality.Comment: In this note (to appear in Funct. Anal. Appl.) we present results from our paper at arXiv:1403.3430 (to appear in Ann. Sc. Norm. Super. Pisa Cl. Sci.) in the simplified setting of compact Lie groups. We refer to the above paper for more general formulations in the setting of compact homogeneous manifolds and for the proof

Diversity strengthens competing teams

How does the composition of a collection of individuals affect its outcome in competition with other collections of individuals? Assuming that individuals can be different, we develop a model to interpolate between individual-level interactions and collective-level consequences. Rooted in theoretical mathematics, the model is not constrained to any specific context. Potential applications include research, education, sports, politics, ecology, agriculture, algorithms and finance. Our first main contribution is a game theoretic model that interpolates between the internal composition of an ensemble of individuals and the repercussions for the ensemble as a whole in competition with others. The second main contribution is the rigorous identification of all equilibrium points and strategies. These equilibria suggest a mechanistic underpinning for biological and physical systems to tend towards increasing diversity due to the strength it imparts to the system in competition with others.Peer reviewe

Diversity strengthens competing teams

How does the composition of a collection of individuals affect its outcome in competition with other collections of individuals? Assuming that individuals can be different, we develop a model to interpolate between individual-level interactions and collective-level consequences. Rooted in theoretical mathematics, the model is not constrained to any specific context. Potential applications include research, education, sports, politics, ecology, agriculture, algorithms and finance. Our first main contribution is a game theoretic model that interpolates between the internal composition of an ensemble of individuals and the repercussions for the ensemble as a whole in competition with others. The second main contribution is the rigorous identification of all equilibrium points and strategies. These equilibria suggest a mechanistic underpinning for biological and physical systems to tend towards increasing diversity due to the strength it imparts to the system in competition with others.Peer reviewe

Nikolskii inequality and besov, triebel-lizorkin, wiener and beurling spaces on compact homogeneous manifolds

In this paper we prove Nikolskiis inequality (also known as the reverse Hölder inequality) on general compact Lie groups and on compact homogeneous spaces with the constant interpreted in terms of the eigenvalue counting function of the Laplacian on the space, giving the best constant for certain indices, attained on the Dirichlet kernel. Consequently, we establish embedding theorems between Besov spaces on compact homogeneous spaces, as well as em- beddings between Besov spaces and Wiener and Beurling spaces. We also analyse Triebel-Lizorkin spaces and β-versions of Wiener and Beurling spaces and their embeddings, and interpolation properties of all these spaces

Hardy–Littlewood and Pitts inequalities for Hausdorff operators

In this paper we study transformed trigonometric series with Hausdorff averages of Fourier coefficients. We prove Hardy–Littlewood and Pitts inequalities for such series. The corresponding results for the Hausdorff averages of the Fourier transforms are also obtained. © 2018 Elsevier Masson SA

Hardy–Littlewood and Pitt's inequalities for Hausdorff operators

In this paper we study transformed trigonometric series with Hausdorff averages of Fourier coefficients. We prove Hardy–Littlewood and Pitt's inequalities for such series. The corresponding results for the Hausdorff averages of the Fourier transforms are also obtained. © 2018 Elsevier Masson SA

Hardy-type theorems on Fourier transforms revised

We obtain new conditions on periodic integrable functions so that their transformed Fourier series belong to Lp. This improves the classical Hardy and Bellman results. A counterpart for the Fourier transforms is also established. Our main tool is a new extension of the Hausdorff–Young–Paley inequality for Fourier transforms. © 2018 Elsevier Inc

A Boas-type theorem for α\alpha-monotone functions

We define the class of $\alpha$-monotone functions using fractional integrals.  For such functions we prove a Boas-type result on the summability of the Fourier coefficients

Hardy-type theorems on Fourier transforms revised

We obtain new conditions on periodic integrable functions so that their transformed Fourier series belong to Lp. This improves the classical Hardy and Bellman results. A counterpart for the Fourier transforms is also established. Our main tool is a new extension of the Hausdorff–Young–Paley inequality for Fourier transforms. © 2018 Elsevier Inc