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We prove a dual Boas type result connecting the behavior of a function and the smoothness of its Fourier–Bessel transform. We obtain sufficient conditions for the weighted integrability of Fourier–Bessel transforms in terms of the moduli of smoothness connected with Bessel translation operators. In some particular case we prove the sharpness of this result. Also we prove a Boas type result about integrability of generalized contractions.

This paper gives a systematic study of Sobolev, Besov and Triebel-Lizorkin spaces on a noncommutative d-torus \\mathbb{T}^d_\\theta (with \\theta a skew symmetric real d\\times d-matrix). These spaces share many properties with their classical counterparts. The authors prove, among other basic properties, the lifting theorem for all these spaces and a Poincar type inequality for Sobolev spaces.

Let $\\mathrm {Lip}_0(M)$ be the space of Lipschitz functions on a complete metric space M that vanish at a base point. We prove that every normal functional in ${\\mathrm {Lip}_0(M)}^*$ is weak* continuous; that is, in order to verify weak* continuity it suffices to do so for bounded monotone nets of Lipschitz functions. This solves a problem posed by N. Weaver. As an auxiliary result, we show that the series decomposition developed by N. J. Kalton for functionals in the predual of $\\mathrm {Lip}_0(M)$ can be partially extended to ${\\mathrm {Lip}_0(M)}^*$ .

Embeddings of discrete metric spaces into Banach spaces recently became an important tool in computer science and topology. The purpose of the book is to present some of the most important techniques and results, mostly on bilipschitz and coarse embeddings. The topics include: (1) Embeddability of locally finite metric spaces into Banach spaces is finitely determined; (2) Constructions of embeddings; (3) Distortion in terms of Poincaré inequalities; (4) Constructions of families of expanders and of families of graphs with unbounded girth and lower bounds on average degrees; (5) Banach spaces which do not admit coarse embeddings of expanders; (6) Structure of metric spaces which are not coarsely embeddable into a Hilbert space; (7) Applications of Markov chains to embeddability problems; (8) Metric characterizations of properties of Banach spaces; (9) Lipschitz free spaces. Substantial part of the book is devoted to a detailed presentation of relevant results of Banach space theory and graph theory. The final chapter contains a list of open problems. Extensive bibliography is also included. Each chapter, except the open problems chapter, contains exercises and a notes and remarks section containing references, discussion of related results, and suggestions for further reading. The book will help readers to enter and to work in a very rapidly developing area having many important connections with different parts of mathematics and computer science.

Fix $d\\geq 2$, and $s\\in (d-1,d)$. The authors characterize the non-negative locally finite non-atomic Borel measures $\\mu $ in $\\mathbb R^d$ for which the associated $s$-Riesz transform is bounded in $L^2(\\mu )$ in terms of the Wolff energy. This extends the range of $s$ in which the Mateu-Prat-Verdera characterization of measures with bounded $s$-Riesz transform is known. As an application, the authors give a metric characterization of the removable sets for locally Lipschitz continuous solutions of the fractional Laplacian operator $(-\\Delta )^\\alpha /2$, $\\alpha \\in (1,2)$, in terms of a well-known capacity from non-linear potential theory. This result contrasts sharply with removability results for Lipschitz harmonic functions.

In this work, we present necessary and sufficient conditions for the boundedness of the commutators generated by multilinear fractional maximal operators on the products of Morrey spaces when the symbol belongs to Lipschitz spaces.

This paper gives a characterization of a class of surjective isometries on spaces of Lipschitz functions with values in a finite dimensional complex Hilbert space.

This monograph contains a study on various function classes, a number of new results and new or easy proofs of old results (Fefferman-Stein theorem on subharmonic behavior, theorems on conjugate functions and fractional integration on Bergman spaces, Fefferman's duality theorem), which are interesting for specialists; applications of the Hardy-Littlewood inequalities on Taylor coefficients to (C, ?)-maximal theorems and (C, ?)-convergence; a study of BMOA, due to Knese, based only on Green's formula; the problem of membership of singular inner functions in Besov and Hardy-Sobolev spaces; a full discussion of g-function (all p > 0) and Calderón's area theorem; a new proof, due to Astala and Koskela, of the Littlewood-Paley inequality for univalent functions; and new results and proofs on Lipschitz spaces, coefficient multipliers and duality, including compact multipliers and multipliers on spaces with non-normal weights. It also contains a discussion of analytic functions and lacunary series with values in quasi-Banach spaces with applications to function spaces and composition operators. Sixteen open questions are posed. The reader is assumed to have a good foundation in Lebesgue integration, complex analysis, functional analysis, and Fourier series. Further information can be found at the author's website at http://poincare.matf.bg.ac.rs/~pavlovic [http://poincare.matf.bg.ac.rs/~pavlovic].

In this article, we introduce the Lp -spaces and create the Kantorovich-type operators of Schurer λ-Bernstein-Bézier basis functions, starting with shifted knots polynomials. We describe the convergence of our novel operators in the Lebesgue spaces and the continuous function space for any 1 ≤ p < ∞. The central moments for these operators are determined by computing the test functions. We then examine the properties of the Korovkin’s type approximation with modulus of continuity of order one and two. We also derive the convergence theorems for these new operators using Peetre’s K-functional and the fundamental conditions of Lipschitz continuous functions. Several direct approximation theorems are also derived by us. In last we given a numerical example with a graphical analysis.

We design the Schurer type Kantorovich–Stancu operators by using shifted knots in the quantum calculus. We obtain the convergence and other related approximation properties of these operators. We discuss the degree of convergence of our operators by applying the modulus of continuity. In addition, we give some basic direct theorems and obtain the approximation in Lipschitz spaces.