Humberto Rafeiro

Humberto Rafeiro
United Arab Emirates University | UAEU · Department of Mathematical Sciences

Doctor of Philosophy

About

110
Publications
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984
Citations
Citations since 2017
40 Research Items
740 Citations
2017201820192020202120222023050100150200
2017201820192020202120222023050100150200
2017201820192020202120222023050100150200
2017201820192020202120222023050100150200

Publications

Publications (110)
Article
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This review article is dedicated to Professor Stefan Samko on the occasion of his 80th birthday. It contains some biographic snippets of S. Samko, as well as a bird’s-eye view on his research during the last decade.
Chapter
In the framework of bounded quasi-metric measure spaces \((X,d,\mu )\), we prove that the fractional operator of variable order \( \alpha (x) \) is bounded from vanishing variable exponent Morrey space \( V\! L ^{p(\cdot ), \lambda (\cdot )} (X) \) to vanishing variable exponent Campanato space , when \(\gamma \leqslant \alpha (x) p(x) + \lambda (x...
Article
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We introduce local grand Lebesgue spaces, over a quasi-metric measure space (X,d,μ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ ( X,d, \mu ) $$\end{document}, where...
Article
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We prove that variable exponent Morrey spaces are closely embedded between variable exponent Stummel spaces. We also study the boundedness of the maximal operator in variable exponent Stummel spaces as well as in vanishing variable exponent Stummel spaces.
Article
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We define the grand Lebesgue space corresponding to the case p=∞$ p = \infty$ and similar grand spaces for Morrey and Morrey type spaces, also for p=∞$ p = \infty$, on open sets in Rn$ \mathbb {R}^n$. We show that such spaces are useful in the study of mapping properties of the Riesz potential operator in the borderline cases αp=n$ \alpha p = n$ fo...
Article
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In this paper, we prove the boundedness of multilinear Calderón-Zygmund operators on product of grand variable Herz spaces. These results generalize the boundedness of multilinear Calderón-Zygmund operators on product of variable exponent Lebesgue spaces and variable Herz spaces.
Article
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We introduce generalized local and global Herz spaces where all their characteristics are variable. As one of the main results we show that variable Morrey type spaces and complementary variable Morrey type spaces are included into the scale of these generalized variable Herz spaces. We also prove the boundedness of a class of sublinear operators i...
Article
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A new characterization of the weighted Taibleson's theorem for generalized Hölder spaces is given via a Hadamard-Liouville type operator (Djrbashian's generalized fractional operator).
Article
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In the paper mentioned in the title, it is proved the boundedness of the Riesz potential operator of variable order α(x) from variable exponent Morrey space to variable exponent Campanato space, under certain assumptions on the variable exponents p(x) and λ(x) of the Morrey space. Assumptions on the exponents were different depending on whether α (...
Article
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In this paper, we prove the boundedness of the fractional maximal and the fractional integral operator in the -adic variable exponent Lebesgue spaces. As an application, we show the existence and uniqueness of the solution for a nonhomogeneous Cauchy problem in the -adic variable exponent Lebesgue spaces. 1. Introduction The field of -adic numbers...
Article
We study fractional potential of variable order on a bounded quasi-metric measure space \((X,d,\mu )\) as acting from variable exponent Morrey space \( L ^{p(\cdot ), \lambda (\cdot )} (X) \) to variable exponent Campanato space \( \mathscr {L } ^{p(\cdot ), \lambda (\cdot )} (X) \). We assume that the measure satisfies the growth condition \( \mu...
Article
We study embeddings of Morrey type spaces \({\varvec{M}}^{p,q,\omega } ({\mathbb {R}}^n ) \), \(1 \leqslant p<\infty \), \(1 \leqslant q<\infty \), both local and global, into weighted Lebesgue spaces \( {\varvec{L}}^p({\mathbb {R}}^n ,w) \), with the main goal to better understand the local behavior of functions \( f \in {\varvec{M}}^{p,q,\omega }...
Article
For a class of sublinear operators, we find conditions on the variable exponent Morrey-type space Lp(⋅),q,ω(⋅,⋅)(Rn) ensuring the boundedness in this space. A priori assumptions on this class are that the operators are bounded in Lp(⋅)(Rn) and satisfy some size condition. This class includes in particular the maximal operator, singular operators wi...
Article
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For the Riesz potential of variable order over bounded domains in Euclidean space, we prove the boundedness result from variable exponent Morrey spaces to variable exponent Campanato spaces. A special attention is paid to weaken assumptions on variability of the Riesz potential.
Article
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We introduce local and global generalized Herz spaces. As one of the main results we show that Morrey type spaces and complementary Morrey type spaces are included into the scale of these Herz spaces. We also prove the boundedness of a class of sublinear operators in generalized Herz spaces with application to Morrey type spaces and their complemen...
Article
The proofs of Theorems 3.4 and 3.5 of (Castillo et al. 2016) contain a flaw. We provide the corrected full proofs.
Article
In this paper we introduce variable exponent Lebesgue spaces where the underlying space is the field of the p-adic numbers. We prove many properties of the spaces and also study the boundedness of the maximal operator as well as its application to convolution operators.
Article
In the limiting case of Sobolev-Adams theorem for Morrey spaces of variable order we prove that the fractional operator of variable order maps the corresponding vanishing Morrey space into VMO.
Article
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Abstract In this paper, we introduce grand variable Herz type spaces using discrete grand spaces and prove the boundedness of sublinear operators on these spaces.
Article
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We give a proof of the boundedness of the Bergman projection in generalized variable-exponent vanishing Morrey spaces over the unit disc and the upper half-plane. To this end, we prove the boundedness of the Calderón—Zygmund operators on generalized variable-exponent vanishing Morrey spaces. We give the proof of the latter in the general context of...
Article
We characterize the linear functionals on variable exponent Bochner–Lebesgue spaces in terms of the variable exponent Riesz bounded variation spaces for vector measures, which are introduced in this paper.
Article
We prove the boundedness of the fractional integration operator of variable order α(x) in the limiting Sobolev case α(x)p(x)=n−λ(x) from variable exponent Morrey spaces Lp⋅,λ⋅Ω into BMO (Ω), where Ω is a bounded open set. In the case α(x)≡ const, we also show the boundedness from variable exponent vanishing Morrey spaces VLp⋅,λ⋅Ω into VMO (Ω). The...
Article
In this paper we establish the boundedness of commutators of sublinear operators in weighted grand Morrey spaces. The sublinear operators under consideration contain integral operators such as Hardy-Littlewood and fractional maximal operators, Calderón-Zygmund operators, potential operators etc. The operators and spaces are defined on quasi-metric...
Article
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We give a simple proof of the boundedness of Bergman projection in various Banach spaces of functions on the unit disc in the complex plain. The approach of the paper is based on the idea of Zaharyuta and Yudovich (Uspekhi Mat Nauk 19(2):139–142, 1964) where the boundedness of the Bergman projection in Lebesgue spaces was proved using Calderón–Zygm...
Article
In this paper we show the validity of Stein’s interpolation theorem on variable exponent Morrey spaces.
Article
We introduce grand Lebesgue sequence spaces and study various operators of harmonic analysis in these spaces, e.g., maximal, convolution, Hardy, Hilbert, and fractional operators, among others. Special attention is paid to fractional calculus, including the density of the discrete version of a Lizorkin sequence test space in vanishing grand spaces.
Article
In this paper we introduce a function space with some generalization of bounded variation and study some of its properties, like embeddings, decompositions and others.
Book
This textbook on functional analysis offers a short and concise introduction to the subject. The book is designed in such a way as to provide a smooth transition between elementary and advanced topics and its modular structure allows for an easy assimilation of the content. Starting from a dedicated chapter on the axiom of choice, subsequent chapte...
Article
In this paper, we obtain the boundedness of the Marcinkiewicz integral on continual Herz spaces with variable exponent, where all parameters defining the space are variable.
Article
We calculate the measure of non-compactness of the multiplication operator \(M_u\) acting on non-atomic Köthe spaces. We show that all bounded below multiplication operators acting on Köthe spaces are surjective and therefore bijective and we give some new characterizations about closedness of the range of \(M_u\) acting on Köthe spaces.
Article
The boundedness of sublinear integral operators in grand Morrey spaces defined by means of measures generated by the Muckenhoupt weights is established. The operators under consideration involve operators of Harmonic Analysis such as Hardy–Littlewood and fractional maximal operators, Calderoń–Zygmund operators, potential operators etc.
Article
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In this article we define and characterize Carleson measures for the setting of variable exponent Bergman spaces. We also estimate the norm of the reproducing kernels under this context.
Article
In this article we use the atomic decomposition of a Herz-type Hardy space of variable smoothness and integrability to obtain the boundedness of the central Calderón-Zygmund operators on Herz-type Hardy spaces with variable smoothness and integrability.
Article
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In the frameworks of some non-standard function spaces (viz. Musielak–Orlicz spaces, generalized Orlicz–Morrey spaces, generalized variable Morrey spaces and variable Herz spaces) we prove the boundedness of the maximal operator with rough kernel. The results are new even for p constant.
Article
In this note we show that, in the case of bounded sets in metric spaces with some additional structure, the boundedness of a family of Lebesgue p-summable functions follow from a certain uniform limit norm condition. As a byproduct, the well known Riesz-Kolmogorov compactness theorem can be formulated only with one condition.
Article
In this paper we give an historical synopsis of various Taylor remainders and their different proofs (without being exhaustive). We overview the formulas and the proofs given by such names as Bernoulli, Taylor, MacLaurin, Lagrange, Lacroix, Cauchy, Schlömilch, Roche, Cox, Turquan, Bourget, Koenig, Darboux, Amigues, Teixeira, Peano, Blumenthal, Wolf...
Article
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In this note, we characterize global Lipschitz Nemytskii operators in the Riesz-bounded variation space with variable exponent, and in the course of the proof, we also show that this space is a Banach algebra.
Article
In this note we correct a misprint in the formulation of Theorem 4.5 of paper Górka (2016). We give the correct formulation and provide a friendlier proof.
Article
In this paper, we show the validity of a Riesz–Thorin type interpolation theorem for linear operators acting from variable exponent Lebesgue spaces into variable exponent Campanato spaces of order k.
Article
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In this article we study the boundedness and compactness of Toeplitz operators defined on variable exponent Bergman spaces. A characterization is given in terms of Carleson measures. http://rdcu.be/kJgk
Article
In this paper we study totally bounded sets in Banach function spaces (BFS), from which we characterize compact sets (via Hausdorff criterion) in some non-standard function spaces which fall under the umbrella of BFS. We obtain a Riesz–Kolmogorov compactness theorem for the grand variable exponent Lebesgue spaces.
Article
Generalized weighted Morrey spaces defined on spaces of homogeneous type are introduced by using weight functions in the Muckenhoupt class. Theorems on the boundedness of a large class of sublinear operators on these spaces are presented. The classes of sublinear operators under consideration contain a whole series of important operators of harmoni...
Chapter
In this chapter, we will introduce the so-called Lebesgue sequence spaces, in the finite and also in the infinite dimensional case. We study some properties of the spaces, e.g., completeness, separability, duality, and embedding. We also examine the validity of Hölder, Minkowski, Hardy, and Hilbert inequality which are related to the aforementioned...
Chapter
In this chapter we study the so-called weak Lebesgue spaces which are one of the first generalizations of the Lebesgue spaces and a prototype of the so-called Lorentz spaces which will be studied in a subsequent chapter. In the framework of weak Lebesgue spaces we will study, among other topics, embedding results, convergence in measure, interpolat...
Chapter
Integral operator theory is a vast field on itself. In this chapter we briefly touch some questions that are related to Lebesgue spaces. We prove the Hilbert inequality, we show the Minkowski integral inequality, and with that tool we show a boundedness result of an integral operators having a homogeneous kernel of degree − 1. We introduce the Hard...
Chapter
In recent years, it had become apparent that the plethora of existing function spaces were not sufficient to model a wide variety of applications, e.g., in the modeling of electrorheological fluids, thermorheological fluids, in the study of image processing, in differential equations with nonstandard growth, among others. Thus, naturally, new fine...
Chapter
In this chapter we study the convolution which is a very powerful tool and some operators defined using the convolution. We first start with a detailed study about the translation operator and after that we introduce the convolution operator and give some immediate properties of the operator. As an immediate application we show that the convolution...
Chapter
Lebesgue spaces are without doubt the most important class of function spaces of measurable functions. In some sense they are the prototype of all such function spaces. In this chapter we will study these spaces and this study will be used in the subsequent chapters. After introducing the space as a normed space, we also obtain denseness results, e...
Chapter
In this chapter we study the distribution function which is a tool that provides information about the size of a function but not about its pointwise behavior or locality; for example, a function f and its translation are the same in terms of their distributions. Based on the distribution function we study the nonincreasing rearrangement and establ...
Chapter
Inequalities play an important role in Analysis, and since many inequalities are just convexity in disguise, we get that convexity is one of the most important tools in Analysis in general and not only in Convex Analysis. In this chapter we will introduce the notion of convexity in its various formulations, and we give some characterizations of con...
Chapter
The spaces considered in the previous chapters are one-parameter dependent. We now study the so-called Lorentz spaces which are a scale of function spaces which depend now on two parameters. Our first task therefore will be to define the Lorentz spaces and derive some of their properties, like completeness, separability, normability, duality among...
Chapter
In this chapter we study one of the most important operators in harmonic analysis, the maximal operator. In order to study this operator we need to have covering lemmas of Vitali type. After the covering lemmas we will study in some detail the maximal operator in Lebesgue spaces and show the Lebesgue differentiation theorem as well as a Theorem of...
Chapter
In this chapter we overview the technique of interpolation of operators, which is widely used in harmonic analysis in connection with Lebesgue spaces. The underlying idea is to obtain boundedness of an operator based on the available information in the endpoints. In the first section we will deal with the Riesz-Thorin interpolation theorem, also kn...
Article
This survey is aimed at the audience of readers interested in the information on mapping properties of various forms of fractional integration operators, including multidimensional ones, in a large scale of various known function spaces. As is well known, the fractional integrals defined in this or other forms improve in some sense the properties o...
Chapter
In this chapter our aim is to introduce grand Morrey spaces in the framework of quasimetric measure spaces with doubling measures and study the boundedness of maximal, fractional, and singular integral operators. We explore also mapping properties of commutators of Calderón–Zygmund singular integrals and potentials with BMO functions. Moreover, we...
Chapter
In this chapter we focus on Morrey and Stummel spaces in the case when the exponents defining the spaces are constant.
Chapter
This chapter deals with mapping properties of maximal, singular integral, and potential operators in weighted generalized grand Lebesgue spaces. Weak and strong type weighted inequalities criteria for these operators (including similar integral transform with product kernels) are established. Single-weight criteria in grand Lebesgue spaces for the...
Chapter
In this chapter we introduce grand Lebesgue spaces on open sets Ω of infinite measure in \( \mathbb{R}^n \), controlling the integrability of \(\vert{f}(x)\vert^{p-\varepsilon}\) at infinity by means of a weight (depending also on ε); in general, such spaces are different for different ways to introduce dependence of the weight on ε.
Chapter
This chapter deals with integral operators with positive product kernels from the two-weight boundedness viewpoint in the classical Lebesgue spaces. We study the two-weight problem in the classical Lebesgue spaces for functions defined on the cone of nonnegative non-increasing functions.
Chapter
We already dealt in Volume 1 with Hölder spaces Hλ(·)(Ω) of variable order, in Sections 8.2.1 and 8.2.3 in the case of open sets \( \Omega \subseteq \mathbb{R}^n \), and in Section 8.3 in the general case of quasimetric measure spaces, where embeddings of variable exponent Sobolev spaces into Hölder spaces were established.
Chapter
One of the purposes of this chapter is to introduce the generalized Morrey spaces with variable exponent defined in Euclidean space and to explore the boundedness of potentials, various types of maximal functions, and singular integrals in these spaces.
Chapter
In this chapter we present estimations for maximal, singular, and potential operators in variable exponent Lebesgue spaces with oscillating weights.
Chapter
In this chapter necessary and sufficient conditions for boundedness of the fractional maximal functions \((\mathcal{M}_{\alpha(.)}f)(x) := \mathop{\rm{sup}}\limits_{{Q}\ni{x}} \frac{1}{\vert{Q}\vert^{1-\alpha(x)/{n}}} \int\limits_{Q} \vert{f}(y)\vert{dy}, \, \, \, 0 < \alpha_{-} \leqslant \alpha_{+} < {n}\), and Riesz potentials \((I^{\alpha(.)} {f...
Chapter
We give an application of the weighted results obtained in Theorem 2.45 with power weights to the theory of Fredholm solvability of singular integral equations (10.1) with piecewise continuous coefficients. As is well known to researches in this field, to investigate such equations in a specific function space, it is important to know precise neces...
Chapter
In this chapter two-weight boundedness problems for various integral operators in variable exponent Lebesgue spaces defined on Euclidean, as well as quasimetric measure spaces are explored.
Chapter
In this chapter we give a complete characterization of the range I α [L p (.)(ℝn )] in terms of the convergence of hypersingular integrals of order α. The proof is based, in particular, on the results on denseness in L p (.)(ℝn ) of Schwartz functions orthogonal to polynomials, and the inversion of the Riesz potential operator by means of hypersing...
Chapter
In this chapter we present in Section 9.1 two general compactness results convenient for applications. One is the so-called dominated compactness theorem for integral operators. We give it in a general context of Banach Function Spaces (BFS) in the well-known sense (see Bennett and Sharpley [27])and recall that L p (.)(Ω) is a BFS, as verified in E...
Chapter
In this chapter we consider the Hardy-type operators \({H}^{\alpha,\mu} f(x) = {x}^{\alpha(x)+\mu(x)-1} \int\limits_{0}^{x} \frac{{f}(y) {dy}} {y^{\alpha(y)}}, \, \, \mathcal{H}_{\beta,\mu}f(x) = {x}^{\beta(x)+\mu(x)} \int\limits_{x}^{\infty} \frac{{f}(y) {dy}}{y^{\beta(y)+1}},\) with variable exponents, in variable exponent Lebesgue spaces.
Chapter
This chapter is devoted mainly to the study of the boundedness/compactness of the weighted Volterra integral operators \({K}_{v} {f}(x) = v(x) \int\limits_{0}^{x} k(x, t)f(t)dt, \, \, \, \, x > 0,\) and \(\mathcal({K}_{v}{f})(x) = {v}(x) \int\limits_{-\infty}^{x} {k}(x, t){f}(t){dt}, \, \, \, {x} \in \mathbb{R} \), in variable exponent Lebesgue spa...
Chapter
This chapter is devoted to the study of the behavior of one-sided maximal functions, Calderón–Zygmund integrals, and potentials in L p (.)(I) spaces, where I is an interval of ℝ.
Chapter
In this chapter we present results on hypersingular operators of order α < 1 acting on some Sobolev type variable exponent spaces, where the underlying space is a quasimetric measure space. The proofs are based on some pointwise estimations of differences of Sobolev functions. These estimates lead also to embeddings of variable exponent Hajlasz-Sob...
Article
In this paper we introduce variable exponent bounded variation spaces in the Riesz sense, prove some embedding results and finally we show that a type of Riesz representation lemma is valid in the newly introduced spaces.
Article
In the framework of variable exponent Lebesgue and Morrey spaces we prove some boundedness results for operators with rough kernels, such as the maximal operator, fractional maximal operator, sharp maximal operators and fractional operators. The approach is based on some pointwise estimates.
Book
This book is devoted exclusively to Lebesgue spaces and their direct derived spaces. Unique in its sole dedication, this book explores Lebesgue spaces, distribution functions and nonincreasing rearrangement. Moreover, it also deals with weak, Lorentz and the more recent variable exponent and grand Lebesgue spaces with considerable detail to the pro...
Book
This book, the result of the authors' long and fruitful collaboration, focuses on integral operators in new, non-standard function spaces and presents a systematic study of the boundedness and compactness properties of basic, harmonic analysis integral operators in the following function spaces, among others: variable exponent Lebesgue and amalgam...
Book
This book, the result of the authors’ long and fruitful collaboration, focuses on integral operators in new, non-standard function spaces and presents a systematic study of the boundedness and compactness properties of basic, harmonic analysis integral operators in the following function spaces, among others: variable exponent Lebesgue and amalgam...
Article
In this paper, we show the validity of a Riesz-Thorin type interpolation theorem for linear operators acting from variable exponent Lebesgue spaces into variable exponent Morrey space in the framework of quasi-metric measure spaces.
Article
p>En este artículo se introduce una generalización del concepto de p-variación de Riesz y se construye un espacio de funciones que es normalizable y además es tanto espacio de Banach como un álgebra de Banach. Adicionalmente, usando el enfoque dado por Medved'ev se obtiene una caracterización integral de las funciones en dicho espacio funcional.</p
Article
In this note we will characterize the boundedness, invertibility, compactness and closedness of the range of multiplication operators on vari- able Lebesgue spaces.
Article
Full-text available
We introduce a function space with some generalization of bounded variation in the sense of de la Vallée Poussin and study some of its properties, like embeddings and decompositions, among others.
Article
In this paper we study compactness of subsets of grand Lebesgue spaces (also known as Iwaniec–Sbordone spaces) and grand Sobolev spaces. Namely, we prove a Kolmogorov–Riesz compactness theorem for grand Lebesgue spaces and the corresponding version of a result of Sudakov. In addition, the validity of a Rellich–Kondrachov type compact embedding is s...
Article
In this paper we introduce a generalization of the concept of Riesz p-variation and construct a function space which is normalizable and moreover is a Banach space as well as a Banach algebra. Furthermore, using Medved'ev approach we obtain an integral characterization of the functions in this function space.
Article
In this paper, we introduce bounded variation spaces in the Wiener sense with p-variable and study some of its basic properties.
Article
We find conditions on the variable parameters and , defining the Herz space , for the validity of Sobolev type theorem for the Riesz potential operator to be bounded within the frameworks of such variable exponents Herz spaces. We deal with a “continual” version of Herz spaces (which coincides with the “discrete” one when q is constant).
Article
In this paper, we establish the regularity, in generalized grand Morrey spaces, of the solution to elliptic equations in non-divergence form with VMO coefficients by means of the theory of singular integrals and linear commutators.
Article
In this paper we show the validity of some embedding results on the space of (φα)-bounded variation, which is a generalization of the space of Riesz p-variation.
Article
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In this paper we show that if the Nemytskii operator maps the (φ, α)-bounded variation space into itself and satisfies some Lipschitz condition, then there are two functions g and h belonging to the (φ, α)-bounded variation space such that f(t, y) = g(t)y + h(t) for all t ∈ [a, b], y ∈ R. Resumen. En este artículo demostramos que si el operador de...
Article
Our aim is to introduce the grand Bochner–Lebesgue space in the spirit of Iwaniec–Sbordone spaces, also known as grand Lebesgue spaces, and prove some of its properties. We will also deal with the associate space for grand Bochner–Lebesgue spaces.
Article
We study the existence of minimizers of a regularized non-convex functional in the context of variable exponent Sobolev spaces by application of the direct method in the calculus of variations. The results are new even in the framework of classical Lebesgue spaces.
Article
We survey known and recently obtained results on Morrey-Campanato spaces with respect to the properties of the spaces themselves, that is, we do not touch the study of operators in these spaces. In particular, we survey equivalent definitions of various versions of the spaces, the so-called φ- and θ-generalizations, the structure of the spaces, emb...
Article
Full-text available
In the setting of homogeneous spaces (X,d,{\mu}), it is shown that the commutator of Calder\'on- Zygmund type operators as well as commutator of potential operator with BMO function are bounded in generalized Grand Morrey space. Interior estimates for solutions of elliptic equations are also given in the framework of generalized grand Morrey spaces...
Article
Full-text available
In this paper we introduce generalized grand Morrey spaces in the framework of quasimetric measure spaces, in the spirit of the so-called grand Lebesgue spaces. We prove a kind of reduction lemma which is applicable to a variety of operators to reduce their boundedness in generalized grand Morrey spaces to the corresponding boundedness in Morrey sp...
Article
In this note we introduce grand grand Morrey spaces, in the spirit of the grand Lebesgue spaces. We prove a kind of \textit{reduction lemma} which is applicable to a variety of operators to reduce their boundedness in grand grand Morrey spaces to the corresponding boundedness in Morrey spaces. As a result of this application, we obtain the boundedn...
Article
Full-text available
Dedicated to Prof. Stefan Samko on the occasion of his 70th Anniversary After recalling some definitions regarding the Chen fractional integrodifferentiation and discussing the pro et contra of various ways of truncation related to Chen fractional differentiation, we show that, within the framework of weighted Lebesgue spaces with variable exponent...

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