Scaled Global Operators and Fueter Variables on Non-zero Scaled Hypercomplex Numbers

This book deals with spaces of functions and distributions of Holder, Sobolev, Hardy, Campanato, Besov type and their descendants. So it is quite clear that we restrict our attention also in this introductory chapter to spaces of these types. There is hardly any doubt that these types of spaces were and are the backbone (in the widest sense of the word) of the theory of function spaces from the very beginning up to our time. On the other hand, in the last few decades many other types of function spaces have been treated extensively, some of them closely connected with the spaces considered here (anisotropic and weighted spaces, spaces with dominating mixed derivatives, spaces on non-smooth domains and on general structures), while others have been based on different principles (Lorentz and Orlicz spaces, etc.). The following books cover a large variety of different aspects of the theory of function spaces in the widest sense: [Sob4, Nik2, BIN, KJF, Pee6, Ste1, StW2, Ada, Maz, MaS, JoW, Triα, Tris, ScT]. Finally, we refer to the recent surveys [BKLN, KuN1] (1988) concentrating mainly, but not exclusively, on developments in the Soviet Union.

Function spaces of type S are introduced and investigated in the literature. They are also applied to study the Cauchy problem. In this paper we shall extend the concept of these spaces to the context of Boehmian spaces and study the Fourier transform theory on these spaces. These spaces enable us to combine the theory of Fourier transform on these function spaces as well as their dual spaces.

Multi-term equations with Riemann–Liouville derivatives and Hölder type function spaces

The present paper deals with the Riemann boundary value problem for analytic functions in the framework of the new function spaces introduced by the first two authors, the so-called grand variable exponent Lebesgue spaces which unify two non-standard type function spaces: variable exponent Lebesgue spaces and grand Lebesgue spaces.

Function spaces and its various applications are subjects of investigation in many research centers and we are glad that this special issue has received the remarkable attention by researchers in these areas. We have received papers from wide range of topics of function spaces such as interpolation methods for stochastic processes spaces, vector optimization problems in Banach spaces, Marcinkiewicz integral operator in Morrey spaces, Hardy’s operator in the variable exponent Lebesgue spaces, and differential equations of weakly parabolic type in Banach spaces. A number of papers are devoted to investigating different kinds of problems in Hardy spaces or Hardy subspaces, for example, Calderon-Zygumnd singular integral operators in Hardy spaces, weighted Hardy spaces related to the Schrodinger operator, and composition operator in hyperbolic Bloch-type spaces. Geometric problems such as modulus of convexity in some Banach sequence spaces, p-uniform convexity, and quniform smoothness in C spaces are also covered by this issue.

One defines a non-homogeneous space (X,μ) as a metric space equipped with a non-doubling measure μ so that the volume of the ball with center x, radius r has an upper bound of the form rn for some n>0. The aim of this paper is to study the boundedness of Calderon–Zygmund singular integral operators T on various function spaces on (X,μ) such as the Hardy spaces, the Lp spaces, and the regularized BMO spaces. This article thus extends the work of X. Tolsa (Math. Ann. 319:89–149, 2011) on the non-homogeneous space (ℝn,μ) to the setting of a general non-homogeneous space (X,μ). Our framework of the non-homogeneous space (X,μ) is similar to that of Hytonen (2011) and we are able to obtain quite a few properties similar to those of Calderon–Zygmund operators on doubling spaces such as the weak type (1,1) estimate, boundedness from Hardy space into L1, boundedness from L∞ into the regularized BMO, and an interpolation theorem. Furthermore, we prove that the dual space of the Hardy space is the regularized BMO space, obtain a Calderon–Zygmund decomposition on the non-homogeneous space (X,μ), and use this decomposition to show the boundedness of the maximal operators in the form of a Cotlar inequality as well as the boundedness of commutators of Calderon–Zygmund operators and BMO functions.

Hardy spaces with variable exponents and generalized Campanato spaces

Approximation by rational functions in hardy space

Since the transfer function (or matrix) of a stable linear system is analytic in |z| ≥ 1, the transformation z → z −1 guarantees its analyticity in the unit disk |z| ≤ 1. In this chapter we will study approximation by polynomials and “stable” rational functions in |z| < 1, using the Hardy spaces H 2 and H ∞ norms. Here, due to the reciprocal transformation, “stability” requires location of poles in |z| > 1. Approximation in the Hankel norm will be considered in the next chapter. In order to achieve a firm understanding of the rudiments of these important mathematical tools for optimal system reduction, filter design, model matching, feedback control, and disturbance rejection, etc., we will first consider SISO systems. Standard methods are sometimes applicable to generalize our discussions to MIMO systems which will be studied in Chap. 6. Here and throughout, P n will always denote the collection of all polynomials of degree no greater than n and R n the collection of all rational functions of degree n.KeywordsHardy SpaceFilter DesignBlaschke ProductExtremal FunctionInterpolation ConditionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this paper, we will study the Hardy and BMO spaces associated to the generalized Hardy operator L_{alpha }= (-Delta )^{alpha /2}+a|x|^{-alpha }. Similarly to the classical Hardy and BMO spaces, we will prove that our new function spaces will enjoy some important results such as molecular decomposition and duality. As applications, we show the boundedness of the spectral multiplier of Laplace transform type and the Sobolev norm inequalities involving the generalized Hardy operator.

Let X be a ball quasi-Banach function space satisfying some mild assumptions and H X (R n ) the Hardy space associated with X. In this article, the authors introduce both the Hardy space H X (R n+1 + ) of harmonic functions and the Hardy space H X (R n+1 + ) of harmonic vectors, associated with X, and then establish the isomorphisms among , where H X,2 (R n+1 + ) and H X,2 (R n+1 + ) are, respectively, certain subspaces of H X (R n+1 + ) and H X (R n+1 + ). Using these isomorphisms, the authors establish the first order Riesz transform characterization of H X (R n ). The higher order Riesz transform characterization of H X (R n ) is also obtained. The results obtained in this article have a wide range of generality and can be applied to the classical Hardy space, the weighted Hardy space, the Herz-Hardy space, the Lorentz-Hardy space, the variable Hardy space, the mixed-norm Hardy space, the local generalized Herz-Hardy space, and the mixed-norm Herz-Hardy space.

Besov type function spaces defined on metric-measure spaces

Polyanalytic Type Function Spaces

In this chapter, we describe several function spaces on homogeneous groups. The origins of the extensive use of homogeneous groups in analysis go back to the book [FS82] of Folland and Stein where Hardy spaces on homogeneous groups have been thoroughly analysed. It turns out that several other function spaces can be defined on homogeneous groups since their main structural properties essentially depend only on the group and dilation structures. Thus, in this chapter we carry out such a construction for Morrey and Campanato spaces and analyse their main properties. Moreover, we describe a version of Sobolev spaces associated to the Euler operator. We call such spaces the Euler–Hilbert–Sobolev spaces.

In this note we give generalisations of Livsic's result that a priori measurable solutions to cocycle equations must in fact be more regular. We go beyond the original continuous hyperbolic examples of Livsic to consider examples of this phenomenon in the context of: (a) 0-transformations; (b) rational maps; and (c) planar maps with indifferent periodic points. Such examples are not immediately covered by Livsic's original approach either due to a lack of continuity or hyperbolicity. 0. INTRODUCTION In 1972 Livsic showed that given a mixing subshift of finite type T: X -* X and a H6lder continuous function c: X -* IR then any solution to the equation (1) c(x) u(Tx) u(x) with u: X ---1iR measurable and essentially bounded (with respect to an equilibrium measure for a H6lder continuous function) must have a continuous version i.e. eu': X -* R continuous such that u(x) = u'(x) a.e. [2], [3] This is an elementary type of rigidity result. Livsic also applied this result to Anosov systems. Another approach to this result for subshifts of finite type and using Perron-Frobenius type operators appeared in [6]. In this note we shall discuss some simple generalisations of Livsic's result to other examples of dynamical systems. We shall be interested in systems with discontinuities or which are not uniformly hyperbolic and thus are not covered by Livsic's original results. In this context the Perron-Frobenius type operator method adapts easily to prove the generalisations of Livsic's original results. To illustrate this approach we shall apply this method to 3-transformations (section 1), rational maps (section 2) and certain multi-dimensional maps (section 3). 1. }5-TRANSFORMATION We begin by recalling the definition of the well-known 3-transformation on the unit interval. Let / > 1 and define T: [0,1) -* [0,1) by T(x) = fx (mod 1). Let v denote Lebesgue measure on the interval [0,1] and Ll ([0, 1], 13, v) denotes the space of integrable functions on [0,1]. Received by the editors August 5, 1996. 1991 Mathematics Subject Classification. Primary 58Fxx. (?)1999 American Mathematical Society 559 This content downloaded from 157.55.39.255 on Wed, 25 May 2016 04:56:50 UTC All use subject to http://about.jstor.org/terms 560 M. POLLICOTT AND M. YURI Definition. We define a Perron-Frobenius operator Li : Ll ([O, 1], 6,B v) --Ll ([O, 1], 6,B v) by Ty= 13 where the summation is over the [,3] or [a1] + 1 pre-images of x. We denote by BBV the space of functions of bounded variation i.e. if f E BBV then we have that var(f) :=sup{ZEif(Xi+1)-f(xi)I: O xoX 0 for x E [0,1). We shall also need the following more general operator. Definition. Given a Lipschitz function c: [0,11 --? R. We define a c-weighted Perron-Frobenius operator L2: L' ([O, 1], B, v) -* L1 ([O, 1], 13, v) by L2 f (x) = E ecp f (y) TY=X (When c= 0 then L2 clearly reduces to Lp.) It is convenient to denote cn (x) = c(x) + c(Tx) +.. + c(Tn lx) for n > 1. The following result is important in proving Theorem 1. Lemma 2. (i) Vf E L1([0, 1], B, v) we have I2nf h(f fdv) in the L1 norm, (ii) Let c: [0,11 -? IR be a Lipschitz function. There exists w E BBV with w > 0 and A > 0 such that the sequence of functions An (T-y=x BBV. The convergence Ijf -* h(J f dv), as n -4 +oo is proved in [9, Theorem 10] when f is uniformly continuous. Since any function in L1 ([O,1], B, v) can be approximated from above and below by such functions, this is sufficient. This content downloaded from 157.55.39.255 on Wed, 25 May 2016 04:56:50 UTC All use subject to http://about.jstor.org/terms REGULARITY OF SOLUTIONS l'O THE MEASURABLE LIVSIC EQUATION 561 (ii) The operator L2: BBV -BBV has a simple maximal eigenvalue A, say, with positive eigenvector w E BBV ([9, Lemma 9]). Moreover, there exists a measure ,t on [0, 1] such that fn =~ --> w f dti, as n +oo [9, Theorem 10]. 1] Our main result on the measurable Livsic equation for fl-transformations is the following. Theorem 1. If c: [0, 1t -* IR is Lipschitz then for any measurable and essentially bounded solution u to (1') c(x) = u(Tx) u(x) there exists uo(x) = v(x) log h(x), where v(x) E BBV such that u == uo a.e. (nt). Proof. We can make the choice f 1 and then iterating the operator 12 introduced above we can write (L n1)(X)= ASn fn (X) Tny3/X In particular, substituting from the identity (1') we get that f Ix eCn(y) fn (x) An E , n