ON FRECHET SPACES OF DISTRIBUTIONS AND MULTIPLIER OPERATORS

Abstract

Many results in Fourier Analysis, known for LP(' < p < QD), C, M and Orlicz spaces etc., have been obtained for Banach spaces of distributions and convolutable Banach spaces of distributions by Vishnu Kant and J.K. Nath in their Ph.D. theses "On the Banach Spaces of Distributions" and "On Convolutable Banach Spaces of Distributions" respectively. In the present thesis we define Frechet Spaces of Distributions (briefly written as FD-spaces), generalize the previously known results to FD-spaces and state some new results. The thesis consisting of six chapters is summarised as below. In Chapter 1 we explain the various assumptions and terminology used, give various definitions and examples, and state some useful known results. In Chapter 2 we: define FD-spaces as follows. Let D denote the space of all distributions on G = R/2nZ. As a dual space of Cam, let D have the strong*topology. A Frechet space E is called an FD-space if it can be continuously embedded into D; and, regarded as a subset of D, it satisfies the following properties (i) Cc° C E; (ii) fGE TxfGE and ITx1xGR is an equicontinuous family of operators on E; (iii) fGE => fGE, where 1(u) = f(U) *12GC 00 ii To claim our extension fruitful, we give an example of a non-normable FD-space which is different from Cam. Throughout the thesis E* (the dual space of E) is endowed with the strong*topology. Chapter 3 deals with homogeneous FD-spaces. An FD-space E is said to be homogeneous if, for every f in E, the function x*Txf is continuous from G to E. If E is homogeneous FD-space, pEM, fGE and FEE*, then we prove that F*fEC, P*fEE and p*FEE*. We show that E and E* are (C,1)-complementary if and, only if E is homogeneous. We also show that if g belongs to a homogeneous FD-space E and p is a Radon measure, then p*g is the limit in E of finite linear combinations of translates of g. In Chapter 4 we define the (C,1)-complementary space E' of an FD-space E as E' = {FGE*Ilimn-Ko F(6 nf) = F(f) for all fEEl. It turns out that E' is continuously embedded into D and that it is a closed subspace of E*. Moreover E is homo- geneous if and only if E' = E*. If FEE' and pEM, then we show that p*FEE'. If FEE', fEE and g(x) = F(Txi) for each xeG, then g generates the distribution F*f. We say that E is perfect if E = (E')'. We give a sufficient condition for an FD-space to be perfect. We also show that if E is a homogeneous FD-space then, E" can be continuously embedded into D. iii Chapter 5 contains some more results on FD-spaces. An FD-space E is said to admit conjugation if 0.0 f, the conjugate of f, is in E for every f in E. We state a necessary and sufficient condition for a distribution to have its conjugate in the same space. Translation invariance and homogeneity of the dual space E* are discussed and it is proved that reflexivity of E implies the homo- geneity of E as well as E*. We show that a homogeneous FD-space admits convergence if and only if {SnIneN}(Snf is the n-th partial sum of Fourier series of f) is an equiconti- nuous family of operators on E. An FD-space E is said to be a character FD-space if f in E implies that enf is in E for every integer n, where enf(u) = f(enu) for each ueC . If E is a homogeneous character FD-space is and {enInGZ} E, then we show an equicontinuous family of operators on that E admits conjugation if and only if E admits conver-gence. Chapter 6 is devoted to the theory of multiplier operators. Suppose F and G are two spaces of distributions, a complex valued function 4) on Z is aaid to be a. multiplier A of type (F,G) if feF '=>4.fe5G. A linear operator U from F to G is said to be a multiplier operator of type (F,G) if there exists a unique 4) on Z (of type (F,G)) such that feF ---*(Uf)^ = 4).ieyG. We denote by M(F,G) the set of all multiplier operators of type (F,G). If E is a homogeneous FD-space then we prove that (E,C) = and (1,1,E*) = If E is a weakly' sequentially complete homogeneous FD-space, iv then it is proved that (E*,C) = and (E,E) = (E*,E*). For a perfect BD-space we prove that (E,E) = (E',E'). Further, if E is homogeneous also we show that (E,E) and (E*,E*) are isometrically isomorphic to each other. If E is a homogeneous FD-space, and UGM(E,E), then we show that each closed invariant subspace of E is stable under U. If E is a homogeneous BD-space, we define A(E,E) as the space consisting of those functions h of C which can be expressed as CO h=Zf.-kg.withf,1GEarldg.GE*, and i=1 as lifillEllgillE* < co; the norm of h in A(E,E) is taken as i=1 Ilhll 'A(E,E) inf IlifillEllgillE* where the infimum is taken over all representations of h. We show that A(E,E) is a homogeneous BD-space. It is also proved that M(E,E) is isometrically isomorphic to the dual space of A(E,E) and that the span of translation operators is weak*ly dense in M(E,E).