# Fourier Analysis on Local Fields. (MN-15)

M. H. TAIBLESON
Pages: 306
https://www.jstor.org/stable/j.ctt130hkh3

1. Front Matter
(pp. i-iv)
2. Preface
(pp. v-vi)
3. Introduction
(pp. vii-x)

With the appearance of the paper of Gelfand and Graev [1] on representations of SL(2,K) and special functions on local fields, and the observation that most of the techniques used in that paper were, in form, rather straightforward extensions of well known methods in euclidean theory, Paul Sally and I set out on a program of developing and extending a few basic results that were needed for an elementary treatment of representation theory (Sally and Taibleson [1]). A review of related work in representation theory and algebraic number theory (See Lang’s treatment of Tate’s results, Lang [1], and other references...

(pp. xi-xii)
5. Chapter I: Introduction to local fields
(pp. 1-19)

This chapter contains a description of several examples of local fields and a review of basic facts about the classification and structure of local fields.

The Rademacher functions are defined on [0,1] by the rule, φk(x) = sgn(sin 2k+1πx), k=1,2,3,····. It is easy to see that the sequence {φk}k=1is orthonormal on [0,1] with respect to ordinary Borel-Lebesgue measure. In the language of probability theory we would say that the sequence of Rademacher functions is a sequence of uncorrelated random variables, each with mean zero and variance 1. While it is not crucial for the development that follows, it is...

6. Chapter II: Fourier analysis on K, the one-dimension case
(pp. 20-114)

This chapter, in §1-§3, contains a treatment of the Fourier transform on K, as an additive group, which includes the theory of distributions on K. In §4 the Mellin transform (the Fourier transform on K-{0} as a multiplicative group is discussed and then in §5 the additive and multiplicative structures are meshed in a treatment of special functions on K. In §6 we treat a special topic; Fourier series on the ring of integers of K.

We assume, in general, that all functions are complex-valued and (Borel) measurable.

Lp= Lp(K), 1 ≤ p ≤ ∞, are the usual spaces,...

7. Chapter III. Fourier analysis on Kn.
(pp. 115-167)

In §l-§3 of this chapter we extend the fundamental results of Fourier analysis from the one-dimensional to the n-dimensional case, covering the L¹ and L² theory in §1-§2 and the theory of distributions in §3. In §4-§9 we treat the theory of fractional integration, Riesz and Bessel in some detail. In §10 we take a brief look at the theory of operators that commute with translations.

We will record here the main facts needed for analysis on Kn, the n-dimensional vector spaces over K. For the most part, the material in §1-§3 of this chapter is a transplantation of §1-§3...

8. Chapter IV. Regularization and the theory of regular and sub-regular functions
(pp. 168-194)

In §l of this chapter we systematically investigate the regularization kernels R(x,k) and the regularizations f(x,k) for f ∈ *. We study the kernels R(x,k) as analogues of the Poisson kernels and the regularizations f(x,k) as analogues of harmonic functions. After a brief look (in §2) at Lipschitz theory we turn in §3 to notions of sub-regular functions, least regular majorants and of non-tangential convergence.

Consider the Poisson kernel P(x,y), x ∈ Rn, y > 0. The properties of P(x,y) that are of crucial interest are:

(a) P(x, y) = y-nP(xy-1, 1), y > 0; (b) P(x, y) ≥ 0; (c)...

9. Chapter V. The Littlewood-Paley function and some applications
(pp. 195-216)

In §l we introduce the Littlewood-Paley functions gp(·;f), and study the relation between the LPproperties of f and gs(·;f). In §2 we introduce a truncated version of g2(·;f), Sf, and study the local equivalence of the n.t. convergence of f(x,k), n.t. boundedness of f(x,k), and existence of Sf(x). (n.t. = “non-tangential”)

Definition. If f(x,k) is regular on Kn× Z we define the Littlewood-Paley functions gp(·;f) by

\mathrm{g_{\infty}(x;f) = sup_{k \epsilon z}|f(x,k) - f(x,k-1)|}

\mathrm{g_{p}(x;f) = [\Sigma _{k\epsilon z}|f(x,k) - f(x,k-1)|^{p}]^{1/p},}1 ≤ p < ∞.

If F is the distribution to which f(x,k) converges we will also write gP(x;F) = gP(x;f).

Lemma (1.1). If f ∈ L² then...

10. Chapter VI. Multipliers and singular integral operators
(pp. 217-240)

In §l and §2 we give a brief overview of the theory of LP-multipliers on Knand in §3 give an important application of the multiplier theory for Fourier series. In §4 we record, without proofs, the important facts about the singular integral theory.

1. Multipliers We say that the measurable function m is a multiplier on LP(1 ≤ P ≤ ∞) if\mathrm{(m\hat{\varphi })^{\vee }}∈ LPfor all φ ∈ * and there is a constant A > 0, independent of φ such that\mathrm{\left \| (m\hat{\varphi }^{\vee }) \right \|_{P}\leq A\left \| \varphi \right \|_{P}}.

Since the map\mathrm{\varphi \rightarrow (m\hat{\varphi })^{\vee }}commutes with translations it follows (III (7.5)) that m is also...

11. Chapter VII. Conjugate systems of regular functions and an F. and M. Riesz theorem
(pp. 241-261)

Lemma (1.1). Let F(x, k) = (f0(x, k), f1(x, k),…,fm(x, k)) be a vector-valued function with each component fj(x, k), j = 0, 1,…,m being regular on Kn× Z. Suppose there is a P0, 0 < P0< 1, such that |F(x,k)|P0is sub-regular, with\mathrm{|F(x,k)| = [\Sigma _{j=0}^{m}|f_{j}(x,k)|^{2}]^{1/2}}. Suppose further that for some P > P0, and A > 0,

(1.2) ∫Kn|F(x, k)|Pdx ≤ A < ∞ for all k ∈ Z,

then,

(a)\mathrm{f_{j}(x)=\underset{k\rightarrow \infty }{lim} f_{j}(x,k)exists a.e., j = 0, 1,…,m.

(b)\mathrm{\underset{k\rightarrow \infty }{lim} \int _{K^{n}}|F(x,k)-F(x)|^{P}}dx = 0, with F(x) = (f0(x), f1(x),…,fm(x)).

Moreover,

(c)\mathrm{f_{j}^{*}(x) = sup_{k\epsilon Z}|f_{j}(x,k)| \epsilon L^{P}(K^{n}),j = 0,1,\cdots ,m.}

Proof. Let P1= P/P0> 1....

12. Chapter VIII. Almost everywhere convergence of Fourier series
(pp. 262-285)

Theorem (1.1). If f ∈ LP(\mathfrak{D}), 1 < p < ∞, and\mathfrak{M}f(x) = supn|Snf(x)|, then Mf∈ LPand there is a constant AP> 0 independent of f such that ||\mathfrak{M}f||P≤ AP||f||P.

Corollary (1.2). If f ∈ LP(\mathfrak{D}), 1 < P ≤ ∞, then Snf(x) → f(x) a.e. as n → ∞

Proof. We may assume 1 < p < ∞. Since * is dense in LP, we approximate f in * and then use (1.1). Thus f = b+g, ||b||P< ∈, g ∈ *. Sng = g for in large enough, and thus for n large enough, lim...

13. Bibliography
(pp. 286-294)
14. Back Matter
(pp. 295-295)