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Nets Oо Subgroups in Locally Compact Groups

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Nets Oо Subgroups in Locally Compact Groups ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XX (1978) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATYCZNE XX (1978) J ose L. B u b io * (Princeton) Nets oî subgroups in locally compact groups Abstract. The approximation of the integral of a function / in a locally compact group by average functions / # defined by subgroups H of the group is studied in some detail, with other related questions and a few applications. 0. Introduction. An old well-known result, due to Kolmogorov, states that given a function f e L1 (T) =и([0,1)), the functions /«0*0 = —n J-JУ /\ р + П— I, n = 1 >2,3,..., i converge in L1 to I = J f(x)dx (see [6.]; УП.4 for a related result). More 0 precisely, if cop denotes the modulus of continuity in L P1 one finds (see [4] or [5]) (0.1) !l/„~i\\r < <opif-, -i-j (1 < P s; со). On the other hand, Jessen proved later (see [2]) that (0.2) /2n (*)->! (a.e.). This type of results also holds if we replace the torus T — [0, 1 ) by the real line B, defining for each / e Ll (B) f r(x) = r j£ f(x + hr) (r> 0) h e Z and making r->0. The convergence in L1 or Lp is local in this case. Our aim is to give a treatment of these questions in the general setting of locally compact groups. Besults of the type (0.1) are Theorems 2,3 and 4 below, and Corollary 3, while Corollary 4 provides the natural extension of Jessen’s result (0.2). All this is studied^'n Section 2. Pre- * Dedicated to Prof. Luis Vigil (Univ. of Zaragoza, Spain) who introduced me to this problem. 454 J. L. Rubio viously, we give the definition and basic facts of fundamental domains for quotient groups in Section 1 . In the definition of fn and fr above, the f 1 n - 1) subgroups Hn = <0,— -------> and Hr = rZ give rise to funda- [ n n j mental domains (0,1 jn) in T and (0, r) in B. Similarly for Tn (or Тш) and B n. If / e L l{T) and ektn(x) = (f(x)e~2nikx)n one easily verifies П (0.3) f(x) = 2 скш+1(х)еык*. k=—n Thus, (0.1) and (0.2) seem to lead (only formally) to convergence of Fourier series. However, this approach only gives estimates like \\sn -flip < (const) cop 1 ogn which for p = 1 or p — oo means the Dini-Lipschitz test and its integral analogue (see [ 6]; I I .6 and TV. Example 7). Elementary inversion formulae like (0.3) are also valid for Tn and Bn. The extension to other groups is given in Section 3, Theorem 7. Finally, the fundamental domains associated to an increasing sequence of subgroups behave in some sense like the rectangles in Bn, and Section 4 is devoted to the study of the corresponding maximal function. The notation to be used is as follows : G will denote a locally compact group with identity e and left Haar measure m. A normal closed sub­ group of G will always be writtenH (or Hj), mH will be a left Haar measure for H and, given a function / defined on G / я И = jf{xt)dmH(t) H will be defined whenever the right-hand side exists (a.e.). The function f(n{x)) =/н(ж) is then well defined on G/Н, and there is a left Haar measure m on GjH such that Weil’s identity holds: (0.4) J fdm — J fdm (f e L x((r)) G G/H (see [3]; III.4.5). We shall assume m, mH and m adjusted in this way. If the subgroup is щ we write fj and fa, and similarly mj, щ . Moreover, any compact group will be assumed to have total measure 1 . If f e IP (G) and V is a neighbourhood of e, the modulus of con­ tinuity is defined in an obvious way V) = SUP Wf(™)-f(x)\\p- veV Locally compact groups 455 Finally, when G is Abelian, G will be its dual group, with Haar measure m* related to.m so that Plancherel’s theorem has the form: ||/||2 = ||/||2-. 1. Fundamental domains. An open subset V of G is called a fun­ damental domain for the quotient group G/H if these two conditions hold: (1.1) VV~lr\E = {(e}, (1 .2 ) the complement of VH in G is a locally null set. Observe that (1.1) means that the restriction to V of the projection 7i : G-+E/H must be one-to-one. It is clear that no fundamental domain can exist for G/H unless H is discrete. On the other hand, we have an existence result for discrete H. T h eo rem 1. If H is discrete, each open subset W of G such that WH = G, contains a fundamental domain V for G/H. L emma 1 . Let be the family of all neighbourhoods U of e in G which satisfy (a) TJV~lrsH = {e}, (b) the boundary of tz{TJ) has measure zero in GjH. Then % is a basis of neighbourhoods of e in G. Proof. We can start with a neighbourhood U0 of e such that I70 Z7^1n nH = {e}. Let g be a continuous function on GjH with compact support and O < 0 < 1 ; g(e) = 1 ; supp(^) с n( TJ0). For each r, with 0 < r < 1, define Ür = {xeG/H: r<g(x)}. The boundary of Vr is contained in g~x{r). But (J g~l(r) is contained 0<r< l in supp(gf), so that at most countably many of the sets g~l{r) have positive measure. Choose r such that ^-1(r) has measure zero. Then U = U0r\ r\7i~l{Vr) is a neighbourhood of e which satisfies (a) and (b). Q.E.D. Before we prove the theorem, observe that we can reduce it to the case of G being <r-compact, because we can consider in any case a ^-com­ pact open subgroup 8 of G, and a fundamental domain for S/H nS would be also a fundamental domain for G[H. Another observation is that (1.2) is equivalent to (1.2') The complement of ti{V) in G/H is locally null. The equivalence follows from Weil’s identity (0.4). 456 J. L. Rubio Proof of Theorem 1. For each xeG/H, let x e W, TJxeW be such that x = n(x), xUx c W. As G/H is assumed to be cr-compact, choose a sequence {TJj} of open sub­ sets of G satisfying conditions (a) and (b) of Lemma 1, and such that G IE = TJj c W. Now, define =*№ )4U«W }, *=2,3,... i-i oo Then V = (J Vk is open in GjH, and its complement is contained in the union of the boundaries of л( Uk), which is locally null (actually, it has measure zero) due to Lemma 1. Finally, let Vk ^ n - 1(t h)r\Uk, Tc =1,2,3,'..., OO v = u v k- fc=1 The sets Vk are pairwise disjoint and the restrictions of л to each of them are one-to-one. Thus, V satisfies (1.1). Condition (1.2') has already been 'proved because л(У) = V, and V <= U TJk cr W. Q.E.D. *=i The measure mH on H will be the counting measure multiplied by some constant c > 0, namely: c = mH({e}). Then, for any f e L x(G) / ^f{tx)dm{x) “ f fH(a>)dm{x). HV teH tv V teH G Using Weil’s identity and the fact that every g e LX(GIH) has the form g = f for some f e L x(G), we get Corollary 1. Let V be a fundamental domain for GjH. Then 1 j g dm (go л) dm. GjH G f Furthermore, the measure of all fundamental domains is the same: m(V) — oo if GjH is not compact, m(V) = e if G/H is compact, Locally compact groups 457 where, in the second case, we mnst choose c = ты{(е}) so that the total measure of GjE is 1. A trivial remark is that if У is a fundamental domain for GjE, so are F -1, у V and Vy, for any yeG . Assume now that G is Abelian, H is discrete and G/E is compact. The same properties are true for G, E° and GjE°, where E° = [sc eG: S(E) = {1 }). Then the dual measures of m, mH and m are precisely m*, (m*)~ and (m*)Eo (see [3]; V.5.4), where mH and (т*)но are adjusted so that both quotient groups have total measure 1. Combining this with the proceeding corollary, we find Co r o lla r y 2 . Let G be Abelian, E discrete and G IE compact. I f V and W are fundamental domains for G IE and GjE0, respectively, then m(V) m*(W) = 1 . 2. Nets of subgroups. In this section we study the approximation of the integral of a function / (or the function itself) by functions of type /я? when the subgroup E becomes very large (or very small). T h eo r em 2 . Let V be a relatively compact open neighbourhood of e such that Y E = G. Let f e L 1n L p(G), with integral I — J fdm.
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