THE DIRICHLET-JORDAN TEST AND MULTIDIMENSIONAL EXTENSIONS

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1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 129, Number 4, Pages S (00) Article electronically published on October 10, 2000 THE DIRICHLET-JORDAN TEST AND MULTIDIMENSIONAL EXTENSIONS MICHAEL TAYLOR (Communicated by Christopher D. Sogge) Abstract. If F is a foliation of an open set Ω R n by smooth (n 1)- dimensional surfaces, we define a class of functions B(Ω, F), supported in Ω, that are, roughly speaing, smooth along F and of bounded variation transverse to F. We investigate geometrical conditions on F that imply results on pointwise Fourier inversion for these functions. We also note similar results for functions on spheres, on compact 2-dimensional manifolds, and on the 3-dimensional torus. These results are multidimensional analogues of the classical Dirichlet-Jordan test of pointwise convergence of Fourier series in one variable. Suppose f L 1 (R n ), with Fourier transform (1) ˆf(ξ) =(2π) n/2 f(x)e ix ξ dx. We set (2) S R f(x) =(2π) n/2 ξ R ˆf(ξ)e ix ξ dξ. When n = 1, the Dirichlet-Jordan test for pointwise convergence of S R f(x) as R states that, if f has bounded variation, then for each x R, (3) lim S Rf(x) = 1 R 2 lim [f(x + ε)+f(x ε)]. ε 0 This can be established as follows. Pic a function h(t),equalto0fort<0, 1 for 0 <t 1, smooth on (0, ), and equal to 0 for t 2. Set h(0) = 1/2. By Riemann s localization principle there is no loss of generality in assuming f has compact support. If f has bounded variation, its distributional derivative f = µ is a (signed) measure, and we have (4) f(x) = h(x y) dµ(y)+g(x), with g C0 (R). If f(x) is adjusted to equal the right side of (3) at each point of discontinuity, then (4) holds for all x R. Thenwehave (5) S R f(x) = S R h(x y) dµ(y)+s R g(x). Received by the editors April 29, 1999 and, in revised form, June 22, Mathematics Subject Classification. Primary 42B08, 35P10. Key words and phrases. Fourier series, Dirichlet-Jordan test, Gibbs phenomenon. The author was partially supported by NSF grant DMS c 2000 American Mathematical Society

2 1032 MICHAEL TAYLOR Obviously S R g(x) g(x) for all x. The Dirichlet-Jordan result can then be proven using the following two properties of S R h: (6) S R h(x) h(x), for every x R (including x = 0), and, for some C<, independent of x, R, (7) S R h(x) C. To establish (6), one can appeal to the Dini test, or use localization and smoothness for x 0, plus a symmetrization argument to cover the case x = 0. The bound (7) is a consequence of the analysis of the Gibbs phenomenon for S R h.fromthis, the Dirichlet-Jordan result can be deduced via Lebesgue s dominated convergence theorem. Let us state an abstract version of this last segment of the argument. Let (Y,B) be a set with sigma algebra, let µ be a finite signed measure on B, and let X be a set. Let h R : X Y C be given, for each R (0, ). Assume that h R (x, ) isb-measurable, for each x X, R (0, ), that (8) h R (x, y) C, x X, y Y,R (0, ), and that (9) lim h R(x, y) =h(x, y), x X, y Y. R Then (10) lim h R (x, y) dµ(y) = h(x, y) dµ(y), x X. R Y Y As mentioned, this is simply a consequence of the dominated convergence theorem. The role played by X here is, in essence, trivial, except for the fact that it arises in nontrivial contexts. Multidimensional analogues of functions for which (6) (7) hold arise as follows. Let Σ be a smooth (n 1)-dimensional surface in R n.letc 1 (Σ) denote the set of caustic points of order 1, in the terminology used in 10 of [PT]. (This follows Definition of [Dui], in the case where Λ is the Lagrangian flow-out of the unit normal bundle of Σ.) Let O Σ be an open neighborhood of C 1 (Σ). Let h(x) bea piecewise smooth function, with compact support, with simple jump across Σ. For x Σ, set h(x) equal to the mean value of its limits from each side. The fact that (11) S R h(x) h(x), x R n \O Σ, follows from Proposition 26 in 10 of [PT] (the result for x Σ holding by the analysis in 11). The fact that, for any compact K R n \O Σ, (12) S R h(x) C K, R (0, ),x K, follows from the analysis of the Gibbs phenomenon in 11 of [PT] (cf. also [CV]). We note that C 1 (Σ) is empty when n =2. Also,whenn =3,C 1 (Σ) is empty if Σ is real analytic and not part of a sphere (as noted by [K]). Now suppose we have a foliation of an open set Ω R n by such surfaces. More precisely, suppose we have smooth functions u 1,...,u n 1,v on Ω, producing a diffeomorphism (13) (u 1,...,u n 1,v):Ω Q R n,

3 THE DIRICHLET-JORDAN TEST AND MULTIDIMENSIONAL EXTENSIONS 1033 where Q istheopencube(, π) (, π). We consider the family of surfaces Σ c = {x Ω:v(x) =c}. Assume that O is an open neighborhood of the union of the sets C 1 (Σ c ). Fix ϕ C0 (Ω). Let h t :Ω R be given by 1 if v(x) >t, h t (x) = 1 2 if v(x) = t, 0 if v(x) <t. Let K be any compact set in R n \O. Then, for each g C (Ω), we have (14) S R (gϕh t )(x) C K (g), R (0, ),x K, t I =(, π). Hence, if we set Φ(g)(R, x, t) =S R (gϕh t )(x), we have (15) Φ:C (Ω) L ((0, ) K (, π)). Now, if we compose this with the inclusion ι : L ((0, ) K (, π)) L loc ((0, ) K (, π)), it is easy to see that the map ι Φ:C (Ω) L loc((0, ) K (, π)) is continuous. It follows that the map Φ in (15) has closed graph. Hence, we can apply the closed graph theorem and deduce that (16) sup S R (gϕh t )(x) C K g H (Ω), l x K,t I,R (0, ) for some finite l. This estimate can also be demonstrated by a recollection of what maes geometrical optics constructions wor, up to any given finite order, and its implementation for the analysis of the Gibbs phenomenon in [PT]. (It would be of interest to study the optimal value of l, but we will not pursue this here. We will stipulate that l>n/2.) Now, if µ is a finite (signed) measure on I we can say that, for each g H l (Ω), (17) f(x) = g(x)ϕ(x)h t (x) dµ(t) S R f(x) f(x), x R n \O. I This class of synthesized functions is somewhat constrained, but it will serve as a starting point for an analysis of a much more natural class of functions, which we will now introduce. Let Ω R n be open and let F = {Σ c : c I} be a foliation of Ω by smooth (n 1)-dimensional surfaces. Let M(Ω) denote the space of finite (signed) Borel measures on Ω. We say (18) f B(Ω, F) if f is a compactly supported element of L (Ω) with the property that (19) X 1 X f M(Ω), for any, and any smooth vector fields X 1,...,X on Ω, provided that at most one of them is not tangent to F. One would have the same class of functions if one insisted the one exceptional vector field be X 1 (or that it be X ). The following is our main result. Theorem 1. Given f B(Ω, F), there exists a Borel measurable f, equaltof a.e., such that, as R, (20) S R f(x) f(x), x R n \O, where O is a neighborhood of the union of C 1 (Σ c ), c I.

4 1034 MICHAEL TAYLOR To begin the proof, we note that B(Ω, F) is clearly a module over C0 (Ω). Hence, using a partition of unity, we can assume that Ω is as in (13), and Σ c = {v = c}. Use the inverse of the diffeomorphism in (13) to pull f bac to a compactly supported element g L (Q), with the property on ω = g/ x n that (21) M T ω M(Q), M =0, 1, 2,..., where (22) T = 2 x x 2. n 1 For the first n 1 factors of (, π) inq, throw in the endpoints and identify them, to regard ω as a compactly supported measure on T n 1 (, π). We have, for ϕ continuous on [, π], (23) ϕ(t)e i x, M T ω C M ϕ L, with x =(x 1,...,x n 1 ) T n 1, Z n 1,so (24) ϕ(t)e i x,ω C M M ϕ L. Hence, we have measures µ on (, π), supported on [ a, a] forsomea<π,such that (25) µ M(I) C M 2M, ω = e i x µ, where the norm denotes the total variation of µ. Hence y (26) g(x,y)= e i x dµ (t), so (27) f(x) = v(x) e i u(x) dµ (t) = ϕ(x) v(x) e i u(x) dµ (t) = v(x) ϕ(x)g (x) dµ (t), where we choose ϕ C0 (Ω) equal to 1 on the support of f, andsetg (x) =e i u(x), with u(x) =(u 1 (x),...,u n 1 (x)). The estimates done above imply convergence in sup-norm of the infinite series, to a function f(x) equal a.e. to f(x). The analysis done above also shows that, for (28) f (x) =ϕ(x)g (x) v(x) dµ (t), we have (29) S R f (x) f (x), x R n \O, and, for each compact K R n \O, (30) sup R (0, ),x K S R f (x) C K g H (Ω) µ l M(I).

5 THE DIRICHLET-JORDAN TEST AND MULTIDIMENSIONAL EXTENSIONS 1035 Now (31) g H l (Ω) C l, so, given N, we can produce M = M(l, N) and apply (25) to obtain (32) sup R (0, ),x K S R f (x) C KN N. Thus, from (27), we have, for x K, (33) lim S Rf(x) = f (x) = f(x), R and the theorem is proven. It is clear what sort of representative of the class of f B(Ω,F) the function f(x) is.ifx 0 Σ c Ω, then f(x 0 ) is the mean of the limit of f(x) asx x 0 from within {v(x) >c} and as x x 0 from within {v(x) <c}. In particular, for each x 0 Ω, 1 (34) f(x 0 ) = lim r 0 V n r n f(x 0 + y) dy, y <r where V n is the volume of the unit ball in R n. There are other Riemannian manifolds M besides R n for which there are analogues of Theorem 1, with (35) S R f(x) =χ R ( )f(x), where is the Laplace-Beltrami operator on M and χ R (λ) is1for λ <R,0 for λ >R,and1/2 for λ = R. Oneclassofexamplesistheclassof strongly scattering manifolds, in the terminology of [PT], 10. Using the compactification tric from 6 of [PT], we can extend Theorem 1 to the case where M is a sphere S n, or other compact ran-one symmetric space. Using results of [BC], we can extend Theorem 1 to compact 2-dimensional manifolds (and then O is empty). Using Theorem 5.4 of [T], we can extend Theorem 1 to the case M = T 3,aslong as all the leaves Σ c of F have nonzero Gauss curvature in Ω. References [BC] L. Brandolini and L. Colzani, Localization and convergence of eigenfunction expansions, J. Fourier Anal. 5 (1999), CMP 2000:12 [CV] L. Colzani and M. Vignati, Gibbs phenomena for multiple Fourier integrals, J.Approximation Theory 80 (1995), MR 95:42021 [Dui] J. J. Duistermaat, Fourier Integral Operators, Birhäuser, Boston, MR 96m:58245 [K] J.-P. Kahane, Le phénomène de Pinsy et la géométrie des surfaces, CRAS Paris 321 (1995), MR 96m:42018 [PT] M. Pinsy and M. Taylor, Pointwise Fourier inversion: a wave equation approach, J. Fourier Anal. 3 (1997), MR 99d:42019 [T] M. Taylor, Pointwise Fourier inversion on tori and other compact manifolds, J. Fourier Anal. 5 (1999), CMP 2000:12 Department of Mathematics, University of North Carolina, Chapel Hill, North Carolina address: met@math.unc.edu