Fourier Expansions of Functions with Bounded Variation of Several Variables

In the first part of the paper we establish the pointwise convergence as t → +∞ for convolution operators ∫ Rd tdK (ty)φ(x− y)dy under the assumptions that φ(y) has integrable derivatives up to an order α and that |K(y)| ≤ c (1 + |y|)−β with α+β > d. We also estimate the Hausdorff dimension of the set where divergence may occur. In particular, when the kernel is the Fourier transform of a bounded set in the plane, we recover a two-dimensional analog of the Dirichlet theorem on the convergence of Fourier series of functions with bounded variation. In the second part of the paper we prove an equiconvergence result between Fourier integrals on euclidean spaces and expansions in eigenfunctions of elliptic operators on manifolds, which allows us to transfer some of the results proved for Fourier integrals to eigenfunction expansions. Finally, we present some examples of different behaviors between Fourier integrals, Fourier series and spherical harmonic expansions. Fourier integrals on Euclidean spaces This section is devoted to the study of the pointwise convergence as t → +∞ for operators of the type ∫ Rd χ ( t−1ξ ) φ̂(ξ) exp(2πiξx)dξ = ∫ Rd tK (ty)φ(x− y)dy. Here φ(x) is an integrable function, φ̂(ξ) = ∫ Rd φ(x) exp(−2πiξx)dx is its Fourier transform, χ(ξ) is an integrable Fourier multiplier andK(x)= ∫ Rd χ(ξ) exp(2πiξx)dξ is the associated convolution kernel. For example, when the multiplier is the characteristic function of a bounded domain which contains the origin, the problem reduces to the pointwise inversion of Fourier transform, lim t→+∞ ∫ tΩ φ̂(ξ) exp(2πiξx)dξ = φ(x). A classical reference for this problem is [3]. Other references more directly related to this paper are [1], [4], [5], [6], [15], [16], [19], [23], [24], [25] which are devoted to Fourier expansions of piecewise smooth functions, [2] and [22], which estimate the capacity of the divergence sets of one-dimensional Fourier series, [7], [12], [17], [26], which contain results on spherical summability of Fourier integrals of functions in Sobolev classes, [18], with a simple proof of the almost everywhere convergence of expansions in eigenfunctions of functions in Sobolev classes. Finally, a special mention should be made to the research tutorial [21]. In order to motivate what Received by the editors April 26, 2004 and, in revised form, November 16, 2004. 2000 Mathematics Subject Classification. Primary 42B08, 43A50. c ©2006 American Mathematical Society Reverts to public domain 28 years from publication 5501 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 5502 LEONARDO COLZANI follows, let us consider two extreme cases. If the function φ(x) has integrable derivatives up to an order α > d, then |φ̂(ξ)| ≤ c (1 + |ξ|) is integrable. Hence, if χ (ξ) is bounded and continuous at the origin, then lim t→+∞ ∫ Rd χ ( t−1ξ ) φ̂(ξ) exp(2πiξx)dξ = χ (0)φ(x). If the kernel K(x) has the decay |K(x)| ≤ c (1 + |x|) with β > d, then K(x) is integrable. Hence, if φ(y) is bounded and continuous at the point x, then lim t→+∞ ∫ Rd tK (ty)φ(x− y)dy = φ(x) ∫ Rd K (y) dy. Here we want to consider an intermediate case between these two extremes and, roughly speaking, our result is that α derivatives of the function expanded and a decay of order β of the summation kernel with α + β > d are sufficient conditions for the pointwise inversion of the Fourier transform, with a possible exception of a set of points with Hausdorff dimension at most d − α. In what follows the smoothness of a function is measured with the Riesz potentials or fractional powers ∆ of the Laplace operator ∆ = − ∑ j ∂ /∂xj , defined spectrally by ∆̂α/2φ(ξ) = (2π |ξ|) φ̂(ξ). However, instead of ∆ one can also use the gradient∇ = {∂/∂xj}. Here it is our basic result. Theorem 1. Let χ(ξ) be an integrable Fourier multiplier, smooth in a neighborhood of the origin with χ(0) = 1. Assume that the associated convolution kernel K(y) has decay |K(y)| ≤ c (1 + |y|)−β. Finally, let φ(y) be an integrable function and assume that ∆φ(y) is a locally finite measure. Then for every γ > 0, ∣∣∣∣∫ Rd χ ( t−1ξ ) φ̂(ξ) exp(2πiξx)dξ − φ(x) ∣∣∣∣ = ∣∣∣∣∫ Rd tK (ty)φ(x− y)dy − φ(x) ∣∣∣∣ ≤ c ∫ Rd t (1 + t |y|) |φ(x− y)− φ(x)| dy +c ∫ Rd td−α (1 + t |y|)−β ∣∣∣∆α/2φ(x− y)∣∣∣ dy. Observe that if χ(ξ) and φ(y) are integrable, then K (y) and φ̂(ξ) are bounded and this guarantees the existence of the integrals ∫ Rd χ ( t−1ξ ) φ̂(ξ) exp(2πiξx)dξ, ∫ Rd tK (ty)φ(x− y)dy. However, this integrability assumption on χ(ξ) and φ(y) can be weakened, as soon as the above two integrals are well defined. It is not difficult to convert the theorem into a convergence result. Theorem 2. Let χ(ξ) be an integrable Fourier multiplier, smooth in a neighborhood of the origin with χ(0) = 1. Assume that the associated convolution kernel K(y) has decay |K(y)| ≤ c (1 + |y|)−β. Finally, let φ(y) be an integrable function and assume that (1 + |y|)α−d ∆φ(y) is a finite measure. If α+ β > d, then the convergence lim t→+∞ ∫ Rd tK (ty)φ(x− y)dy = φ(x) License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use FUNCTIONS WITH BOUNDED VARIATION 5503 holds with the possible exception of a set of points with Hausdorff dimension at most d − α. More precisely, assume that at a point x there exists 0 < ε < 1 such that when r → 0+, ∫ {|y|≤r} ∣∣∣∆α/2φ(x− y)∣∣∣ dy ≤ crd−α+ε. Then, when t → +∞, ∣∣∣∣∫ Rd tK (ty)φ(x− y)dy − φ(x) ∣∣∣∣ ≤ ⎨⎩ ct−ε if ε < α+ β − d, ct−ε log(t) if ε = α+ β − d, ctd−α−β if ε > α+ β − d. A by product of the above result is another proof of the well-known fact that the singularities of functions in Sobolev classes of index α have Hausdorff dimension at most d−α. On the other hand, there are functions in these spaces which are infinite on sets of dimension d− α. Hence the estimate of the dimension of the divergence set is best possible. Later we will show that also the condition α + β > d is best possible. By substituting the square root of the Laplace operator with the gradient, one can apply the theorem to functions with bounded variation, that is, integrable functions whose distributional first derivatives are finite measures. In particular, the theorems apply to piecewise smooth functions, which are linear combinations of functions ψ(x)χD(x), products of smooth functions and characteristic functions of bounded domains with smooth boundary. The content of the following theorem is that in this case convergence holds everywhere, even along the discontinuities of the functions expanded. Theorem 3. Let χ(ξ) be an integrable Fourier multiplier, with compact support, smooth in a neighborhood of the origin and with χ(0) = 1. Assume that the associated convolution kernel K(y) has decay |K(y)| ≤ c (1 + |y|)−β for some β > d− 1. If φ(y) is a piecewise smooth function suitably normalized along the discontinuities, then at every point x, lim t→+∞ ∫ Rd χ ( t−1ξ ) φ̂(ξ) exp(2πiξx)dξ = lim t→+∞ ∫ Rd tK (ty)φ(x− y)dy = φ(x). Classical examples of Fourier multipliers to which the above theorems apply are the Bochner-Riesz multipliers. Corollary 4. The Bochner-Riesz multipliers (1−|ξ|2)+ are associated to the kernels π−γΓ(γ + 1) |y|−γ−d/2 Jγ+d/2 (2π |y|), which have decay c (1 + |y|)−γ−(d+1)/2. The Bochner-Riesz means with index γ > (d − 1)/2 − α of functions with α integrable derivatives converge, with a possible exception of a set of points with Hausdorff dimension at most d− α, lim t→+∞ ∫ {|ξ| 1. If the domain is convex with a smooth boundary with positive Gauss curvature, the exponent becomes −(d + 1)/2. In particular, in dimension one when the domain is an interval, one recaptures the Dirichlet theorem on the convergence of Fourier License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 5504 LEONARDO COLZANI integrals of functions with bounded variation, and in dimension two, if the domain has just a bit of smoothness, one obtains a generalization of this Dirichlet theorem. Corollary 5. Let Ω be a bounded planar domain containing the origin and assume that |χ̂Ω(y)| ≤ c (1 + |y|)−β for some β > 1. If φ(y) is an integrable function with bounded variation, then, with a possible exception of a set of points with Hausdorff dimension at most 1,