Some remarks on restriction of the Fourier transform for general measures

Publicacions Matem`atiques, Vol 43 (1999), 655–664. SOME REMARKS ON RESTRICTION OF THE FOURIER TRANSFORM FOR GENERAL MEASURES Per Sj¨ olin and Fernando Soria Abstract In this paper we establish a formal connection between the average decay of the Fourier transform of functions with respect to a given measure and the Hausdorff behavior of that measure. We also present a generalization of the classical restriction theorem of Stein and Tomas replacing the sphere with sets of prefixed Hausdorff dimension n−1+α, with 0 <α<1. Introduction Let νbe a positive, finite Borel measure in Rn,n≥1. Assume also that νhas compact support. We define for a given f∈L1(Rn) and for R>0 the R-average of the Fourier transform of fwith respect to the measure νas σ(f)(R)=σν(f)(R)=| f(Rξ)|2dν(ξ). In this work we shall study the following problem: Assuming that 0<α≤n, for which values of βdoes there exist a constant C=Cα,β such that the estimate (1) σ(f)(R)≤C RβRn | f(ξ)|2dξ |ξ|n−α,R>1 holds for all fin, say, C∞ c(Rn) with supp f⊂B1? (Here Brdenotes the ball of radius r>0 centered at the origin.) Research partially supported by the European Commission, TMR network “Harmonic Analysis” and by the G¨oran Gustafsson Foundation. 656 P. Sj¨ olin, F. Soria Estimates of this sort have been considered in [5], [1], [9] and [10] for the case dν =dx|Sn−1=dω, the Lebesgue measure on the unit sphere (see also [4]). In these papers the authors study the minimum Hausdorff dimension that a set Amust have to ensure that the “distance set” {|x−y|:x, y ∈A}has positive measure on R+. For that, they have to look at the problem only for the case in which f(x)≥0. This is of significant importance as we will see below. In [2] and [3] a similar problem is considered for dν =dω again, and for β=n−1. However, the right hand side of (1), i.e., the norm of fin the Sobolev space ˙ Hα−n 2is replaced by a larger one. In fact, (1) cannot hold for any value of αif we insist in asking βto be n−1, except when α=n(see Theorem 3). The results of our work, as most of the papers mentioned above, describe some aspects of the behavior of the Fourier transform when restricted to lower dimensional sets. In the classical theory, the curvature of the sets plays a central role. We study here a slightly different problem, the average decay of the restriction with respect to arbitrary measures, and, as we will see, it is the geometry given by the Hausdorff behavior of our sets what really matters in this case. Also, in the last section we investigate a generalization of the usual restriction theorems replacing the sphere with sets of prefixed Hausdorff dimension n−1+α, with 0<α<1. Main results In the following theorems we shall use the condition (2) ν(Bt(x0)) ≤Ct γ,t>0,x 0∈Rn, where γ≥0 and Bt(x0) denotes the ball {x:|x−x0|<t}. Theorem 1. If (2) holds then (1) holds with β=α+γ−nfor all f of the above type and 0<α≤n. As we will see below, this theorem is sharp for certain measures, like the Lebesgue measure associated to the unit sphere. In general we have the following converse statement Theorem 2. Assume that (1) holds for all fof the above type and for some β≥0and some α,0<α≤n. Then (2) holds with γ=β. Restriction of the Fourier transform 657 Proof of Theorem 1: Let us observe that σ(f)(R)=Rn f(Rξ) f(Rξ)dν(ξ) =RnRnRn e−iRξ·(x−y)dν(ξ)f(x)f(y)dx dy =RnRnν(R(x−y))f(x)f(y)dx dy. Choose now φ∈C ∞ c(Rn) so that φ(x)=1for|x|≤2 and with φeven. Then, using the support condition on fwe have σ(f)(R)=RnRnν(R(x−y))φ(x−y)f(x)f(y)dx dy =RnRn KR(y−x)f(x)f(y)dx dy =Rn KR∗f(y)f(y)dy, where KR(x)=ν(−Rx)φ(x). Hence, by Plancherel σ(f)(R)=cRn KR(ξ) f(ξ) f(ξ)dξ and, therefore, suffices to prove (3) | KR(ξ)|≤ C Rβ 1 |ξ|n−α with β=α+γ−n. To do this observe that  KR(ξ)= φ(ξ−Ry)dν(y). Now, if |ξ|≥AR (for Alarge enough depending on the diameter of the support of ν) then | KR(ξ)|≤CN|ξ|−Nfor any positive integer N. This follows easily from the fact that  φis a Schwartz function. Thus (3) holds in this case. It remains to study the case |ξ|≤AR. For that we use 658 P. Sj¨ olin, F. Soria condition (2): | KR(ξ)|≤CN1 1+|ξ−Ry|Ndν(y) ≤CN|ξ−Ry|<1 dν(y)+ ∞  k=1 2k−1≤|ξ−Ry|<2k 2−kN dν(y) ≤CN ∞  k=0 2−kN νB2k R (ξ/R) ≤C ∞  k=0 2−kN 2k Rγ ≤CR −γ, for sufficiently big N. Now (3) follows if R−γ≤CR−β|ξ|α−n. That is, if |ξ|n−α≤CRγ−β. But this holds since n−α=γ−β≥0 and |ξ|≤AR. This finishes the proof of Theorem 1. We are thankful to the referee for pointing out the connection between our result and a previous result of Strichartz in [12] where he shows that if the measure νhas property (2) (what he calls the locally uniformly γ-dimensional property) then one has |x|≤R | fdν(x)|2dx ≤CRn−γ||f||2 L2(ν). As can be seen readily, this corresponds to the dual statement of our theorem in the special case α=n. Originally we were not interested in this end point case. However, after the above observation by the referee and the fact that our arguments trivially covered it, we decided to include it for completeness. Proof of Theorem 2: Choose gan even function in the Schwartz class S such that supp g⊂B1 2and |g(x)|≥1onB1. Fix t, with 0 <t<1, and x0∈Rn. Set R=1 tand define f∈Sby  f(Ry)=g(R(y−x0)). It follows that  f(y)=g(y−Rx0) and f(x)=(2π)−neiRx0·xg(x). Now, assuming that (1) holds for some αand some βwe have ν(Bt(x0)) ≤Rn |g(R(y−x0))|2dν(y) ≤C RβRn |g(y−Rx0)|2dy |y|n−α. We conclude that ν(Bt(x0)) ≤Cgtβ, since Rn|g(y−y0))|2|y|α−ndy is bounded by a constant independent of y0whenever 0 <α≤n. Restriction of the Fourier transform 659 Remarks. If we restrict ourselves to the case in which the functions are assumed to be positive we could get better estimates for (1) than those obtained in Theorem 1, at least for certain values of α. For instance, suppose that we know the rate of decay of the Fourier transform of dν; that is, suppose that (4) |ν(ξ)|≤ C |ξ|η 2 , for some η≥0. Then we have the following estimate for every 0 <α≤η 2 σ(f)(R)≤C RαRn | f(ξ)|2dξ |ξ|n−α provided that f≥0. The proof of this fact is straightforward. Write as in Theorem 1 σ(f)(R)=RnRnν(R(x−y))f(x)f(y)dx dy. Now, using that νis finite and (4) we see that |ν(ξ)|≤Cmin 1,1 |ξ|η 2≤C |ξ|α if α≤η/2. Hence, if f≥0 we conclude σ(f)(R)≤CRnRn f(x)f(y) (R|x−y|)αdx dy =C RαRn | f(ξ)|2dξ |ξ|n−α. The best value of ηfor which (4) holds (the supremum of these values, rather) is sometimes called the Fourier exponent of ν. More precisely, the Fourier dimension of a compact set Eis defined as the supremum of the Fourier exponents of all the probability measures supported in E. For any smooth hypersuface of Rnwith non-vanishing curvature we know that this value is n−1. Thus, if we take dν =dω we have that (1) holds with β=αprovided that 0 <α≤n−1 2and that fis assumed positive. One might think from this that Theorem 1 is far from optimal. However, for “flat” sets, where the Fourier dimension is 0, it gives positive results about the average decay (1). For instance, if we take Lebesgue measure on the surface of the unit cube, then its Fourier exponent is 0 but nevertheless (1) holds with β=α−1 for every 1 <α≤n. (This was observed also in [10].) 660 P. Sj¨ olin, F. Soria Another classical example, now in dimension 1, is provided by the usual Cantor set. As is well known (see the book by P. Mattila [6, p. 168]) this set has Fourier dimension 0, so nothing will be gained in terms of the average decay of L2-functions from this. However, the standard measure on the set satisfies (2) with γ= log 2/log 3 and, therefore, Theorem 1 gives (1) with β=α+ log 2/log 3 −1. Remember that in the above argument it was essential to have f≥0. In fact, as the next result states, Theorem 1 is sharp in the case dν =dω. Theorem 3. Fix 0<α≤nand set β(α) = sup{β: (1) holds for all f,supp f⊂B1}. Then if we consider the usual measure on the unit sphere, dν =dω we have β(α)=α−1. In other words, what Theorem 3 says is that the average properties (1), at least in the two cases considered of the unit sphere and the surface of the unit cube, are determined by the Hausdorff dimension (n−1in this case) rather than by the Fourier exponent of the natural measures associated to them. Proof of Theorem 3: It is clear that (2) holds with γ=n−1 and invoking Theorem 1 we conclude that (1) holds with β=α−1. Hence, β(α)≥α−1 (see also Theorem 2 in [10]). It remains to prove that β(α)≤α−1. Assume that (1) holds and choose φ∈C ∞ c(R) such that supp φ⊂ (−1 4,1 4) and  φ(0) = 0. Take also ψ∈C ∞ c(Rn−1) with supp ψ⊂B1 2and  ψ(0) = 0. Given R>1 we define Fas F(x1,x )=eiRx1φ(x1)Rn−1 2ψ(R1 2x) where we denote x=(x2,... ,x n). It follows that  F(ξ1,ξ)= φ(ξ1−R) ψ(ξR−1 2) and, as one can easily see, σ(F)(R)≥cR−n−1 2. On the other hand, given %>0wehave Rn | F(ξ)|2dξ |ξ|n−α =| φ(ξ1−R)|2| ψ(ξR−1 2)|21 |ξ|n−αdξ1dξ ≤|ξ1−R|≤R|ξ|≤R1/2+ | φ(ξ1−R)|2| ψ(ξR−1 2)|21 |ξ|n−αdξ1dξ+CR−N ≤CRn−1 2+n+α−n Restriction of the Fourier transform 661 and (1) implies that R−n−1 2≤CR−βRn−1 2+n+α−n. Hence, since %can be taken arbitrarily small we obtain β≤α−1 and β(α)≤α−1. This completes the proof of Theorem 3. An example of Lp−L2restriction. Let n≥2. Recall the classical result of Stein-Tomas about restriction of the Fourier transform to spheres of Rn(see for example [11]) which states that (5) Sn−1 | f(ω)|2dω1/2 ≤C||f||p,1≤p≤2(n+1) n+3 . On the other hand it is easy to see from the Hausdorff-Young inequality that (6) B1 | f(y)|2dy1/2 ≤C||f||p,1≤p≤2. We shall here find some estimates which are intermediate to these results. We could do this for instance by means of an interpolating sequence of measures, defining an analytic family of operators and invoking then the interpolation theorem of Stein in this context. An example of this could be given by the family of measures µz=1 Γ(z)1−|z|2z−1 +. Our example however will be rather on “intermediate sets” between the unit sphere and the unit ball and is more related in spirit to the kind of problems about Hausdorff dimension that we have considered before. More precisely, we shall prove the following Theorem 4. For every α, with 0<α<1, and for 1<p 0<2(n+1−α) n+3−3α one can find a set E⊂(0,1] with dim E=αand such that if we define S=S(E)={x∈Rn:|x|∈E} then there exists a natural probability measure νon Sfor which one has S | f(y)|2dν(y)1/2 ≤C||f||p,1≤p≤p0. 662 P. Sj¨ olin, F. Soria We will obtain this as a consequence of results by Mockenhaupt in [7]. For fixed α,0<α<1, and small %>0 the idea is to consider as above a compact subset Eof [1 2,1] of dimension αwith the property that there exists a probability measure µsupported in Esuch that (7) |µ(ξ)|≤ C |ξ|α 2−, and (8) µ(Bt(x0)) ≤Ct α. A particular construction, using Cantor-like sets and extending previous work of Salem [8], can be found in [7], where the following theorem is also proved Theorem A. Let νbe a compactly supported positive measure in Rn which satisfies the estimates |ν(ξ)|≤ C |ξ|η 2 ,η>0, and ν(Bt(x0)) ≤Ct γ,γ>0. Then Rn | f(y)|2dν(y)1/2 ≤C||f||p,1≤p<2(2n−2γ+η) 4(n−γ)+η. Proof of Theorem 4: Let us consider a set Eand a measure µsatisfying (7) and (8) as above and let us look at the product measure defined on S(E)bydν =cdµ(r)dω (cis just a normalizing constant so that ν becomes a probability). It is then easy to see ν(Bt(x0)) ≤Ct n−1+α,x 0∈Rn. We also have ν(ξ)=cESn−1 e−iξ·(sω)dωdµ(s) =CE Jn−2 2(rs)(rs)2−n 2dµ(s) =c1E eirs(rs)1 2−n 2dµ(s)+c2E e−irs(rs)1 2−n 2dµ(s) +O(r−n 2−1 2) =O(r−α 2+−n 2+1 2), Restriction of the Fourier transform 663 where r=|ξ|. Here we have used (7) and well known asymptotic formulas for the Bessel function Jn−2 2. Therefore, νsatisfies the hypothesis of the above theorem with exponents γ=n−1+αand η=n−1+α−2%. Observing that 2(2n−2γ+η) 4(n−γ)+η=2(n+1−α−2%) n+3−3α−2% in our case, we then conclude that Rn | f(y)|2dν(y)1/2 ≤C||f||p,1≤p≤p0, if p0<2(n+1−α) n+3−3αby choosing an appropriate, small %. Acknowledgment. The second author wants to express his gratitude to Anthony Carbery for many enlightening conversations about this subject. Bibliography 1. J. Bourgain, Hausdorff dimension and distance sets, Israel J. Math. 87 (1994), 193–201. 2. A. Carbery and F. Soria, Pointwise Fourier inversion and localisation in Rn,J. Fourier Anal. Appl. 3(1997), 847–858. 3. A. Carbery, F. Soria and A. Vargas, In preparation. 4. K. Falconer, On the Hausdorff dimension of distance sets, Mathematika 32 (1985), 206–212. 5. P. Mattila, Spherical averages of Fourier transforms of measures with finite energy; dimension of intersections and distance sets, Mathematika 34 (1987), 207–228. 6. P. Mattila,“Geometry of Sets and Measures in Euclidean Spaces,” Cambridge Studies in Advanced Mathematics 44, Cambridge University Press, Cambridge, 1995. 7. G. Mockenhaupt,“Bounds in Lebesgue spaces of oscillatory integral operators,” Habilitationsschrift, Siegen, 1996. 8. R. Salem, On singular monotonic functions whose spectrum has a given Hausdorff dimension, Ark. Mat. 1(1951), 353–365. 9. P. Sj¨ olin, Estimates of spherical averages of Fourier transforms and dimensions of sets, Mathematika 40 (1993), 322–330.