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Restricted Radon Transforms and Unions of Hyperplanes RESTRICTED RADON TRANSFORMS AND UNIONS OF HYPERPLANES Daniel M. Oberlin Department of Mathematics, Florida State University October 2004 1 Introduction x (n 1) n If Σ − is the unit sphere in R , the Radon transform Rf of a suitable function f on Rn is defined by (n 1) Rf(σ; t) = Z f(p + tσ) dmn 1(p) σ Σ − ; t R; − σ? 2 2 where the integral is with respect to (n 1)-dimensional Lebesgue measure on the − hyperplane σ?. We also define, for 0 < δ < 1, 1 Rδf(σ; t) = δ− Z f(x + tσ) dmn(x): [σ B(0;1)]+B(0,δ) ?\ The paper [5] is concerned with the mapping properties of R from Lp(Rn) into mixed norm spaces defined by the norms q=r 1=q 1 r Z hZ g(σ; t) dti dσ : Σ(n 1) j j − −∞ (n 1) Here dσ denotes Lebesgue measure on Σ − . The purpose of this paper is mainly to study the possibility of analogous mixed norm estimates when dσ is replaced (n 1) by measures dµ(σ) supported on compact subsets S Σ − having dimension < n 1. We are usually interested in the case r =⊆ and will mostly settle for estimates− of restricted weak type in the indices p and1 q and those only for f supported in a ball. The following theorem, which we regard as an estimate for a restricted Radon transform, is typical of our results here: Theorem 1. Fix α (1; n 1). Suppose µ is a nonnegative and finite Borel (n 1) 2 − measure on Σ − satisfying the Frostman condition dµ(σ1)dµ(σ2) Z Z α < : (n 1) (n 1) σ1 σ2 1 Σ − Σ − j − j Typeset by -TEX AMS 1 2 DANIEL M. OBERLIN Then (n 1) 1/α 1=2 (1) λ µ σ Σ − : sup RχE(σ; t) > λ . E f 2 t R g j j 2 for λ > 0 and Borel E B(0; 1). That is, ⊆ α, 1=2 RχE Lµ 1(L ) . E : k k 1 j j Suppose that α (0; n 1). Say that a Borel set E Rn satisfies the (Besicov- 2 − ⊆ (n 1) itch) condition B(n 1; α) if there is a compact set S Σ − having Hausdorff dimension α such that− for each σ S there is a translate⊆ of an (n 1)-plane or- thogonal to σ which intersects E in2 a set of positive (n 1)-dimensional− Lebesgue measure. It is well-known that, given (0; α), such an− S supports a probability measure µ satisfying the hypothesis of Theorem2 1, but with α in place of α. In conjunction with Theorem 1, standard arguments imply that− such an E must have positive n-dimensional Lebesgue measure. That is, B(n 1; α) sets in Rn have positive Lebesgue measure if α > 1. As will be pointed out in− 2, the next theorem implies that, for α (0; 1) and in certain cases, B(n 1; α)x sets have Hausdorff dimension at least n2 1+α. (Here is a notational comment:− E will usually denote the Lebesgue measure− of E with the appropriate dimensionj beingj clear from the context.) Theorem 2. Suppose α (0; 1). Suppose µ is a nonnegative measure on a compact interval J R which satisfies2 the condition ⊆ e µ(I) I α . j j e for subintervals I J. Let µ be the image of µ under a one-to-one and bi-Lipschitz ⊆ (n 1) mapping of J into Σ − . If 0 < γ < β < α ande 1 1 + β γ 1 1 + γ 1 γ = − ; = ; η = − p 1 + 2β γ q 1 + 2β γ 1 + 2β γ − − − then there is the estimate q; 1=p η RδχE Lµ 1(L ) . E δ− k k 1 j j for Borel E B(0; 1) and δ (0; 1). ⊂ 2 Contrasting with Theorems 1 and 2, the next result provides a global estimate for a restricted Radon transform: Theorem 3. Suppose n 4. Let S be the (n 2)-sphere ≥ − n 1 − σ = (σ : : : ; σ ) Σ(n 1) : σ2 = σ2 1 n − X j n f 2 1 g and let µ be Lebesgue measure on S. Then there is an estimate (n 1)=n Rχ n 2 E E L − (L ) . − k k µ 1 j j RESTRICTED RADON TRANSFORMS AND UNIONS OF HYPERPLANES 3 for Borel E Rn. ⊆ This result is an analogue of (3) in [5] (which is a similar estimate but with µ (n 1) replaced by Lebesgue measure on Σ − and q = n). The proof of Theorem 3 parallels the proof in [5] but requires the L2 Fourier restriction estimates for the light cone in Rn in place of an easier L2 estimate used in [5]. The remainder of this paper is organized as follows: 2 contains the proofs of Theorems 2 and 3 and the statement and proof of a similarx result in the case when d is an integer strictly between 1 and n 1 and µ is Lebesgue measure on (n 1) − a suitable d-manifold in Σ − ; 3 contains the proof of Theorem 3; 4 contains some miscellaneous observations andx remarks: an analogue for Kahane'sx notion of Fourier dimension of Theorem 2 when n = 2 and examples bearing on the question of whether B(n 1; 1) sets in Rn must have positive measure or only full dimension (the answer depends− on the set S of directions). 2 Proofs of Theorems 1 and 2 x As Theorem 1 is a consequence of its analogue, uniform in δ (0; 1), for the 2 operators Rδ, we will restrict our attention to these operators. A standard method for obtaining restricted weak type estimates is to estimate E from below. We will do this with a particularly simple-minded strategy based onj j two observations and originally employed in [3] and [4]. The first observation is that N N n=1 En En Em En : Xn=1 X1 m<n N j [ j ≥ j j − ≤ ≤ j \ j (n 1) δ The second is the well-known fact that if σ Σ − and if, for t R, P denotes 2 2 σ a plate [σ? B(0; 1)] + B(0; δ) + tσ, then \ C(n)δ2 P δ P δ σ1 σ2 j \ j ≤ σ1 σ2 j − j (so long as σ1 and σ2 are not too far apart, an hypothesis we tacitly assume since it can be acheived by multiplying the measures µ appearing below by an appropriate partition of unity). Thus if, for n = 1;:::;N, we have plates P δ satisfying E σn δ j \ P C1λδ, it follows that σn j ≥ 2 1 (2) E C1Nλδ C(n)δ : j j ≥ − X σm σn 1 m<n N ≤ ≤ j − j Our strategy, then, will be to choose N and (n 1) σn σ Σ − : sup RδχE(σ; t) > λ 2 f 2 t R g 2 so that (2) gives, for example, 2 (n 1) 2/α E & λ µ σ Σ − : sup RδχE(σ; t) > λ ; j j f 2 t R g 2 which is the analogue of (1) for the operator Rδ. For Theorem 1 the following lemma will facilitate this choice: 4 DANIEL M. OBERLIN Lemma 1. Let µ be as in Theorem 1. There is C = C(µ) such that given n N (n 1) 2 and a Borel S Σ − with µ(S) > 0, one can choose σn S, 1 n N, such that ⊆ 2 ≤ ≤ 1 CN 2 : 2/α X σm σn ≤ µ(S) 1 m<n N ≤ ≤ j − j Proof of Lemma 1: Suppose σ1; : : : ; σN are chosen independently and at random from the probability space (S; µ ). Then, for 1 m < n N, µ(S) ≤ ≤ 1 1 1 E = 2 Z Z dµ(σm) dµ(σn) σm σn µ(S) σm σn ≤ j − j S S j − j 1 1 1/α 1 1/α C 1 dµ(σ )dµ(σ ) − dµ(σ )dµ(σ ) ; 2 Z Z m n Z Z α m n 2/α µ(S) σm σn ≤ µ(S) S S S S j − j by the hypothesis on µ. Thus 1 CN 2 E 2/α X1 m<n N σm σn ≤ µ(S) ≤ ≤ j − j and the lemma follows. Proof of Theorem 1: Let S be the set (n 1) σ Σ − : sup RδχE(σ; t) > λ f 2 t R g 2 δ so that if σ S then there is t R such that if Pσ = [σ? B(0; 1)] + B(0; δ) + tσ 2δ 2 \ then E P C1λδ. The conjunction of Lemma 1 and (2) yields j \ σ j ≥ 2 2 2α (3) E C1Nλδ C2δ N µ(S)− : j j ≥ − : 2/α We consider two cases (noting that N = N0 = λC1µ(S) =C2δ makes the RHS of (3) equal to 0): Case I: Assume N0 > 10. In this case choose N N such that 2 λC µ(S)2/α λC µ(S)2/α 1 N 1 : 2C2δ ≥ ≥ 3C2δ Then it follows from (3) that 2/α 2 2 4/α λC1µ(S) 2 λ C1 µ(S) 2/α 2 2/α E C1 λδ C2δ 2 2 µ(S)− = κλ µ(S) j j ≥ 3C2δ − 4C2 δ 2 1/α 1=2 for κ = C =(12C2). This gives λµ(S) E as desired. 1 . j j Case II: Assume N0 10. In this case (unless S≤is empty) we estimate 2 2 2/α λ C1 µ(S) E C1λδ j j ≥ ≥ 10C2 RESTRICTED RADON TRANSFORMS AND UNIONS OF HYPERPLANES 5 which again yields λ µ(S)1/α E 1=2 and so completes the proof of Theorem 1.
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