Restricted Radon transforms and projections of planar sets
aa r X i v : . [ m a t h . C A ] M a y RESTRICTED RADON TRANSFORMS ANDPROJECTIONS OF PLANAR SETS
DANIEL M. OBERLIN
Abstract.
We establish a mixed norm estimate for the Radon trans-form in R when the set of directions has fractional dimension. Thisestimate is used to prove a result about an exceptional set of directionsconnected with projections of planar sets. That leads to a conjectureanalogous to a well-known conjecture of Furstenberg. Introduction
For each ω ∈ S , fix ω ⊥ with ω ⊥ ⊥ ω . Define a Radon transform R forfunctions f on R by Rf ( t, ω ) = Z − f ( t ω + s ω ⊥ ) ds. Suppose 0 < α < λ on S which is α -dimensional in the sense that λ ( B ( ω, δ )) . δ α for ω ∈ S . We are interestedin mixed norm estimates for R of the following form:(1.1) h Z S (cid:16) Z − | Rf ( t, ω ) | s dt (cid:17) q/s dλ ( ω ) i /q . k f k p . Here are some conditions which are necessary for (1.1): testing on f = χ B (0 ,δ ) shows that(1.2) 2 p ≤ s ;if there is ω ∈ S such that λ ( B ( ω , δ )) & δ α for small positive δ , thentesting on 1 by δ rectangles centered at the origin in the direction ω ⊥ gives(1.3) 1 p ≤ s + αq ;if the Lebesgue measure in S of the δ -neighborhood in S of the supportof λ is . δ − α , then testing on unions of 1 by δ rectangles in the directionsof the support of λ gives(1.4) 1 − αp ≤ s . Date : May, 2007.This work was supported in part by NSF grant DMS-0552041.
Our first result is that these necessary conditions are almost sufficient:
Theorem 1.1.
Suppose p, q, r ∈ [1 , ∞ ] satisfy the conditions (1.2) , (1.3) ,and (1.4) with strict inequality. Then the estimate (1.1) holds. Now suppose that µ is a nonnegative Borel measure on R . If ω ∈ S ,define the projection µ ω of µ in the direction of ω by Z R f ( y ) dµ ω ( y ) . = Z R f ( x · ω ) dµ ( x ) , where x · ω denotes the inner product in R . Fix α ∈ (0 ,
1) and suppose that λ is an α -dimensional measure on S . Then, for ǫ >
0, there is C = C ( ǫ )such that Z S dλ ( ω ) | ω · ω | α − ǫ ≤ C ( ǫ )for all ω ∈ S . The computation Z S I α − ǫ ( µ ω ) dλ ( ω ) = Z S Z R Z R dµ ω ( y ) dµ ω ( y ) | y − y | α − ǫ dλ ( ω ) = Z R Z R Z S dλ ( ω ) | ω · x − x | x − x | | α − ǫ dµ ( x ) dµ ( x ) | x − x | α − ǫ ≤ C ( ǫ ) I α − ǫ ( µ )is due to Kaufman [2]. Refining an earlier result of Marstrand [3], it showsthat if E ⊂ R has dimension β ≤ p ω ( E ) is the projection of E ontothe line through the origin in the direction of ω , then(1.5) dim { ω ∈ S : dim p ω ( E ) < α } ≤ α whenever α ≤ β . (In this note “dim” stands for Hausdorff dimension.) Inparticular,(1.6) dim { ω ∈ S : dim p ω ( E ) < β } ≤ β. The next theorem, whose analog for Minkowski dimension is trivial, com-plements Kaufman’s results (1.5) and (1.6):
Theorem 1.2. If dim E = β ≤ then (1.7) dim { ω ∈ S : dim p ω ( E ) < β/ } = 0 . The estimates (1.6) and (1.7) lead naturally to the conjecture that if α ≤ β ≤ { ω ∈ S : dim p ω ( E ) < ( α + β ) / } ≤ α. One may view this conjecture as an analog of the conjecture that Furstenberg α -sets have dimension at least (3 α + 1) /
2, with (1.5) being the analog of theknown 2 α lower bound for the dimension of Furstenberg sets and with (1.7)being the analog of the known ( α + 1) / β = 1 would imply the Furstenberg conjecture for a certain class of modelFurstenberg sets. (Information about Furstenberg’s conjecture is containedin [5].) The link between Theorems 1.1 and 1.2 is the fact that, formally, µ ω = Rµ ( · , ω ). ESTRICTED RADON TRANSFORMS AND PROJECTIONS OF PLANAR SETS 3 Proof of Theorem 1.1
The lines bounding the regions defined by (1.2) and (1.4) intersect at( p , s ) = ( α , − α α ). Then equality in (1.3) gives q = α , so the importantestimate is an L α → L α ( L (1+ α ) / (1 − α ) ) estimate. Easy estimates com-bined with an interpolation argument show that Theorem 1.1 will follow ifwe establish (1.1) for f = χ E and a collection of triples ( p, q, r ) which arearbitrarily close to (cid:0) α, α, (1 + α ) / (1 − α ) (cid:1) . Standard arguments thenshow that it is enough to prove that if Rχ E ( t, ω ) ≥ µ for( t, ω ) ∈ F = { ( t, ω ) : ω ∈ A, t ∈ B ( ω ) ⊂ [ − , } , where there is some B such that B ≤ m ( B ( ω )) ≤ B for ω ∈ A , then µ p λ ( A ) p/q B p/s ≤ C ( δ ) m ( E )if p = α + δα + 1 δα + 1 , q = α + δα + 1 , s = α + δα + 1 δα + 1 − α for small δ > ω ∈ A let E ( ω ) = { t ω + s ω ⊥ ∈ E : t ∈ B ( ω ) , s ∈ [ − , } . Since Rχ E ( t, ω ) ≥ µ and m ( B ( ω )) ≥ B , it follows that(2.1) m ( E ( ω )) ≥ µ B. Using the change of coordinates x ( x · ω , x · ω ), one can check that(2.2) m (cid:0) E ( ω ) ∩ E ( ω ) (cid:1) . B | ω − ω | . We will bound m ( E ) from below by using(2.3) m ( E ) ≥ m (cid:0) ∪ Nj =1 E ( ω j ) (cid:1) ≥ N X j =1 m ( E ( ω j )) − X ≤ j Then the left member of (2.6) gives (2.7) again.3. Proof of Theorem 1.2 For ρ > 0, let K ρ be the kernel defined on R d by K ρ ( x ) = | x | − ρ χ B (0 ,R ) ( x )where R = R ( d ) is positive. Suppose that the finite nonnegative Borelmeasure ν is a γ -dimensional measure on R d in the sense that ν (cid:0) B ( x, δ ) (cid:1) ≤ C ( ν ) δ γ for all x ∈ R d and δ > 0. If ρ < γ it follows that ν ∗ K ρ ∈ L ∞ ( R d ) . Also ν ∗ K ρ ∈ L ( R d )so long as ρ < d . Thus, for ǫ > ν ∗ K ρ ∈ L p ( R d ) , ρ = γ + 1 p ( d − γ ) − ǫ by interpolation. The following lemma is a weak converse of this observation. Lemma 3.1. If (3.1) holds with ǫ = 0 and p > , then ν is absolutelycontinuous with respect to Hausdorff measure of dimension γ − ǫ for any ǫ > . Thus the support of ν has Hausdorff dimension at least γ .Proof. Recall from [1] (see p. 140) that, for s ∈ R and 1 ≤ p, q ≤ ∞ , thenorm k f k sp,q of a distribution f on R d in the Besov space B sp,q can be definedby k f k spq = k ψ ∗ f k L p ( R d ) + (cid:16) ∞ X k =1 (cid:0) sk k φ k ∗ f k L p ( R d ) (cid:1) q (cid:17) /q ESTRICTED RADON TRANSFORMS AND PROJECTIONS OF PLANAR SETS 5 for certain fixed ψ ∈ S ( R d ), φ ∈ C ∞ c ( R d ), and where φ k ( x ) = 2 kd φ (2 k x ). If ν ∗ K ρ ∈ L p ( R d ), then k ν ∗ χ B (0 ,δ ) k L p ( R d ) . δ ρ . It follows that k ν k spq < ∞ if s < ρ − d = ( γ − d ) /p ′ . Now, for t > < p ′ , q ′ < ∞ , the Besovcapacity A t,p ′ ,q ′ ( K ) of a compact K ⊂ R d is defined by A t,p ′ ,q ′ ( K ) = inf {k f k tp ′ ,q ′ : f ∈ C ∞ c ( R d ) , f ≥ χ K } . It is shown in [4] (see p. 277) that A t,p ′ ,q ′ ( K ) . H d − tp ′ ( K ). Thus it followsfrom the duality of B sp,q and B − sp ′ ,q ′ that ν ( K ) . k ν k spq A − s,p ′ ,q ′ ( K ) . H d + sp ′ ( K ) = H γ − ǫ ( K )if s = ( γ − d − ǫ ) /p ′ . (cid:3) Now suppose that µ is a nonnegative and compactly supported Borelmeasure on R which is β -dimensional in the sense that µ (cid:0) B ( x, δ ) (cid:1) . δ β . Ifthe radii R (1) and R (2) (in the definition of K ρ ) are chosen so that R (1) = 1and R (2) is large enough, depending on the support of µ , then one can verifydirectly that µ ω ∗ K ( ρ − ( t ) . Z R (2) − R (2) µ ∗ K ρ ( tω + sω ⊥ ) ds. If p, q, s are such that (1.1) holds and if ρ = β + (2 − β ) /p − ǫ , so that (3.1)implies that µ ∗ K ρ ∈ L p ( R ), then a rescaling of (1.1) gives(3.2) Z S k µ ω ∗ K ( ρ − k qL s ( R ) dλ ( ω ) < ∞ . If we could take ( p, q, s ) = (cid:0) α, α, (1 + α ) / (1 − α ) (cid:1) and ǫ = 0 then(3.2) would yield Z S k µ ω ∗ K τ k αL (1+ α ) / (1 − α ) ( R ) dλ ( ω ) < ∞ with τ = (1 − α + αβ ) / (1 + α ). Adjusting for the fact that (3.2) actuallyholds only for ( p, q, s ) close to (cid:0) α, α, (1 + α ) / (1 − α ) (cid:1) and with ǫ > Z S k µ ω ∗ K τ k α − ǫL (1+ α − ǫ ) / (1 − α ) ( R ) dλ ( ω ) < ∞ with τ = (1 − α + αβ ) / (1 + α ) − ǫ for any ǫ > 0. With ν = µ ω , p =(1 + α − ǫ ) / (1 − α ), and d = 1, Lemma 3.1 then shows that, for any ǫ > µ ω ’s support exceeds β/ − ǫ for λ -almost all ω ’s.Since this is true for any α -dimensional measure λ and for any α ∈ (0 , { ω ∈ S : dim p ω ( E ) < β/ } = 0 as desired. RESTRICTED RADON TRANSFORMS AND PROJECTIONS OF PLANAR SETS References [1] J. Bergh, J. L¨ofstr¨om, Interpolation Spaces, Grundlehren der mathematischen Wis-senschaften, bd. 223, Springer-Verlag, Berlin, (1976).[2] R. Kaufman, On Hausdorff dimension of projections, Mathematika (1968), 153–155.[3] J. Marstrand, Some fundamental geometrical properties of plane sets of fractionaldimension, Proc. London. Math. Soc. (1954), 257–302.[4] B.-M. Stocke, Differentiability properties of Besov potentials and Besov spaces, Arkivf¨or Mat. (1984), 269–286.[5] T. Wolff, Recent work connected with the Kakeya problem, in: Prospects in mathe-matics (Princeton, NJ, 1966), Amer. Math. Soc., Providence, RI, (1966), 129–162. D. M. Oberlin, Department of Mathematics, Florida State University, Tal-lahassee, FL 32306 E-mail address ::