On the maximum principle for the Riesz transform

Dedicated to a memory of remarkable mathematician and man Victor Petrovich Havin ON THE MAXIMUM PRINCIPLE FOR THE RIESZ TRANSFORM

arXiv:1701.04500v1 [math.CA] 17 Jan 2017

VLADIMIR EIDERMAN AND FEDOR NAZAROV Abstract. Let µ be a measure in Rd with compact support and continuous density, and let Z y−x Rs µ(x) = dµ(y), x, y ∈ Rd , 0 < s < d. |y − x|s+1 We consider the following conjecture: sup |Rs µ(x)| ≤ C

|Rs µ(x)|,

sup

C = C(d, s).

x∈supp µ

x∈Rd

This relation was known for d − 1 ≤ s < d, and is still an open problem in the general case. We prove the maximum principle for 0 < s < 1, and also for 0 < s < d in the case of radial measure. Moreover, we show that this conjecture is incorrect for non-positive measures.

1. Introduction Let µ be a non-negative finite Borel measure with compact support in Rd , and let 0 < s < s d. The truncated Riesz operator Rµ,ε is defined by the equality Z y−x s f (y) dµ(y), x, y ∈ Rd , f ∈ L2 (µ), ε > 0. Rµ,ε f (x) = s+1 |y − x| |y−x|>ε s For every ε > 0 the operator Rµ,ε is bounded on L2 (µ). By Rµs we denote a linear operator on L2 (µ) such that Z y−x s f (y) dµ(y), Rµ f (x) = |y − x|s+1 whenever the integral exists in the sense of the principal value. We say that Rµs is bounded on L2 (µ) if s kRµs k := sup kRµ,ε kL2 (µ)→L2 (µ) < ∞. ε>0 s Rµ 1(x) is

In the case f ≡ 1 the function said to be the s-Riesz transform (potential) of µ s and is denoted by R µ(x). If µ has continuous density with respect to the Lebesgue measure md in Rd , that is if dµ(x) = ρ(x) dmd (x) with ρ(x) ∈ C(Rd ), then Rs µ(x) exists for every x ∈ Rd . By C, c, possibly with indexes, we denote various constants which may depend only on d and s. We consider the following well-known conjecture. Conjecture 1.1. Let µ be a nonnegative finite Borel measure with compact support and continuous density with respect to the Lebesgue measure in Rd . There is a constant C such that sup |Rs µ(x)| ≤ C sup |Rs µ(x)|. (1.1) x∈supp µ

x∈Rd

For s = d − 1 the proof is simple. Obviously, Rs µ(x) = ∇Uµs (x), 1

(1.2)

2

VLADIMIR EIDERMAN AND FEDOR NAZAROV

where Uµs (x)

1 = s−1

Z

dµ(y) , s 6= 1, Uµ1 (x) = − s−1 |y − x|

Z

log |y − x| dµ(y).

Thus each component of the vector function Rs µ(x), s = d − 1, is harmonic in Rd \ supp µ. Applying the maximum principle for harmonic functions we get (1.1). For d − 1 < s < d, the relation (1.1) was established in [2] under stronger assumption that ρ ∈ C ∞ (Rd ). In fact it was proved that (1.1) holds for each component of Rs µ with C = 1 as in the case s = d − 1. The proof is based on the formula which recovers a density ρ from Uµs . But this method does not work for s < d − 1. The problem under consideration has a very strong motivation and also is of independent interest. In [2] it is an important ingredient of the proof of the following theorem. By Hs we denote the s-dimensional Hausdorff measure. Theorem 1.2 ([2]). Let d − 1 < s < d, and let µ be a positive finite Borel measure such that Hs (supp µ) < ∞. Then kRs µkL∞ (md ) = ∞ (equivalently, kRµs k = ∞). If s is integer, the conclusion of Theorem 1.2 is incorrect. For 0 < s < 1 Theorem 1.2 was proved by Prat [10] using different approach. The obstacle for extension of this result to all noninteger s between 1 and d − 1 is the lack of the maximum principle. The same issue concerns the quantitative version of Theorem 1.2 obtained by Jaye, Nazarov, and Volberg [3]. The maximum principle is important for other problems on the connection between geometric properties of a measure and boundedness of the operator Rµs on L2 (µ) – see for example [3], [5], [6], [7]. All these results are established for d − 1 < s < d or s = d − 1. The problem of the lower estimate for kRµs k in terms of the Wolff energy (a far going development of Theorem 1.2) which is considered in [3], [5], was known for 0 < s < 1. And the results in [6], [7] are (d − 1)-dimensional analogs of classical facts known for s = 1 (in particular, [7] contains the proof of the analog of the famous Vitushkin conjecture in higher dimensions). For 0 < s ≤ 1, the proofs essentially use the Melnikov curvature techniques and do not require the maximum principle. But this tool is absent for s > 1. At the same time the validity of the maximum principle itself remained open even for 0 < s < 1. It is especially interesting because the analog of (1.1) does not hold for each component of Rs µ when 0 < s < d − 1 unlike the case d − 1 ≤ s < d – see Proposition 2.1 below. We prove Conjecture 1.1 for 0 < s < 1 in Section 2 (Theorem 2.3). The proof is completely different from the proof in the case d − 1 ≤ s < d. In Section 3 we prove Conjecture 1.1 in the special case of radial density of µ (that is when dµ = h(|x|) dmd (x)), but for all s ∈ (0, d). Section 4 contains an example showing that Conjecture 1.1 is incorrect for nonpositive measures, even for radial measures with C ∞ -density (note that in [14, Conjecture 7.3] Conjecture 1.1 was formulated for all finite signed measures with compact support and C ∞ density).

2. The case 0 < s < 1 We start with a statement showing that the maximum principle fails for every component of Rs µ if 0 < s < d − 1.

ON THE MAXIMUM PRINCIPLE FOR THE RIESZ TRANSFORM

3

Proposition 2.1. For any d ≥ 2, 0 < s < d − 1, and any M > 0, there is a positive measure µ in Rd with C ∞ -density such that sup |R1s µ(x)| > M sup |R1s µ(x)|,

x∈Rd

(2.1)

x∈supp µ

where R1s µ is the first component of Rs µ. Proof. Let E = {(x1 , . . . , xd ) ∈ Rd : x1 = 0, x22 + · · · + x2d ≤ 1}, and let Eδ , δ > 0, be a δ-neighborhood of E in Rd . Let µ = µδ be a positive measure supported on Eδ with µ(Eδ ) = 1 and with C ∞ -density ρ(x) such that ρ(x) < 2/vol(Eδ ) ≤ Cd /δ. Then |R1s µ(x′ )| > Ad , where x′ = (1, 0, . . . , 0), 0 < δ < 1/2. On the other hand, for x ∈ supp µ integration by parts yields Z Z δ 1 s dµ(y) + dµ(y) |R1 µ(x)| < s s+1 |y−x|≥δ |y − x| |y−x|2ε |y−x|≤2ε

4


One can easily see that ∂ y−x 1 C 1 1 ∂xi φε (|y − x|) |y − x|s+1 < C ε |y − x|s + |y − x|s+1 < |y − x|s+1 , |y − x| ≤ 2ε. Hence, Z 2ε Z 1 1 dµ(y) < C dµ(B(x, t)) |I1 (x)| < C s+1 s+1 t 0 |y−x|≤2ε |y − x| Z 2ε µ(B(x, 2ε)) µ(B(x, t)) ≈ + dt. s+1 (2ε) ts+2 0 d Since µ has a continuous density with respect to md , we have µ(B(x, t)) < R Aµ,B t as t ≤ 2ε < 1, x ∈ B. Taking into account that s < d − 1, we obtain the relation B I1 (x) dmd (x) → 0 as ε → 0. To estimate the integral of I2 (x) we use the equality ∇ · |x|xs+1 = d−s−1 . Thus, |x|s+1 Z Z Z dµ(y) I2 (x) dmd (x) = C dmd (x) s+1 |y−x|>2ε |y − x| B B Z Z dmd (x) dµ(y) =C s+1 B(x0 ,r+2ε) B∩{|y−x|>2ε} |y − x| Z Z dmd (x) dµ(y) =: C(J1 + J2 ). + s+1 Rd \B(x0 ,r+2ε) B |y − x|

Obviously,

Z   

r

td−1 dt ≈ r d−s−1 , |y − x0 | ≤ r, s+1 dmd (x) t 0 ≈ d s+1 r  |y − x| B  , |y − x0 | > r.  |y − x0 |s+1 In order to estimate J1 we note that for sufficiently small ε, Z Z dmd (x) dmd (x) ≈ ≈ r d−s−1 , y ∈ B(x0 , r + 2ε). s+1 s+1 |y − x| |y − x| B∩{|y−x|>2ε} B Z

Hence, J1 ≈ r d−s−1 µ(B(x0 , r + 2ε)). Moreover, Z ∞ Z dµ(B(x0 , t)) rd d dµ(y) = r . J2 ≈ s+1 ts+1 r+2ε Rd \B(x0 ,r+2ε) |y − x0 | Passing to the limit as ε → 0, we get (2.2)

Now we are ready to prove our main result. Theorem 2.3. Let µ be a non-negative measure in Rd with continuous density and compact support. Let 0 < s < 1. Then (1.1) holds with a constant C depending only on d and s. Proof. Let us sketch the idea of proof. Let a measure µ be such that µ(B(y, t)) ≤ Cts , y ∈ Rd , t > 0. For Lipschitz continuous compactly supported functions ϕ, ψ, define the form hRs (ψµ), ϕiµ by the equality ZZ 1 y−x s hR (ψµ), ϕiµ = (ψ(y)ϕ(x) − ψ(x)ϕ(y)) dµ(y) dµ(x); 2 Rd ×Rd |y − x|s+1 the double integral exists since |ψ(y)ϕ(x)−ψ(x)ϕ(y)| ≤ Cψ,ϕ |x−y|. If we assume in addition R that ψ dµ = 0, we may define hRs (ψµ), ϕiµ for any (not necessarily compactly supported)


5

bounded Lipschitz continuous function ϕ on Rd ; here we follow [4]. Let supp ψ ∈ B(0, R). For |x| > 2R we have Z Z y − x x y − x d |y − x|s+1 ψ(y) dµ(y) = d |y − x|s+1 + |x|s+1 ψ(y) dµ(y) R R Z Cψ C |yψ(y)| dµ(y) = s+1 . ≤ s+1 |x| |x| Rd Choose a Lipschitz continuous compactly supported function ξ which is identically 1 on B(0, 2R). Then we may define the form hRs (ψµ), ϕiµ as Z Z y−x s s hR (ψµ), ϕiµ = hR (ψµ), ξϕiµ + ψ(y) dµ(y) (1 − ξ(x))ϕ(x) dµ(x). s+1 Rd Rd |y − x| The repeated integral is well defined because Z ∞ Z ∞ Z µ(B(0, t)) 1 dµ(x) ≤C dt ≤ C dt. s+1 s+2 2 t 2R 2R t |x|>2R |x| Assuming that Theorem 2.3 is incorrect and using the Cotlar inequality we establish the existence of a positive measure ν such that ν has no point masses, the operator RνsRis bounded on L2 (ν), and hRs (ψν), 1iν = 0 for every Lipschitz continuous function ψ with ψ dµ = 0. It means that ν is a reflectionless measure, that is a measure without point masses with s the following properties: Rνs is bounded on L2 (ν), and hR R (ψν), 1iν = 0 for every Lipschitz continuous compactly supported function ψ such that ψ dµ = 0. But according to the recent result by Prat and Tolsa [11] such measures do not exist for 0 < s < 1. We remark that the proof of this result contains estimates of an analog of the Melnikov’s curvature of a measure. This is the obstacle to extent the result to s ≥ 1. We now turn to the details. Suppose that C satisfying (1.1) does not exists. Then for every n ≥ 1 there is a positive measure µn such that sup |Rs µn (x)| = 1,

sup

|Rs µn (x)| ≤

x∈supp µn

x∈Rd

1 . n

Let θµ (x, r) :=

µ(B(x, r)) , rs

θµ := sup θµ (x, r). x,r

We prove that 0 < c < θµn < C.

(2.3)

The estimate from above is a direct consequence of Lemma 2.2. Indeed, for any ball B(x, r) (2.2) implies the estimate Z s d−1 (R µn · n) dσ ≥ Cr d−s−1 µn (B), cd r ≥ ∂B

which implies the desired inequality. The estimate from below follows immediately from a Cotlar-type inequality sup |Rs µn (x)| ≤ C sup |Rs µn (x)| + θµn x∈Rd

x∈supp µn

(see [8, Theorem 7.1] for a more general result).

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Let B(xn , rn ) be a ball such that θµn (xn , rn ) > c = c(s, d), and let νn (·) = rn−s µn (xn + rn ·). Then x − xn s s R µn (x) = R νn , θνn (y, t) = θµn (rn y + xn , rn t). rn In particular, νn (B(0, 1)) = θµn (xn , rn ) > c. Choosing a weakly converging subsequence of {νn }, we obtain a positive measure ν. If we prove that (a) ν(B(y, t)) ≤ Cts , s R (b) hR ν, ψiν = 0 for every Lipschitz continuous compactly supported function ψ with ψ dν = 0, (c) the operator Rνs is bounded on L2 (ν), then ν is reflectionless, and we come to contradiction with Theorem 1.1 in [11] mentioned above. Thus, the proof would be completed. The property (a) follows directly from (2.3). For weakly converging measures νn with θµn < C we may apply Lemma 8.4 in [4] which yields (b). To establish (c) we use the inequality s Rs,∗ µ(x) := sup |Rµ,ε 1(x)| ≤ kRs µkL∞ (md ) + C, ε>0

x ∈ Rd , C = C(s),

for any positive Borel measure µ such that µ(B(x, r)) ≤ r s , x ∈ Rd , r > 0, – see [12, Lemma 2] or [13, p. 47], [1, Lemma 5.1] for a more general setting. Thus, Rεs νn (x) := Rνsn ,ε 1(x) ≤ C for every ε > 0. Hence, Rεs ν(x) ≤ C for ε > 0, x ∈ Rd , and the nonhomogeneous T 1-theorem [9] implies the boundedness of Rνs on L2 (ν). 3. The case of radial density Lemma 2.2 allows us to prove the maximum principle for all s ∈ (0, d) in the special case of radial density. Proposition 3.1. Let dµ(x) = h(|x|) dmd (x), where h(t) is a continuous function on [0, ∞), and let s ∈ (0, d − 1). Then (1.1) holds with a constant C depending only on d and s. We remind that for s ∈ [d−1, d) Conjecture 1.1 is proved in [2] for any compactly supported measure with C ∞ density. Thus, for compactly supported radial measures with C ∞ density (1.1) holds for all s ∈ (0, d). Proof. Because µ is radial, by (2.2) we have Z s d−1 s (R µ · n) dσ cd r |R µ(x)| = ∂B(0,r)

≈r

d−s−1

µ(B(0, r)) + r

Z

d

∞

r

Thus, µ(B(0, r)) +r |R µ(x)| ≈ rs s

Z

r

∞

dµ(B(0, t)) , ts+1

dµ(B(0, t)) . ts+1

Fix w 6∈ supp µ, and let r = |w|. If µ(B(0, r)) ≥r rs

Z

∞ r

dµ(B(0, t)) , ts+1

r = |x|.


7

then there is r1 ∈ (0, r) such that {y : |y| = r1 } ⊂ supp µ and µ(B(0, r)) = µ(B(0, r1 )). Hence, µ(B(0, r)) µ(B(0, r1)) |Rs µ(w)| ≈ ≤ C|Rs µ(x1 )|, |x1 | = r1 . < s s r r1 If Z ∞ µ(B(0, r)) dµ(B(0, t)) r such that {y : |y| = r2 } ⊂ supp µ and µ(B(0, r)) = µ(B(0, r2 )). Hence, Z ∞ µ(B(0, r)) dµ(B(0, t)) µ(B(0, r2 )) < 0, we construct a signed measure ν = ν(ε) in R5 with the following properties: (a) ν is a radial signed measure with C ∞ -density; (b) supp ν ∈ Dε := {1 − ε ≤ |x| ≤ 1 + ε}; (c) |R2 ν(x)| < ε for x ∈ supp ν; |R2 ν(x)| > a > 0 for |x| = 2, where a is an absolute constant. Here R2 ν means Rs ν with s = 2. Let ∆2 := ∆ ◦ ∆, and let  2/3, |x| ≤ 1, 1 1 u(x) = − , |x| > 1.  |x| 3|x|3

1 ) = 0 in R5 \ {0}. Hence, ∆2 u(x) = 0, |x| = 6 1. Moreover, Note that ∆( |x|1 3 ) = 0 and ∆2 ( |x| 5 ∇u is continuous in R and ∇u(x) = 0, |x| = 1. For δ ∈ (0, ε), let ϕδ (x) be a C ∞ -function in R5 such that ϕδ > 0, supp ϕδ = {x ∈ R R5 : |x| ≤ δ}, and ϕδ (x) dm5 (x) = 1 (for example, a bell-like function on |x| ≤ δ}). Let Uδ := u ∗ ϕδ . Then ∆2 Uδ (x) = 0 as x 6∈ Dδ . Also, ∆Uδ (x) → 0 as |x| → ∞. Hence, the function ∆Uδ can be represented in the form ∆Uδ = c( |x|1 3 ∗ ∆2 Uδ ) (here and in the sequel by c we denote various absolute constants). Set dνδ = ∆2 Uδ dm5 . Then supp νδ ∈ Dδ , and (b) 1 1 1 ) = |x|c 3 , we have ∆Uδ = c∆( |x| ∗ ∆2 Uδ ), that is Uδ = c( |x| ∗ ∆2 Uδ ) + h, is satisfied. Since ∆( |x| 1 where h is a harmonic function in R5 . Since both Uδ and |x| ∗ ∆2 Uδ tend to 0 as x → ∞, we have 1 2 Uδ = c ∗ ∆ Uδ . |x| Thus, Z y−x 2 dνδ (y) = c∇Uδ (x). R ν(x) = c |y − x|3 Obviously, ∇Uδ = ∇(u∗ϕδ ) = (∇u)∗ϕδ , and hence maxx∈Dδ |∇Uδ (x)| → 0 as δ → 0. On the other hand, for fixed x with |x| > 1 (say, |x| = 2) we have limδ→0 |∇u ∗ ϕδ | = |∇u(x)| > 0. Thus, (c) is satisfied if δ is chosen sufficiently small.

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Remark. R It is well-known that the maximum principle (with a constant C) holds for potentials K(|x − y|) dµ(y) with non-negative kernels K(t) decreasing on (0, ∞), and nonnegative finite Borel measures µ. Our arguments show that for non-positive measures the analog of (1.1) fails even for potentials with positive Riesz kernels. In fact we have proved that for every ε > 0, there exists a signed measure η = η(ε) in R5 with C ∞ -density and such that supp η ∈ Dε := {1 −ε ≤ |x| ≤ 1 + ε}, |uη (x)| < ε for x ∈ supp η, but |uη ((2, 0, . . . , 0))| > R dη(y) b > 0, where uη (x) := |y−x| , and b is an absolute constant. Indeed, for the first component R12 ν of R2 ν we have ∂ ∂ 1 ∂ 2 2 Uδ = c ∗ (∆ Uδ ) = cuη , where dη = (∆2 Uδ ) dm5 . R1 ν = c ∂x1 |x| ∂x1 ∂x1 References [1] D. R. Adams, V. Ya. Eiderman, Singular operators with antisymmetric kernels, related capacities, and Wolff potentials, Internat. Math. Res. Notices 2012, first published online January 4, 2012, doi:10.1093/imrn/rnr258. [2] V. Eiderman, F. Nazarov, A. Volberg, The s-Riesz transform of an s-dimensional measure in R2 is unbounded for 1 < s < 2, J. Anal. Math. 122 (2014), 1–23; arXiv:1109.2260. [3] B. Jaye, F. Nazarov, A. Volberg, The fractional Riesz transform and an exponential potential, Algebra i Analiz 24 (2012), no. 6, 77–123; reprinted in St. Petersburg Math. J. 24 (2013), no. 6, 903–938; arXiv:1204.2135. [4] B. Jaye, F. Nazarov, Reflectionless measures for Calder´ on-Zygmund operators I: general theory, to appear in J. Anal. Math., arXiv:1409.8556. [5] B. Jaye, F. Nazarov, M. C. Reguera, X. Tolsa, The Riesz transform of codimension smaller than one and the Wolff energy, arXiv:1602.02821. [6] F. Nazarov, X. Tolsa, A. Volberg, On the uniform rectifiability of AD regular measures with bounded Riesz transform operator: the case of codimension 1, Acta Math. 213 (2014), no. 2, 237–321; arXiv:1212.5229, 88 p. [7] F. Nazarov, X. Tolsa, A. Volberg, The Riesz transform, rectifiability, and removability for Lipschitz harmonic functions, Publ. Mat. 58 (2014), no. 2, 517–532, arXiv:1212.5431, 15 p. [8] F. Nazarov, S. Treil, and A. Volberg, Weak type estimates and Cotlar inequalities for Calder´ onZygmund operators on nonhomogeneous spaces, Internat. Math. Res. Notices 1998, no. 9, 463–487. [9] F. Nazarov, S. Treil, and A. Volberg, The T b-theorem on non-homogeneous spaces, Acta Math. 190 (2003), no. 2, 151–239. [10] L. Prat, Potential theory of signed Riesz kernels: capacity and Hausdorff measure, Internat. Math. Res. Notices, 2004, no. 19, 937–981. [11] L. Prat, X. Tolsa, Non-existence of reflectionless measures for the s-Riesz transform when 0 < s < 1, Ann. Acad. Sci. Fenn. Math. 40 (2015), no. 2, 957–968. [12] M. Vihtil¨a, The boundedness of Riesz s-transforms of measures in Rn , Proc. Amer. Math. Soc. 124 (1996), no. 12, 3797–3804. [13] A. Volberg, Calder´ on-Zygmund capacities and operators on nonhomogeneous spaces. CBMS Regional Conference Series in Mathematics, 100. AMS, Providence, RI, 2003. [14] A. L. Volberg, V. Ya. Eiderman, Nonhomogeneous harmonic analysis: 16 years of development, Uspekhi Mat. Nauk 68 (2013), no. 6(414), 3–58 (Russian); translation in Russian Math. Surveys 68 (2013), no. 6, 973–1026. Vladimir Eiderman, Department of Mathematics, Indiana University, Bloomington, IN E-mail address: [email protected] Fedor Nazarov, Department of Mathematics, Kent State University, Kent, OH E-mail address: [email protected]