ON THE FRACTIONAL LANDIS CONJECTURE

ANGKANA R ¨ULAND AND JENN-NAN WANG

Abstract. In this paper we study a Landis-type conjecture for fractional Schr¨odinger equa-
tions of fractional power s ∈ (0, 1) with potentials. We discuss both the cases of differentiable
and non-differentiable potentials. On the one hand, it turns out for differentiable potentials
with some a priori bounds, if a solution decays at a rate e^{−|x|}^{1+}, then this solution is triv-
ial. On the other hand, for s ∈ (1/4, 1) and merely bounded non-differentiable potentials,
if a solution decays at a rate e^{−|x|}^{α} with α > 4s/(4s − 1), then this solution must again be
trivial. Remark that when s → 1, 4s/(4s − 1) → 4/3 which is the optimal exponent for the
standard Laplacian. For the case of non-differential potentials and s ∈ (1/4, 1), we also derive
a quantitative estimate mimicking the classical result by Bourgain and Kenig.

1. Introduction

In this work, we study a Landis-type conjecture for the fractional Schr¨odinger equation,

(1) ((−∆)^{s}+ q)u = 0 in R^{n}

with s∈ (0, 1) and

(2) |q(x)| ≤ 1.

Roughly speaking, we are interested in the maximal vanishing rate of solutions to this equation
at infinity. For s = 1, in Section 3.5 in [KL88] V.A. Kondratev and E.M. Landis conjectured
that if |q(x)| ≤ 1 and |u(x)| ≤ C^{0} satisfies |u(x)| ≤ exp(−C|x|^{1+}), then u ≡ 0. The Landis
conjecture was disproved by Meshkov [Mes91], who constructed a potential q and a nontrivial
uwith |u(x)| ≤ C exp(−C|x|^{4}^{3}). He also showed that if |u(x)| ≤ C exp(−C|x|^{4}^{3}^{+}), then u≡ 0.

In their seminal work, Bourgain and Kenig [BK05] derived a quantitative form of Meshkov’s result in their resolution of Anderson localization for the Bernoulli model in higher dimensions.

It should be pointed out that in Meshkov’s counterexample both q and u are complex -valued functions. In other words, the exponent 4/3 is optimal in the complex case (which corresponds to the situation of systems). The proof in [BK05] is based on the Carleman method. In the spirit of the Carleman method, several extensions have been made in [CS99,Dav14,DZ17,DZ18,LW14], which also take singular drift coefficients and potentials into account.

In view of Meshkov’s counterexample, Kenig [Ken06] refined the Landis conjecture and asked whether this conjecture is true for real -valued potentials and solutions. In 2005, Kenig, Silvestre and the second author [KSW15] confirmed the Landis conjecture (in a quantitative form) when n = 2 and q ≥ 0. This result was later extended to the more general situation with ∆ being replaced by any second order elliptic operator [DKW17]. In the very recent preprint [DKW18], this is further improved by also allowing for (exponentially) small negative contributions in the potential. The Landis conjecture in the real case with n = 1 was recently studied by Rossi [Ros18].

The main theme of this paper is to investigate a Landis-type conjecture for fractional Schr¨odinger equations. We will consider both qualitative and quantitative estimates when the potentials are

Wang is supported in part by MOST 105-2115-M-002-014-MY3.

1

either differentiable or simply bounded. Similar to the original Landis conjecture (for the Laplace operator), we are concerned with the maximal decay rate of non-trivial solutions. We are espe- cially interested in understanding how the decay rate depends on the fractional power s∈ (0, 1).

Detailed statements of our results are described below.

1.1. Qualitative estimates. We first discuss the qualitative behaviour of solutions to (1). Here we will show that if the potential q satisfies an additional regularity estimate, then independently of the value of s∈ (0, 1) there are no super-exponentially decaying solutions to (1).

Theorem 1. Let s∈ (0, 1) and assume that u ∈ H^{s}(R^{n}) is a solution to (1) such that q satisfies
q∈ C^{1}(R^{n}), (2), and in addition

(3) |x · ∇q(x)| ≤ 1

holds. Suppose that u further satisfies the following decay behaviour: there exists α >1 such that ˆ

R^{n}

e^{|x|}^{α}|u|^{2}dx≤ C < ∞.

(4)

Then u≡ 0.

For s = 1, a similar qualitative estimate as in Theorem 1 with a differentiable potential satisfying (2) and (3) was proved by Meshkov [Mes89]. Without the additional regularity result on q, it is still possible to prove a qualitative decay result. However, as our argument for this does not distinguish between the real and complex situation, the obtained decay deteriorates.

Theorem 2. Let s∈ (1/4, 1) and assume that u ∈ H^{s}(R^{n}, C) is a solution to (1). Suppose that
(2) holds and u further satisfies the following decay behaviour: there exists α > _{4s−1}^{4s} such that

ˆ

R^{n}

e^{|x|}^{α}|u|^{2}dx≤ C < ∞.

(5)

Then u≡ 0.

We emphasize that as s→ 1 in both of our main results, the identified critical decay exponents correspond to the ones from the case s = 1. Moreover, in the first result, Theorem1, the critical decay rate does not depend on the value of s ∈ (0, 1). It is thus natural to ask whether the derived decay exponents are optimal or rather an artifact of our argument. Let us comment on this. For real-valued problems (i.e. scalar equations), we expect that the exponential decay (independent of the value of s∈ (0, 1)) as the critical decay behaviour is sharp. Indeed, as in [KSW15] and [BK05] it is possible to relate the decay behaviour at infinity to the local maximal vanishing rate at zero (if growth conditions are assumed, which are necessary due to the global character of the problem). Analogous arguments as in the classical case s = 1 would lead to the conjecture that when considering

• the equation (1) with potentials of the size|q(x)| ≤ M (instead of |q(x)| ≤ 1),

• and solutions u(x) which satisfy the growth bounds kuk^{L}^{∞}(R^{n})≤ C^{0}andkuk^{L}^{∞}^{(B}1)≥ 1,
a lower bound of the form

(6) kuk^{L}^{∞}^{(B}r)≥ r^{CM}

1 2s,

holds. Here C = C(C0) > 0 and r∈ (0, r^{0}) for some sufficiently small constant r0 >0. Results
of this flavour have been proved for eigenfunctions or equations with differentiable potentials
(with dependences on the C^{1} norm of the potentials, that is, M in (6) is the size of kqk^{C}^{1}(R^{n}))
in [R¨ul17a,Zhu15] (on compact manifolds or bounded domains, respectively). For the spectral
fractional Laplacian and its eigenfunctions on compact manifolds these dependences are indeed
immediate consequences from the corresponding ones of the Laplacian.

We recall that∇q satisfies (3). Taking this expected quantitative maximal rate of vanishing
(6) for granted and relying on a scaling argument as in [BK05,KSW15], i.e. considering solutions
uR,x_{0}(x) for x0∈ B^{R}\ B^{R/2} of the correspondingly rescaled version of (1), then suggest global
lower bounds of the type

|zinf_{0}|=R sup

|z−z0|<1|u(z)| ≥ exp(−CR log(R)) for R 1,

where C = C(C0) > 0. Hence, this strongly suggests that for the class of potentials under con-
sideration, Theorem1 is essentially sharp (possibly up to logarithmic contributions). However,
we further believe that as in [KSW15], at least under sign conditions on the potential and in
one dimension (which on the level of the Cafferelli-Silvestre extension corresponds to the two-
dimensional setting from [KSW15] in which complex analysis tools are available), it might be
possible to reduce the necessary regularity for q to L^{∞}regularity.

In view of Meshkov’s example [Mes91], in the case of Theorem2, at least the growth behaviour for s → 1 is expected to be optimal. As in the case s = 1, the Carleman estimate of Theorem 5 which lies at the core of the argument for Theorem2 is of perturbative character. It hence does not distinguish between the real vs the complex, i.e. the scalar vs the systems cases.

It would be interesting to investigate whether also for s ∈ (1/4, 1) there are Meshkov-type examples saturating the proposed exponents from Theorem2. We remark that the restriction to s∈ (1/4, 1) seems necessary as long as we only consider radial weight functions in our Carleman estimates (due to the subelliptic nature of these estimates). We seek to prove the results of Theorems1 and2by combining elliptic estimates with Carleman estimates.

1.2. Quantitative estimates. In this subsection, we present a quantitative version of Theorem
2. Note that here q∈ L^{∞}(R^{n}) and that (2) is satisfied.

Theorem 3. Let s∈ (1/4, 1) and assume that u ∈ H^{s}(R^{n}, C) is a solution to (1). Suppose that
u further satisfieskuk^{L}^{2}({|x|<1})= 1 and that there exists a constant C0>0 such that

kuk^{L}^{∞}(R^{n})≤ C^{0}.

Then there exists a constant C = C(n, s, C0) > 0 such that for R > 0 large

|xinf0|=Rkuk^{L}^{∞}^{({|x−x}0|<1})≥ Ce^{−CR}

4s 4s−1log R.

We remark that by virtue of the ellipticity of the problem, it does not really matter in which topology one works for the lower bounds. For instance, it would have equally been possible to derive similar results under the assumption that|u(0)| = 1.

This result is similar in flavour to the decay estimates in [BK05]. However both the arguments in the qualitative and the quantitative settings involve new intricacies and technical challenges due to the nonlocal character of the equations at hand. Since lower bound estimates for nonlocal equations pose serious difficulties, as in various other works on (quantitative) unique continuation [FF14,FF15,BG17,Gar17,R¨ul15,R¨ul17a,R¨ul17b,RS17], we opt for working with the Caffarelli- Silvestre extension [CS07], c.f. (7), (8) in Section2, instead of dealing with the nonlocal equation (1) directly. This allows us to investigate a local (degenerate) elliptic equation. It however comes at the expense of having to study this equation in n+1 instead of n dimensions. In the additional dimension, the control on the solution can hence only be derived through the equation. This implies that we always have to transfer information from the boundary to the bulk and vice versa. In the qualitative estimates, we for instance have to show that (exponential) decay on the boundary implies (exponential) decay in the bulk. Similarly, we have to transfer upper and lower bounds in the quantitative results from the boundary into the bulk and vice versa. This poses non-trivial challenges, which however are overcome by an ingredient which was used in [RS17]:

As one of our key tools which allows us to switch between the bulk and the boundary we rely on a boundary-bulk interpolation inequality, c.f. Proposition2.4.

1.3. Organization of the article. This paper is organized as follows. In Section 2, we will
collect several preliminary results that are needed in the proofs of the main theorems. We
then derive Carleman estimates for the fractional Laplacian (−∆)^{s} with differentiable or non-
differentiable potentials in Section3. The proofs of the qualitative estimates, Theorem 1and2,
are given in Section4. Finally, in Section5, we will prove Theorem3.

2. Auxiliary Results

2.1. The Cafferelli-Silvestre extension. In the sequel, it will be convenient to localize the problem at hand. This will be achieved by means of the Caffarelli-Silvestre extension [CS07]

which allows us to address the problem at hand by relying on tools for local equations. To this
end, for s∈ (0, 1) and u ∈ H^{s}(R^{n}) we consider a solution ˜u∈ ˙H^{1}(R^{n+1}+ , x^{1−2s}_{n+1}) :={v : R^{n+1}+ →
R :

´

R^{n+1}_{+}

x^{1−2s}_{n+1}|∇v|^{2}dx≤ C < ∞} of the degenerate elliptic equation (whose weight however still
is in the Muckenhoupt class),

∇ · x^{1−2s}n+1∇˜u = 0 in R^{n+1}+ ,

˜

u= u on R^{n}× {0}.

(7)

We recall that by the observations in [CS07],
(−∆)^{s}u(x) = cn,s lim

x_{n+1}→0x^{1−2s}_{n+1}∂_{n+1}u(x)˜

for some constant cn,s 6= 0. In particular, the equation (1) can be reformulated as the local, degenerate elliptic equation

∇ · x^{1−2s}n+1∇˜u = 0 in R^{n+1}+ ,

˜

u= u on R^{n}× {0},
cn,s lim

x_{n+1}→0x^{1−2s}_{n+1}∂n+1u˜= qu on R^{n}× {0}.

(8)

This however comes at the expense of adding a new variable, in which we have to infer control by exploiting the equation.

When dealing with this equation, it will be convenient to also introduce the following notations
for the underlying domains and the related weighted function spaces. For Ω⊂ R^{n+1}+ , x0∈ R^{n+1}+ ,
r, R >0, we denote

B_{r}^{+}(x0) :={x ∈ R^{n+1}+ : |x − x^{0}| ≤ r}, B^{0}r(x0) :={x ∈ R^{n}× {0} : |x − x^{0}| ≤ r},
B_{r}^{+}:= B_{r}^{+}(0), B_{r}^{0} := B_{r}^{0}(0),

A^{+}_{r,R}:={x ∈ R^{n+1}+ : r≤ |x| ≤ R}, A^{0}r,R:={x ∈ R^{n}× {0} : r ≤ |x| ≤ R},
H^{1}(Ω, x^{1−2s}_{n+1}) :={v : Ω ∩ R^{n+1}+ → R :

ˆ

Ω∩R^{n+1}+

x^{1−2s}_{n+1}(|v|^{2}+|∇v|^{2})dx≤ C < ∞},

H^{1}(S_{+}^{n}, θ^{1−2s}_{n} ) :={v : S+^{n} → R :
ˆ

S_{+}^{n}

θ_{n}^{1−2s}(|v|^{2}+|∇^{S}+^{n}v|^{2})dθ≤ C < ∞}, θ^{n}= xn+1

|x| . As an important elliptic estimate which we will be using frequently we recall Caccioppoli’s inequality.

Lemma 2.1. Let s ∈ (0, 1) and ˜u ∈ H^{1}(B_{4r}^{+}, x^{1−2s}_{n+1}) be a solution to (7). Then, there exists
C= C(n, s) > 0 such that

kxn+1^{1−2s}^{2} ∇˜ukL^{2}(B_{r}^{+})≤ C

r^{−1}kxn+1^{1−2s}^{2} u˜kL^{2}(B_{2r}^{+})+kukL^{1}^{2}^{2}(B_{2r}^{0} )k lim_{x}

n+1→0x^{1−2s}_{n+1}∂_{n+1}u˜kL^{1}^{2}^{2}(B^{0}_{2r})

.
Proof. The proof follows as for instance in Lemma 4.5 in [RS17], where however the boundary
terms are estimated by an L^{2}− L^{2}estimate instead of an H^{s}− H^{−s}estimate.
2.2. Boundary decay implies bulk decay. In order to deal with the original nonlocal problem
(1), we seek to apply methods which were developed for (quantitative) unique continuation results
for the local equation (8). Hence, we first translate the decay behaviour that is valid on R^{n} to
decay behaviour which also holds on R^{n+1}+ . To this end, we heavily rely on interior and boundary
three balls estimates for the degenerate elliptic equation (8).

Proposition 2.2. Let s∈ (0, 1) and u ∈ H^{s}(R^{n}) be a solution to (1). Assume that (2) holds
and there exist constants C, β≥ 1 such that

(9) ke^{|x|}^{β}^{/2}uk^{L}^{2}(R^{n})≤ C.

Then, there exist constants C_{1}, c_{1} > 0 such that for all x = (x^{0}, x_{n+1}) ∈ R^{n+1}+ the Caffarelli-
Silvestre extension u(x) satisfies˜

|˜u(x^{0}, xn+1)| ≤ C^{1}e^{−c}^{1}^{|(x}^{0}^{,x}^{n+1}^{)|}^{β}.

In order to infer the claimed interior decay, we rely on propagation of smallness estimates.

Here we make use of two types of propagation of smallness results: The first being an interior propagation of smallness while the second one is a boundary-bulk propagation of smallness estimate. In order to use tools from the quantitative analysis of elliptic equations, in the sequel, we view (1) in terms of its Caffarelli-Silvestre extension (8).

Proposition 2.3. Let s ∈ (0, 1) and ˜u ∈ H^{1}(B^{+}_{4}, x^{1−2s}_{n+1}) be a solution to (7). Assume that
r∈ (0, 1) and x^{0}= (x^{0}_{0},5r)∈ B2^{+}. Then, there exists α= α(n, s)∈ (0, 1) such that

k˜ukL^{∞}(B^{+}_{2r}(x_{0})) ≤ Ck˜uk^{α}_{L}^{∞}_{(B}_{r}^{+}_{(x}_{0}_{))}k˜uk^{1−α}_{L}∞(B_{4r}^{+}(x0)).

Proof. As (x0)n+1= 5r, this follows from a standard interior L^{2}three balls estimate (c.f. Propo-
sition 5.4 in [RS17]) together with L^{2}− L^{∞} estimates for uniformly elliptic equations.
Proposition 2.4. Let s ∈ (0, 1) and let ˜u ∈ H^{1}(R^{n+1}+ , x^{1−2s}_{n+1}) be a solution to (8) with q ∈
L^{∞}(R^{n}). Assume that x0∈ R^{n}× {0}. Then,

(a) there exist α = α(n, s)∈ (0, 1) and c = c(n, s) ∈ (0, 1) such that
kxn+1^{1−2s}^{2} u˜kL^{2}(B^{+}_{cr}(x_{0}))

≤C

kxn+1^{1−2s}^{2} u˜kL^{2}(B^{+}_{16r}(x_{0}))+ r^{1−s}kuk^{L}^{2}^{(B}^{0}_{16r}^{(x}0))

^{α}

×

×

r^{s+1}k lim_{x}

n+1→0x^{1−2s}_{n+1}∂n+1u˜k^{L}^{2}^{(B}16r^{0} (x0))+ r^{1−s}kuk^{L}^{2}^{(B}^{0}16r(x0))

^{1−α}

+ C

kxn+1^{1−2s}^{2} u˜kL^{2}(B_{16r}^{+} (x_{0}))+ r^{1−s}kuk^{L}^{2}^{(B}^{0}_{16r}^{(x}0))

1+s^{2s}

×

×

r^{s+1}k lim_{x}

n+1→0x^{1−2s}_{n+1}∂n+1u˜k^{L}^{2}^{(B}16r^{0} (x0))+ r^{1−s}kuk^{L}^{2}^{(B}^{0}16r(x0))

^{1−s}_{1+s}
.

(b) there exist α = α(n, s)∈ (0, 1) and c = c(n, s) ∈ (0, 1) such that

k˜ukL^{∞}(B^{+}cr
2

)≤Cr^{−}^{n}^{2}

r^{s−1}kxn+1^{1−2s}^{2} u˜kL^{2}(B_{16r}^{+} )+kuk^{L}^{2}^{(B}^{0}16r)

^{α}

r^{2s}kquk^{L}^{2}^{(B}^{0}16r)+kuk^{L}^{2}^{(B}16r^{0} )

1−α

+ Cr^{−}^{n}^{2}

r^{s−1}kxn+1^{1−2s}^{2} u˜kL^{2}(B^{+}_{16r})+kuk^{L}^{2}^{(B}^{0}_{16r}^{)}1+s^{2s}

r^{2s}kquk^{L}^{2}^{(B}^{0}_{16r}^{)}+kuk^{L}^{2}^{(B}^{0}_{16r}^{)}^{1−s}_{1+s}
+ Cr^{−}^{n}^{2}r^{s}kqukL^{1}^{2}^{2}(B_{16r}^{0} )kukL^{1}^{2}^{2}(B^{0}_{16r}).

(10)

Proof. The proof relies on a splitting argument and the boundary-bulk interpolation estimates
from Propositions 5.10-5.12 (also Proposition 5.6) in [RS17]. In order to infer the claim, we argue
in two steps, first deriving a suitable L^{2} estimate and then upgrading this to an L^{∞} estimate.

By scaling, it suffices to prove the estimate for r = 1. Without loss of generality, we can take x0= 0.

Step 1: The L^{2} estimate. For the L^{2} estimate we rely on Propositions 5.10-5.12 in [RS17].

Here we distinguish between the cases s∈ [^{1}2,1) and s∈ (0,^{1}2).

Step 1a: The case s∈ [^{1}2,1). In order to invoke the estimate from [RS17], we split our solution
uinto two parts ˜u= u1+ u2. The function u1 deals with the Dirichlet data

∇ · x^{1−2s}n+1∇u^{1}= 0 in R^{n+1}+ ,
u1= ζu in R^{n}× {0}.

Here ζ∈ C0^{∞}(B_{16}^{0} ) is a smooth cut-off function, which is equal to one in B_{8}^{0}. We will estimate u1

by bounds on the Caffarelli-Silvestre extension. The function u2= ˜u− u^{1} in turn is admissible
in Propositions 5.10-5.12 in [RS17], i.e., u2|^{B}8^{0} = 0.

We begin with the estimate for u1: Invoking Lemma 4.2 in [RS17], we obtain the bound

kxn+1^{1−2s}^{2} u1kL^{2}(R^{n+1}_{+} )≤ Ckζuk^{H}^{s−1}(R^{n})≤ Ckζuk^{L}^{2}(R^{n})≤ Ckuk^{L}^{2}^{(B}_{16}^{0} ^{)}.
(11)

The estimate for u2 follows from Proposition 5.10 in [RS17]. The result assert that for each
s∈ (^{1}2,1) there exists a constant c = c(s, n)∈ (0, 1) and α = α(s, n) ∈ (0, 1) such that

(12) kxn+1^{1−2s}^{2} u_{2}kL^{2}(B^{+}_{c})≤ Ckxn+1^{1−2s}^{2} u_{2}k^{α}L^{2}(B^{+}_{2})k lim_{x}

n+1→0x^{1−2s}_{n+1}∂_{n+1}u_{2}k^{1−α}L^{2}(B_{2}^{0}).

We modify this by interpolation in order to obtain an estimate where the normal derivative of
u2 is measured in the H^{−2s}norm. To this end, we note that for any w by interpolation and the
characterization of the trace map (c.f. Step 1 in the proof of Proposition 5.11 in [RS17]):

kwk^{L}^{2}(R^{n})≤ Ckwk

2s 1+s

H^{1−s}(R^{n})kwk

1−s 1+s

H^{−2s}(R^{n})

≤ C

kxn+1^{2s−1}^{2} wkL^{2}(R^{n+1}_{+} )+kxn+1^{2s−1}^{2} ∇wkL^{2}(R^{n+1}_{+} )

1+s^{2s}

kwk

1−s 1+s

H^{−2s}(R^{n})

≤ C
µ^{1−s}

kxn+1^{2s−1}^{2} wkL^{2}(R^{n+1}_{+} )+kxn+1^{2s−1}^{2} ∇wkL^{2}(R^{n+1}_{+} )

+ µ^{−2s}kwk^{H}^{−2s}(R^{n})

. (13)

Applying this to w = ηx^{1−2s}_{n+1}∂n+1u2, where η is a smooth, radial cut-off function which is equal
to one on B_{2}^{+} and vanishes outside of B_{4}^{+} gives us

(14)

k lim_{x}

n+1→0x^{1−2s}_{n+1}∂n+1u2k^{L}^{2}^{(B}^{0}2)

≤C
µ^{1−s}

kxn+1^{1−2s}^{2} ∂n+1u2kL^{2}(B_{4}^{+})+kxn+1^{2s−1}^{2} ∇(ηx^{1−2s}n+1∂n+1u2)kL^{2}(R^{n+1}_{+} )

+µ^{−2s}k lim_{x}

n+1→0ηx^{1−2s}_{n+1}∂n+1u2k^{H}^{−2s}(R^{n})

.

Similar to the proof of Proposition 5.11 in [RS17], we now estimate each term on the right hand side of (14). The last term gives us

(15) k lim_{x}

n+1→0ηx^{1−2s}_{n+1}∂n+1u2k^{H}^{−2s}(R^{n})≤ Ck lim_{x}

n+1→0x^{1−2s}_{n+1}∂n+1u2k^{H}^{−2s}^{(B}^{0}8).
Applying Caccioppoli’s inequality in Lemma2.1(with zero Dirichlet condition) implies
(16) kxn+1^{1−2s}^{2} ∂_{n+1}u_{2}kL^{2}(B_{4}^{+})≤ Ckxn+1^{1−2s}^{2} u_{2}kL^{2}(B_{8}^{+}).

It remains to estimate the second term on the right hand side of (14). Indeed, for the resulting bulk term we have

(17)

kxn+1^{2s−1}^{2} ∇(ηx^{1−2s}n+1∂n+1u2)kL^{2}(R^{n+1}+ )≤ kxn+1^{1−2s}^{2} (∇η)(∂^{n+1}u2)kL^{2}(R^{n+1}+ )+kxn+1^{1−2s}^{2} η∇^{0}∂n+1u2kL^{2}(R^{n+1}+ )

+kxn+1^{2s−1}^{2} η∂n+1x^{1−2s}_{n+1}∂n+1u2kL^{2}(R^{n+1}_{+} )

≤ kxn+1^{1−2s}^{2} (∂n+1u_{2})kL^{2}(B^{+}_{4})+kxn+1^{1−2s}^{2} ∇^{0}∂_{n+1}u_{2}kL^{2}(B_{4}^{+})

+kxn+1^{1−2s}^{2} η∆^{0}u_{2}kL^{2}(R^{n+1}_{+} )

≤ Ckxn+1^{1−2s}^{2} u2kL^{2}(B^{+}_{8}).

Here we first used the triangle inequality, then the support condition for η and the equation for
u_{2} and finally applied Caccioppoli’s inequality (twice for the last two terms, noting that ∇^{0}u_{2}
solves a similar problem).

Substituting (15)-(17) into (14) and optimizing the resulting estimate in µ > 0 gives
k lim_{x}

n+1→0x^{1−2s}_{n+1}∂_{n+1}u_{2}k^{L}^{2}^{(B}^{0}2)≤ Ckxn+1^{1−2s}^{2} u_{2}k

2s 1+s

L^{2}(B^{+}_{8})k lim_{x}

n+1→0x^{1−2s}_{n+1}∂_{n+1}u_{2}k

1−s 1+s

H^{−2s}(B_{8}^{0}).
Inserting this into (12) leads to

(18) kxn+1^{1−2s}^{2} u2kL^{2}(B^{+}c)≤ Ckxn+1^{1−2s}^{2} u2k^{α}L^{˜}^{2}(B^{+}_{8})k lim_{x}

n+1→0x^{1−2s}_{n+1}∂n+1u2k^{1− ˜}H^{−2s}^{α} (B^{0}_{8})

where ˜α = ^{1−s}_{1+s}α+_{1+s}^{2s} . By slight abuse of notation, in the sequel, we simply drop the tilde.

Combining the two bounds (11), (18) and
k lim_{x}

n+1→0x^{1−2s}_{n+1}∂n+1u1kH^{−2s}(B^{0}_{16})≤ k(−∆)^{s}u1k^{L}^{2}(R^{n})≤ ku^{1}k^{L}^{2}(R^{n})≤ Ckuk^{L}^{2}^{(B}_{16}^{0} ^{)},

and applying the triangle inequality leads to
kxn+1^{1−2s}^{2} u˜kL^{2}(B_{c}^{+})

≤C

kxn+1^{1−2s}^{2} u˜kL^{2}(B^{+}_{16})+kuk^{L}^{2}^{(B}_{16}^{0} ^{)}^{α}
k lim_{x}

n+1→0x^{1−2s}_{n+1}∂n+1u˜kH^{−2s}(B^{0}_{16})+kuk^{L}^{2}^{(B}_{16}^{0} ^{)}

1−α

≤C

kxn+1^{1−2s}^{2} u˜kL^{2}(B^{+}_{16})+kuk^{L}^{2}^{(B}_{16}^{0} ^{)}^{α}
k lim_{x}

n+1→0x^{1−2s}_{n+1}∂n+1u˜k^{L}^{2}^{(B}_{16}^{0} ^{)}+kuk^{L}^{2}^{(B}^{0}_{16}^{)}

1−α

. (19)

This already implies the claim of (a). In order to exploit it for the proof of (b), we strengthen
the estimate slightly. By Caccioppoli’s inequality of Lemma2.1 (now with an L^{2} estimate for
the boundary contributions), we can further upgrade (19) to

kxn+1^{1−2s}^{2} u˜kL^{2}(B_{c}^{+}_{˜})+kxn+1^{1−2s}^{2} ∇˜ukL^{2}(B^{+}_{˜}_{c})

≤C

kxn+1^{1−2s}^{2} u˜kL^{2}(B_{16}^{+})+kuk^{L}^{2}^{(B}_{16}^{0} ^{)}^{α}
k lim_{x}

n+1→0x^{1−2s}_{n+1}∂_{n+1}u˜k^{L}^{2}^{(B}^{0}_{16}^{)}+kuk^{L}^{2}^{(B}^{0}_{16}^{)}

1−α

+ Ck lim_{x}

n+1→0x^{1−2s}_{n+1}∂_{n+1}u˜kL^{1}^{2}^{2}(B_{16}^{0} )kukL^{1}^{2}^{2}(B^{0}_{16})

(20)

with ˜c= c/2.

Step 1b: The case s ∈ (0,^{1}2). The case s ∈ (0, 1/2) is similar as the case discussed above
and relies on a splitting strategy. As above, the estimate for u1 is a direct consequence of the
boundary bulk estimates for the Caffarelli-Silvestre extension. Thus, the main remaining estimate
is the derivation of the corresponding analogue of (18). As in the proof of (18) this follows an
application of the corresponding L^{2}result from [RS17] (Proposition 5.12) and interpolation. More
precisely, Proposition 5.12 in [RS17] implies that for some c = c(n, s) > 0 and α = α(n, s)∈ (0, 1)
we have

kxn+1^{1−2s}^{2} u_{2}kL^{2}(B^{+}_{c})

≤ C(kxn+1^{1−2s}^{2} u2k^{α}L^{2}(B_{2}^{+})k lim_{x}

n+1→0x^{1−2s}_{n+1}∂n+1u2k^{1−α}L^{2}(B^{0}_{2})+k lim_{x}

n+1→0x^{1−2s}_{n+1}∂n+1u2k^{L}^{2}^{(B}2^{0})).

In order to pass from this estimate which involves an L^{2} norm of the weighted Neumann data
to an estimate which involves its H^{−2s} norm, we apply the interpolation estimate (13) as in
the case s∈ (^{1}2,1). With this estimate at hand, the analogues of (19) and (20) then follow by
combining the estimates of the splitting argument as above. Note that (20) now becomes

kxn+1^{1−2s}^{2} u˜kL^{2}(B^{+}_{˜}_{c})+kxn+1^{1−2s}^{2} ∇˜ukL^{2}(B_{c}^{+}_{˜})

≤C

kxn+1^{1−2s}^{2} u˜kL^{2}(B^{+}_{16})+kuk^{L}^{2}^{(B}^{0}16)

^{α}
k lim_{x}

n+1→0x^{1−2s}_{n+1}∂n+1˜uk^{L}^{2}^{(B}16^{0} )+kuk^{L}^{2}^{(B}16^{0} )

1−α

+ C

kxn+1^{1−2s}^{2} u˜kL^{2}(B_{16}^{+})+kuk^{L}^{2}^{(B}^{0}_{16}^{)}1+s^{2s}
k lim_{x}

n+1→0x^{1−2s}_{n+1}∂n+1u˜k^{L}^{2}^{(B}_{16}^{0} ^{)}+kuk^{L}^{2}^{(B}^{0}_{16}^{)}

^{1−s}_{1+s}

+ Ck lim_{x}

n+1→0x^{1−2s}_{n+1}∂_{n+1}u˜kL^{1}^{2}^{2}(B^{0}_{16})kukL^{1}^{2}^{2}(B_{16}^{0} ).
(21)

Step 2: The L^{∞} estimate. In order to pass from the L^{2}-based bounds from step 1 to L^{∞}
based estimates, we rely on an estimate due to Jin, Li, Xiong [JLX11] (Proposition 2.4 (i), c.f.

also Proposition 3.2 in [FF14]), which states that under our conditions on B_{1/2}^{+} it holds
k˜ukL^{∞}(B_{1/2}^{+} )≤ C(kxn+1^{1−2s}^{2} u˜kL^{2}(B^{+}_{1})+kxn+1^{1−2s}^{2} ∇˜ukL^{2}(B_{1}^{+})).

(22)

Combining this with the estimate (21) and inserting the identity cn,s lim

x_{n+1}→0x^{1−2s}_{n+1}∂n+1u˜ = qu
entails

k˜ukL^{∞}(B^{+}_{˜}_{c}

2

)≤C

kxn+1^{1−2s}^{2} ˜ukL^{2}(B_{16}^{+})+kuk^{L}^{2}^{(B}16^{0} )

^{α}

kquk^{L}^{2}^{(B}^{0}16)+kuk^{L}^{2}^{(B}^{0}16)

^{1−α}

+ C

kxn+1^{1−2s}^{2} u˜kL^{2}(B^{+}_{16})+kuk^{L}^{2}^{(B}_{16}^{0} ^{)}1+s^{2s}

kquk^{L}^{2}^{(B}^{0}_{16}^{)}+kuk^{L}^{2}^{(B}_{16}^{0} ^{)}^{1−s}_{1+s}
+ CkqukL^{1}^{2}^{2}(B_{16}^{0} )kukL^{1}^{2}^{2}(B^{0}_{16}).

(23)

Based on this, we can also derive a pure L^{∞}estimate. Indeed, using H¨older’s inequality together
with the L^{2}integrability of the function (0, 1)3 t 7→ t^{1−2s}^{2} ∈ R for s ∈ (0, 1) on bounded domains
results in

k˜ukL^{∞}(B^{+}_{c}_{˜}

2

)≤C(1 + kqk^{L}^{∞}^{(B}_{16}^{0} ^{)})^{1−α}

k˜ukL^{∞}(B^{+}_{16})+kuk^{L}^{∞}^{(B}^{0}_{16}^{)}^{α}

kuk^{1−α}L^{∞}(B_{16}^{0} )

+ C(1 +kqk^{L}^{∞}^{(B}16^{0} ))^{1−s}^{1+s}

k˜ukL^{∞}(B^{+}_{16})+kuk^{L}^{∞}^{(B}^{0}16)

_{1+s}^{2s}
kuk

1−s 1+s

L^{∞}(B^{0}_{16})

+ CkqkL^{1}^{2}^{∞}(B^{0}_{16})kuk^{L}^{∞}^{(B}^{0}16)

≤C(1 + kqk^{L}^{∞}^{(B}_{16}^{0} ^{)})^{1−s}^{1+s}

k˜ukL^{∞}(B^{+}_{16})+kuk^{L}^{∞}^{(B}_{16}^{0} ^{)}^{α}

kuk^{1−α}L^{∞}(B_{16}^{0} )

+

k˜ukL^{∞}(B^{+}_{16})+kuk^{L}^{∞}^{(B}^{0}_{16}^{)}_{1+s}^{2s}
kuk

1−s 1+s

L^{∞}(B^{0}_{16})

+ CkqkL^{1}^{2}^{∞}(B^{0}_{16})kuk^{L}^{∞}^{(B}^{0}16).
Here we recall that

1− α = 1− s

1 + s−1− s

1 + sα < 1− s 1 + s.

Proof of Proposition2.2. Step 1: L^{∞}decay. We first prove that the L^{2} bound in the statement
of the proposition entails a similar L^{∞} bound. In the sequel, we denote by ˜c, ˜Cgeneral positive
constants which may depend on n, s and which are likely to change from line to line. Pick any
R≥ 1 and x^{0}∈ R^{n}× {0} with |x^{0}| = 32R, (9) implies

kuk^{L}^{2}^{(B}16R^{0} (x0))≤ ˜Ce^{−˜}^{cR}^{β}.
(24)

We next recall that by the H^{s}(R^{n}) boundedness of u and the properties of the Caffarelli-Silvestre
extension, we have kxn+1^{1−2s}^{2} ∇˜ukL^{2}(R^{n+1}+ ) ≤ C. Thus, the L^{2} boundedness of u and Poincar´e’s
inequality then also yield a bound for kxn+1^{1−2s}^{2} u˜k^{L}^{2}(R^{n}×(0,C1)) for any C1 > 0. Combining the
bound from Proposition2.4(a), the L^{2}_{loc}(R^{n+1}+ , x^{1−2s}_{n+1}) boundedness of ˜u, the fact thatkqk^{L}^{∞} ≤ 1
and (24), we infer that

kxn+1^{1−2s}^{2} u˜kL^{2}(B^{+}_{cR}(x0)) ≤ ˜Ce^{−˜}^{cR}^{β}.

R 2R

−R

−2R

xn+1

x^{0}

Figure 1. The chain of balls from the proof of Proposition2.2.

Here c > 0 denotes the constant from Proposition2.4. Finally, invoking a translated and rescaled version of (10) then also entails the bound

(25) k˜ukL^{∞}(B^{+}

c2 R 16

(x0))≤ ˜Ce^{−˜}^{cR}^{β}.

This yields a bound for ˜ufor|x| ≥ 32R. We may in particular use this for R = 10. Hence, only
on the compact set B_{C}^{+} with C = 320 an L^{∞} bound has not yet been obtained. This however
follows by applying a rescaled version of (22). Thus, ˜u ∈ L^{∞}(R^{n+1}+ ). The estimate in the
bounded region and the quantitative estimate in the unbounded annuli can finally be combined
to infer that for all R > 0 and x0∈ R^{n}× {0}, |x^{0}| = 32R we have

k˜ukL^{∞}(B^{+}

c2 R 16

(x0))≤ ˜Ce^{−˜}^{cR}^{β}.

In particular, by choosing R = ^{16}_{c}R˜ and keeping x0∈ R^{n}× {0}, |x^{0}| = 32R, we also obtain the
estimate

(26) k˜ukL^{∞}(B^{+}

c ˜R(x_{0})) ≤ ˜Ce^{−¯}^{c ˜}^{R}^{β}.

Step 2: Conclusion. With the bounds from Step 1, Propositions2.3and2.4at hand, the proof
of Proposition2.2follows by a chain of balls argument. More precisely, for x = (x^{0}, xn+1) with

|x| > 2 there exists a value R = 2^{k} for k∈ N such that x ∈ A^{+}R,2R:={x ∈ R^{n+1}+ : |x| ∈ (R, 2R]}.

This annulus can be covered by a finite union of balls and half balls:

A^{+}_{R,2R}⊂

m1

[

j=1

B_{r}^{+}_{j}(xj)∪

m2

[

k=1

B_{cR}^{+}((x^{0}k,0))

with the property that these balls form a chain, i.e. there is sufficient overlap between these to
iterate the following estimates (c.f. Figure1for an illustration of this), where c is the constant
derived in Proposition 2.4. We explain this iteration more carefully. Starting with a half ball
B_{cR}^{+}(¯x) with ¯x= (x^{0}_{1},0)∈ R^{n}× {0} we invoke (26) to infer that

k˜ukL^{∞}(B_{cR}^{+}(¯x))≤ Ce^{−aR}^{β},
where a = a(n, s) > 0.

We now begin with the propagation of decay estimates into R^{n+1}+ along a chain of balls.

We first choose a ball B_{r}^{+}_{1}(x1) ⊂ BcR^{+}(¯x) with r1 = ^{cR}_{5} in such a way that a large part of
B_{r}^{+}_{1}(x1) ⊂ A^{+}R,2R, B_{4r}^{+}_{1} ⊂ R^{n+1}+ and such that |B^{+}2r1(x1)∆B_{cR}^{+} (¯x)| ≥ c^{0}R for some constant

c0= c0(n, s) > 0. In B_{4r}^{+}_{1}(x1) we apply the three balls inequality of Proposition2.3. Therefore,
in combination with the L^{∞}estimates from step 1, we obtain

k˜ukL^{∞}(B_{2r1}^{+} (x_{1}))≤ Ck˜uk^{α}L^{∞}(B^{+}

4r1(x_{1}))k˜uk^{1−α}_{L}∞(B^{+}_{r1}(x_{1}))

≤ Ck˜uk^{1−α}_{L}∞(B_{cR}^{+}(¯x))

≤ Ce^{−a(1−α)R}^{β}.

We iterate this along our chain of balls B^{+}r_{j}(xj) with rj =^{cR}_{5} , which allows us to eventually cover
A^{+}_{R,2R} with a finite number of balls. In particular, it yields the decay estimate

k˜ukL^{∞}(A^{+}_{R,2R})≤ Ce^{−˜}^{aR}^{β}

for some positive constants C(m1, m_{2}, n, s), ˜a(m1, m_{2}, n, s). Finally, the analogous bounds also
follow in arbitrary other dyadic annuli centered at zero by rescaling the previous estimate (the
number of balls in the chain of balls estimate stays constant, since the size of the balls is also

rescaled).

2.3. An interpolation inequality. We need another bulk-boundary interpolation estimate, which will play a relevant role in our Carleman inequalities in the next section. Although this was already introduced in [R¨ul15], we reprove it here for self-containedness.

Proposition 2.5. Let s∈ (0, 1) and u : S+^{n} → R with u ∈ H^{1}(S_{+}^{n}, θ^{1−2s}_{n} ). Then there exists a
constant C = C(s, n) > 0 such that for all τ > 1

kukL^{2}(S^{n−1})≤ C(τ^{1−s}kθ^{n}^{1−2s}^{2} uk^{L}^{2}^{(S}_{+}^{n}^{)}+ τ^{−s}kθ^{n}^{1−2s}^{2} ∇^{S}^{n}uk^{L}^{2}^{(S}_{+}^{n}^{)}).

Proof. We argue in two steps.

Step 1: Derivation of a whole space estimate. Let w∈ H^{s}(R^{n})∩H^{1}(R^{n+1}+ , x^{1−2s}_{n+1}). Then trace
estimates in the space H^{1}(R^{n+1}+ , x^{1−2s}_{n+1}) (c.f. for instance Lemma 4.4 in [RS17]) imply

(27) kwk^{L}^{2}(R^{n})≤ C(kxn+1^{1−2s}^{2} wkL^{2}(R^{n+1}+ )+kxn+1^{1−2s}^{2} ∇wkL^{2}(R^{n+1}+ )).

Starting from (27), scaling x by τ^{−1} with τ > 0 (i.e., x→ τ^{−1}x), we then obtain
kwk^{L}^{2}(R^{n})≤ C(τ^{1−s}kxn+1^{1−2s}^{2} wkL^{2}(R^{n+1}_{+} )+ τ^{−s}kxn+1^{1−2s}^{2} ∇wkL^{2}(R^{n+1}_{+} )).

(28)

Step 2: Conclusion. Considering u ∈ H^{1}(S+^{n}, θ^{1−2s}_{n} ), we first extend this function zero ho-
mogeneously into a neighbourhood of S+^{n} and multiply it by a cut-off function, i.e. we define
w(x) := η(x)u(_{|x|}^{x}), where η(x) = 1 if|x| ∈ (1/2, 3/2) and η(x) = 0 if |x| ∈ (0, 1/4) ∪ (2, ∞). The
resulting compactly supported function still satisfies w ∈ H^{1}(R^{n+1}+ , x^{1−2s}_{n+1}) and further has the
property that

kxn+1^{1−2s}^{2} ∇wkL^{2}(R^{n+1}+ )≤ C(kθ^{n}^{1−2s}^{2} ∇^{S}^{n}uk^{L}^{2}^{(S}^{n}+)+kθ^{n}^{1−2s}^{2} uk^{L}^{2}^{(S}^{n}+)),
kxn+1^{1−2s}^{2} wkL^{2}(R^{n+1}_{+} )≤ Ckθ^{n}^{1−2s}^{2} uk^{L}^{2}^{(S}^{n}_{+}^{)},

kuk^{L}^{2}^{(S}^{n−1}^{)}≤ Ckwk^{L}^{2}(R^{n}).

Inserting these into (28) and choosing τ ≥ τ^{0}>1 (for some uniform τ0) then implies
kuk^{L}^{2}^{(S}^{n−1}^{)}≤ Ckwk^{L}^{2}(R^{n})≤ C(τ^{1−s}kxn+1^{1−2s}^{2} wkL^{2}(R^{n+1}+ )+ τ^{−s}kxn+1^{1−2s}^{2} ∇wk^{L}^{2}(R^{n+1}))

≤ C(τ^{1−s}(1 + τ^{−1})kθ^{n}^{1−2s}^{2} uk^{L}^{2}^{(S}+^{n})+ τ^{−s}kθ^{n}^{1−2s}^{2} uk^{L}^{2}^{(S}^{n}+)).