Discrete and Continuous Dynamical Systems

doi:10.3934/dcds.2024084

OPTIMAL BOUNDARY REGULARITY AND A HOPF-TYPE

LEMMA FOR DIRICHLET PROBLEMS INVOLVING THE
LOGARITHMIC LAPLACIAN

Vı́ctor Hernández-Santamarı́a 1 ,
∗1
Luis Fernando López Rı́os 2 and Alberto Saldaña
1 Instituto de Matemáticas, Universidad Nacional Autónoma de México,

Circuito Exterior, Ciudad Universitaria, 04510 Coyoacán, Ciudad de México, Mexico.

2 Institutode Investigaciones en Matemáticas Aplicadas y en Sistemas,
Universidad Nacional Autónoma de México, Circuito Escolar s/n,
Ciudad Universitaria, C.P. 04510, Ciudad de México, Mexico.

(Communicated by Xavier Ros-Oton)

Abstract. We study the optimal boundary regularity of solutions to Dirich-

let problems involving the logarithmic Laplacian. Our proofs are based on
the construction of suitable barriers via the Kelvin transform and direct com-
putations. As applications of our results, we show a Hopf-type lemma for
nonnegative weak solutions and the uniqueness of solutions to some nonlinear
problems.

1. Introduction. In this paper, we study the optimal boundary regularity of so-

lutions of
L∆ u = f in Ω, u=0 in RN \ Ω, (1)
N ∞
where Ω ⊂ R (N ≥ 1) is a bounded open set and f ∈ L (Ω). Here L∆ denotes the
logarithmic Laplacian, that is, the pseudodifferential operator with Fourier symbol
2 ln |ξ|. This operator can also be seen as a first-order expansion of the fractional
Laplacian (the pseudodifferential operator with symbol |ξ|2s ); in particular, for
ϕ ∈ Cc∞ (RN ),
(−∆)s ϕ = ϕ + sL∆ ϕ + o(s) as s → 0+ in Lp (R) with 1 < p ≤ ∞,
see [10, Theorem 1.1]. Moreover, it has also the following integral representation
u(x) − u(y)
Z Z
u(y)
L∆ u(x) := cN N
dy − cN dy + ρN u(x), (2)
B1 (x) |x − y| N
R \B1 (x) |x − y|N
where cN and ρN are explicit constants given in (14) and u is a suitable function.
The logarithmic Laplacian appears in some interesting applications. For instance,
it is a tool to characterize the differentiability properties of the solution mapping
of fractional Dirichlet problems, see [25, 26]. It is also used to describe the behav-
ior of solutions to linear and nonlinear problems involving the fractional Laplacian
(−∆)s in the small order limit (as s → 0), see [2, 22, 24], where the operator L∆

2020 Mathematics Subject Classification. Primary: 35S15, 35B65; Secondary: 35A02.

Key words and phrases. Hopf lemma, Kelvin transform, uniqueness by convexity.
∗ Corresponding author: Alberto Saldaña.

1
2 V. HERNÁNDEZ-SANTAMARÍA, L. F. LÓPEZ RÍOS AND A. SALDAÑA

naturally appears. We also mention the following very recent works: the Cauchy
problem with L∆ is studied in [9]; higher-order expansions of the fractional Lapla-
cian are considered in [7]; a characterization of the logarithmic Laplacian via a
local extension problem on the (N + 1)-dimensional upper half-space in the spirit
of the Caffarelli-Silvestre extension is obtained in [8]; and a finite element method
is analyzed and implemented in [23] to approximate solutions of (1).
The kernel in (2) is sometimes called of zero-order, because it is a limiting case for
hypersingular integrals. As a consequence, the regularizing properties of this type of
operators are very weak and are a subject of current research. Interior regularity of
(bounded weak) solutions to (1) has been studied in [6,10,21,27]; in particular, it is
known that if f ∈ L∞ (Ω) then u ∈ C(Ω). Furthermore, the modulus of continuity of
u can be characterized: u belongs to the space of α-log-Hölder continuous functions
in RN for some α ∈ (0, 1) (see the definition of Lα (RN ) in (20) below); in particular,
|u(x) − u(y)|
kukLα (RN ) = kukL∞ (RN ) + sup < ∞, (3)
x,y∈RN `α (|x − y|)
x6=y

where ` : [0, ∞) → [0, ∞) is given by

1
`(r) := ,
| ln(min{r, 0.1})|
see Figure 3 below.
This regularity is not enough to evaluate L∆ u pointwisely; however, if the right-
hand side f is 1-log-Hölder continuous in Ω, then u is (1 + α)-log-Hölder continuous
in Ω and, with this regularity, the equation (1) holds pointwisely, that is, u is a
classical solution (see [6, Theorem 1.1]).
It is an interesting question to determine what is the optimal α that characterizes
the regularity of a solution across the boundary of the domain ∂Ω, namely, the
largest α so that (3) holds. In this regard, in [10, Theorem 1.11] it has been
established that, if Ω is a Lipschitz domain with uniform exterior sphere condition,
f ∈ L∞ (Ω), and u is a (bounded weak) solution of (1), then there is C > 0 such
that
|u(x)| ≤ C`τ (dist(x, ∂Ω)) for all x ∈ Ω and for all τ ∈ (0, 21 ). (4)
The proof of this estimate is done by constructing suitable barriers via direct com-
putations and using comparison principles in small domains. Another related result
on boundary regularity is given in [28], where, using probabilistic tools, sharp two-
sided boundary estimates are shown for Green functions in bounded C 1,1 -domains
associated to a subordinate Brownian motion X when the Laplace exponent of the
corresponding subordinator is a Bernstein function. These results yield boundary
estimates for problems that are closely related to (1), for example, they apply to the
logarithmic Shrödinger operator (I − ∆)log , which is the pseudodifferential operator
with symbol ln(1 + |ξ|2 ), see [19, 20] and the references therein. For this operator,
the results in [28] would yield that (4) holds with τ = 12 . This strongly suggests that
the same should be true for the operator L∆ ; however, the results in [28] cannot
be applied directly to L∆ (note that the symbol 2 ln |ξ| is negative for |ξ| < 1 and
tends to −∞ as ξ → 0).
Our main results show that (4) also holds for τ = 12 and that this value is optimal.
Let us introduce some notation. As usual, Br (x) denotes the open ball in RN of
radius r > 0 centered at x and Br := Br (0). We say that a bounded open set
 OPTIMAL BOUNDARY REGULARITY FOR THE LOGARITHMIC LAPLACIAN 3

Ω ⊂ RN satisfies a (uniform) exterior sphere condition if there is δ > 0 such that,

for all x0 ∈ ∂Ω,
there is y0 ∈ RN with Bδ (y0 ) ∩ Ω = {x0 }. (5)
We use H(Ω) to denote the Hilbert space associated to L∆ with Dirichlet exterior
conditions, see (15).
Theorem 1.1. Let Ω ⊂ RN be a bounded open set satisfying an exterior uniform
sphere condition, let f ∈ L∞ (Ω), and let u ∈ H(Ω) ∩ L∞ (RN ) be a weak solution
of (1). Then u ∈ C(RN ) and there is C > 0 such that
1
|u(x)| ≤ C` 2 (dist(x, ∂Ω)) for all x ∈ Ω. (6)
This result is the analogue of [30, Lemma 2.7] for the logarithmic Laplacian.
By studying the case of the torsion function in a (small) ball, we also show that
this boundary regularity in Theorem 1.1 is optimal. For A ⊂ RN , we use |A| to
denote its Lebesgue measure. Let Ψ denote the digamma function and let γ be the
Euler-Mascheroni constant (see the Notation section below).
N N
Theorem 1.2. Let N ≥ 1, r > 0 be such that |Br | < 2N e 2 (Ψ( 2 )−γ) |B1 |, and let τ
be the torsion function of the ball Br , namely, the unique classical solution of
L∆ τ = 1 in Br , τ =0 in RN \Br . (7)
Then,
τ (x0 − tη(x0 ))
∞ > lim inf 1 >0 for x0 ∈ ∂Br , (8)
+
t→0 ` 2 (t)
where η is the outer unit normal vector field along ∂Br . Moreover, there is c > 1
such that
1 1
c−1 ` 2 (r − |x|) ≤ τ (x) ≤ c ` 2 (r − |x|) for x ∈ Br . (9)

We refer to [23, Figure 1] for the numerical approximation of the torsion function
in different intervals.
The proofs of Theorems 1.1 and 1.2 are done by constructing suitable barriers
and by using a comparison principle in small domains. However, in the case of the
logarithmic Laplacian, this is a highly nontrivial task, because there is no easy choice
for a suitable barrier; in particular, a closed formula for the torsion function in any
ball is not available and one cannot argue as in [30] (this is also the reason why (4)
is only established for τ ∈ (0, 12 ) in [10]). To overcome this difficulty, we use the
following strategy. First, we construct a barrier in an open interval I = (0, 2) that
1
behaves as ` 2 (x) for x close to 0 (see (26)). Then, via sharp direct computations, we
show that its logarithmic Laplacian is bounded in I and that it is positive close to
0, see Theorem 2.4. Afterwards, we use this barrier to construct higher-dimensional
barriers in half balls (see Figure 1 and (44)). Using the calculations in the one-
dimensional case and a Leibniz-type formula (see Section A.2), we show that this
new function also has a bounded logarithmic Laplacian that is positive close to the
origin (see Theorem 2.19).
Finally, to obtain a suitable barrier to characterize domains satisfying the (uni-
form) exterior sphere condition, we use the following Kelvin transform formula,
which is of independent interest. Let x, x0 ∈ RN and R > 0, the inversion of x with
4 V. HERNÁNDEZ-SANTAMARÍA, L. F. LÓPEZ RÍOS AND A. SALDAÑA

1.4

1.2

1.0

0.8

0.6

0.4

0.2

-0.5 0.0 0.5 1.0 1.5

Figure 1. The shape of the initial barriers for N = 1 and N = 2.

respect to the sphere SR (x0 ) := ∂BR (x0 ) is given by

x − x0
x∗ = x0 + R 2 , x 6= x0 ,
|x − x0 |2
and the Kelvin transform of a function u : RN → R with respect to the sphere
SR (x0 ) is
u∗ (x∗ ) = |x∗ − x0 |−N u(x), x∗ 6= x0 .
For Ω ⊂ RN we define Ω∗ = {x∗ : x ∈ Ω}.
Proposition 1.3. Let x0 ∈ RN , R > 0, and x∗ denote the inversion of x with
respect to the sphere SR (x0 ). Let Ω be an open bounded Lipschitz set and u : RN →
R be a measurable function with u ≡ 0 on RN \ Ω.
i) If u is Dini continuous in x 6= x0 , then u∗ is Dini continuous in x∗ and
|z|−N − |x|−N
Z
∗ ∗ ∗ −N ∗ −2N
L∆ u (x ) = |x | L∆ u(x) + cN |x | u(x) dz
Ω |z − x|N
+ (hΩ∗ (x∗ ) − hΩ (x))|x∗ |−N u(x), (10)
where hΩ is given in (18).
ii) If u, L∆ u ∈ L∞ (Ω) and BR (x0 ) ⊂ Ωc , then u∗ , L∆ u∗ ∈ L∞ (Ω∗ ).
iii) If BR (x0 ) ⊂ Ωc and u ∈ H(Ω), then u∗ ∈ H(Ω∗ ).
 OPTIMAL BOUNDARY REGULARITY FOR THE LOGARITHMIC LAPLACIAN 5

Note that formula (10) is different from the one obtained for the fractional Lapla-
cian (see, for example, [3] or [1, Proposition 2]). For the logarithmic Laplacian, the
geometry of the domain has a stronger influence in the formula, and this is repre-
sented in the last two summands in (10).
Using the initial barrier and the Kelvin transform, we obtain a new function
which has a bounded logarithmic Laplacian and which has the optimal regularity
at the curved part of the boundary (Proposition 3.4), see Figure 2.

1.0

0.5

-0.5

-1.0

Figure 2. The Kelvin transform can be used to obtain a new barrier.

In the picture, the dotted line represents the boundary of the unitary
circle. The red line segment (representing a subset of a hyperplane)
is sent with the Kelvin transform to the red half-circle (representing a
halfsphere).

With this barrier one has the main ingredient to characterize the optimal bound-
ary regularity. The other important ingredient is the comparison principle. Al-
though the operator L∆ does not satisfy the maximum principle in general, in [10,
Corollary 1.9] (see Theorem 4.1 in Section 4) it is shown that it holds in sufficiently
small domains. With this and with suitable scalings (see Section A.1) one can adapt
the method of barrier functions.
As a byproduct of our approach, we can show the following Hopf-type lemma for
the logarithmic Laplacian (following the ideas in [16]). Let EL be the bilinear form
6 V. HERNÁNDEZ-SANTAMARÍA, L. F. LÓPEZ RÍOS AND A. SALDAÑA

associated to L∆ (see (15)) and let

(
V(Ω) := u ∈ L1loc (RN ) :
)
|u(x)| |u(x) − u(y)|2
Z Z Z Z
2
dx + dx dy + u dx < ∞ ,
RN (1 + |x|)N x,y∈Ω |x − y|N Ω
|x−y|<1
(11)
which is introduced in [10] for the validity of a maximum principle, see Theorem 4.1
below.
We say that an open set Ω ⊂ RN satisfies an interior sphere condition at x0 ∈ ∂Ω
if there is δ > 0 and y0 ∈ RN with Bδ (y0 ) ⊂ Ω and Sδ (y0 ) ∩ ∂Ω = {x0 }.
Theorem 1.4 (A Hopf-type lemma for the logarithmic Laplacian). Let Ω ⊂ RN be
an open set and assume that Ω satisfies the interior sphere condition at x0 ∈ ∂Ω.
If v ∈ C(RN ) ∩ V(Ω) is a nontrivial nonnegative function such that v(x0 ) = 0 and
EL (v, ϕ) ≥ 0 for all ϕ ∈ Cc∞ (Ω) with ϕ ≥ 0 in Ω,
then
v(x0 − tη(x0 ))
lim inf 1 > 0, (12)
t→0+ ` 2 (t)
where η is an outer unit normal vector field along ∂Ω.
As a further application of our regularity results and of Theorem 1.4, we also
show the uniqueness of positive solutions for the nonlinear problem
L∆ v = −µ ln (|v|) v in Ω, v=0 in RN \Ω, (13)
where µ > 0 and Ω is a bounded open set of class C 2 , see Theorem 5.4 in Section 5.3.
The proof of this result relies on a convexity-by-paths argument, following the
approach in [4]. Problem (13) appears naturally in the small-order limit of solutions
to the fractional Lane-Emden equation; to be more precise, if us is a positive solution
of
(−∆)s us = usps −1 in Ω, us = 0 in RN \Ω,
where ps ∈ (1, 2) is such that µ = lims→0+ 2−p q
s , then us → v in L (Ω) for 1 ≤
s

q < ∞ as s → 0+ , where v is a positive solution of (13). See [2] for more details.
Note that these problems are sublinear. In the superlinear case (ps > 2) uniqueness
is known to be false in general (see [12–15, 17, 18] for uniqueness and multiplicity
results for positive solutions of fractional problems). Equation (13) with µ < 0
(which would be the superlinear case in the logarithmic setting) has been studied
in [24], but nothing is known yet about the uniqueness or multiplicity of positive
solutions and our techniques cannot be used in this setting.
To close this introduction, we mention that we believe a similar strategy can also
be used to study the sharp boundary behavior of more general integral operators
associated with zero-order kernels, as those considered in [6,11, 19,21], for instance.
The paper is organized as follows. In Section 2 we construct the initial barriers
described in Figure 1. Section 3 contains the proof of Proposition 1.3 regarding the
Kelvin transform and the construction of the barrier portrayed in Figure 2. Theo-
rem 1.1 is shown in Section 4, whereas the proofs of Theorem 1.2 (optimal regularity
 OPTIMAL BOUNDARY REGULARITY FOR THE LOGARITHMIC LAPLACIAN 7

for the torsion problem), Theorem 1.4 (Hopf-type lemma for the logarithmic Lapla-
cian), and the uniqueness of positive solutions of (13) (Theorem 5.5) are contained
in Section 5. Finally, we include an Appendix with some useful results regarding
scalings and a Leibniz-type formula for the logarithmic Laplacian of a product.
Notation. We use Br (x) to denote the open ball of radius r > 0 centered at
x ∈ RN . If x = 0, then we simply write Br := Br (0). We also set Sr (x) = ∂Br (x),
Sr := Sr (0), and
N
2π 2
σN := |S1 | =
Γ( N2 )
to denote the surface measure of the unit sphere. For U ⊂ RN , we use U c := RN \U.
The constants involved in the definition of L∆ are given by
N
cN := π − 2 Γ( N2 ), ρN := 2 ln 2 + Ψ( N2 ) − γ, and γ := −Γ0 (1). (14)
0
Here γ is also known as the Euler-Mascheroni constant and Ψ := ΓΓ is the digamma
function.
Next, following [10], we introduce the variational framework for L∆ . Let Ω ⊂ RN
be an open bounded set and let H(Ω) be the Hilbert space given by
(
|u(x) − u(y)|2
Z Z
H(Ω) := u ∈ L2 (RN ) : dx dy < ∞
x,y∈RN
|x − y|N
|x−y|≤1
)
and u = 0 in RN \ Ω (15)

with the inner product

(u(x) − u(y))(v(x) − v(y))
Z Z
cN
E(u, v) := dy dx,
2 RN B1 (x) |x − y|N
and the norm
1
kuk := (E(u, u)) 2 . (16)
The operator L∆ has the following associated quadratic form
Z Z Z
u(x)v(y)
EL (u, v) := E(u, v) − cN dx dy + ρ N uv dx. (17)
x,y∈RN |x − y|N RN
|x−y|≥1

Furthermore, for u ∈ H(Ω),

(u(x) − u(y))2
Z Z Z
cN
EL (u, u) = dx dy + (hΩ (x) + ρN )u(x)2 dx,
2 Ω Ω |x − y|N Ω
where
Z Z !
−N −N
hΩ (x) := cN |x − y| dy − |x − y| dy , (18)
B1 (x)\Ω Ω\B1 (x)

see [10, Proposition 3.2].

For f ∈ L2 (Ω), we say that u ∈ H(Ω) is a weak solution of L∆ u = f in Ω, u = 0
in RN \Ω, if
Z
EL (u, ϕ) = f ϕ dx for all ϕ ∈ Cc∞ (Ω).
Ω
8 V. HERNÁNDEZ-SANTAMARÍA, L. F. LÓPEZ RÍOS AND A. SALDAÑA

By [10, Theorem 1.1], it holds that

Z
EL (u, u) = ln(|ξ|2 )|û(ξ)|2 dξ for all u ∈ Cc∞ (Ω),
RN
where û is the Fourier transform of u. Moreover, for ϕ ∈ Cc∞ (Ω), we have that
L∆ ϕ ∈ Lp (RN ) and
Z
EL (u, ϕ) = uL∆ ϕ dx for u ∈ H(Ω), (19)
Ω
see [10, Theorem 1.1].
Let v : Ω → R be a measurable function. The modulus of continuity of v at a
point x ∈ Ω is defined by
ωv,x,Ω : (0, ∞) → [0, ∞), ωv,x,Ω (r) = sup |v(y) − v(x)|.
y∈Ω
|y−x|≤r
R1 ω (r)
The function v is called Dini continuous at x if 0 v,x,Ω
r dr < ∞. If
Z 1
ωv,Ω (r)
dr < ∞ for the uniform continuity modulus ωv,Ω (r) := sup ωv,x,Ω (r),
0 r x∈Ω

then we call v uniformly Dini continuous in Ω.

Let ` : [0, ∞) → [0, ∞) be given by
1
`(r) = − .
ln(min{0.1, r})

`(r)

1
10
| r

Figure 3. The function `.

For α > 0, we also define the α-log-Hölder Banach space (see [6, Lemma 7.1]) by
Lα (Ω) := {u : Ω → R : kukLα (Ω) < ∞}, (20)
where
|u(x) − u(y)|
kukLα (Ω) := kukL∞ (Ω) + [u]Lα (Ω) , [u]Lα (Ω) := sup α
,
x,y∈Ω ` (|x − y|)
x6=y
∞
and k · kL∞ (Ω) is the usual norm in L (Ω). We also set kuk∞ := kukL∞ (RN ) .
We shall use the following semi-homogeneity property for the modulus of conti-
nuity `.
Lemma 1.5 (Lemma 3.2 in [6]). There is c > 0 such that
`(λr) ≥ c `(λ)`(r) for all r, λ > 0.
 OPTIMAL BOUNDARY REGULARITY FOR THE LOGARITHMIC LAPLACIAN 9

2. An initial barrier. The main goal of this section is to show the following result.
Let N ≥ 1 and let
B2+ := B2 ∩ RN
+, where RN
+ := {x ∈ R
N
: x1 > 0}. (21)
1 +
Theorem 2.1. There is a function w ∈ L 2 (RN ) ∩ C ∞ (RN \RN
+ ) ∩ H(B2 ) such that
N +
w = 0 in R \B2 and
L∆ w ∈ L∞ (B2+ ).
Moreover, there is c > 0 and δ > 0 such that
1 1
c ` 2 (x1 ) < w(x) < c−1 ` 2 (x1 ) and L∆ w(x) > c for all x ∈ B2+ ∩ Bδ .
To show Theorem 2.1 we construct explicitly the function w and estimate its
logarithmic Laplacian with direct calculations.
We show first some auxiliary lemmas. Let Ω ⊂ RN be an open bounded set.
Lemma 2.2. Let V ∈ L∞ (RN ) be such that V = 0 in RN \Ω and
V (z) − V (z + y)
Z
sup dy < C (22)
z∈Ω B1 |y|N
for some C > 0. Then L∆ V ∈ L∞ (Ω).
Proof. Let x ∈ Ω, then
V (x) − V (x + y)
Z Z
V (y)
|L∆ V (x)| = cN N
dy − cN dy + ρN V (x)
B1 |y| Ω\B1 (x) |y − x|N
≤ cN C + kV k∞ (cN |Ω| + |ρN |) < ∞.

Lemma 2.3. Let V ∈ C(Ω) be such that V = 0 on ∂Ω and η ∈ [ 43 , 1]. Let U ⊂ Ω

be an open subset such that U ∩ ∂Ω 6= ∅ and
V (z) − V (z + y) |V (z + y)|
Z Z
σ
inf N
dy ≥ (1 + η)σ, sup cN N
dy ≤ (23)
z∈U B
1
|y| z∈U RN \B
1
|y| 2

for some σ > 0. Then there is an open subset U 0 ⊂ U such that U 0 ∩ ∂Ω = U ∩ ∂Ω

and
L∆ V ≥ σ in U 0 .

Proof. Since V = 0 on ∂Ω and U ∩ ∂Ω 6= ∅, there is, by continuity, an open subset

U 0 ⊂ U with U 0 ∩ ∂Ω = U ∩ ∂Ω and such that |ρN V | < σ4 in U 0 . Then, by (23), for
every x ∈ U 0 ,
V (x) − V (x + y)
Z Z
V (x + y)
L∆ V (x) = cN N
dy − c N dy + ρN V (x)
B1 |y| N
R \B1 |y|N
σ σ
≥ (1 + η)σ − − ≥ σ.
2 4

Now, we split the proof of Theorem 2.1 in two cases: the (simpler) one-dimensional
case and the higher-dimensional case.
10 V. HERNÁNDEZ-SANTAMARÍA, L. F. LÓPEZ RÍOS AND A. SALDAÑA

2.1. The one-dimensional case. Let

√ r
ln 2 4 1
0<ζ< ln < , (24)
4 3 4
let ϕ ∈ Cc∞ (R) be an even function such that
ϕ = 1 in (−1 − ζ, 1 + ζ), ϕ = 0 in R\(−1 − 2ζ, 1 + 2ζ), 0 ≤ ϕ ≤ 1 in R,
(25)
and let u : R → R be given by
1
u(x) := p ϕ(x)χ[0,∞) (x), (26)
− ln x2
where χ[0,∞) is the characteristic function of [0, ∞), see Figure 1 (left).
The goal of this section is to show Theorem 2.1 in the case N = 1, which we
state next.
1
Theorem 2.4. The function u given by (26) belongs to H((0, 2))∩C ∞ (0, ∞)∩L 2 (R)
and there is δ ∈ (0, 2) such that
√
L∆ u ∈ L∞ ((0, 2)) and L∆ u ≥ ln 2 in (0, δ).
We show first some auxiliary lemmas.
Lemma 2.5. The function u given by (26) satisfies that
1√
Z
u(ε + y)
dy < ln 2 for ε ∈ (0, 2ζ).
R\(−1,1) |y| 2

Proof. By (24) it follows that, for ε ∈ (0, 2ζ) ⊂ (0, 21 ),

Z Z 1+2ζ
u(ε + y) 1 ϕ(y)
0< dy = p y dy
R\(−1,1) |y| 1+ε − ln 2
|y − ε|
Z 1+2ζ−ε
1 1
≤q dy
− ln 1+2ζ 1 |y|
2
2ζ 2ζ 1√
≤q =q < ln 2.
1+ 1
− ln 34 2
− ln 2 2

Lemma 2.6. It holds that

Z ε
|u(ε) − u(y + ε)|
dy = o(1) as ε → 0.
−ε |y|
In particular,
2ε
u(ε) − u(y)
Z
J1 = J1 (ε) := dy = o(1) as ε → 0.
0 |ε − y|
Proof. Let ε > 0 be small enough so that u is increasing in (0, ε). If y < 0, then
(y + 1)ε < ε and u(ε) > u(ε) − u((y + 1)ε) > 0. Therefore,
Z − 21 Z − 12
|u(ε) − u((y + 1)ε)|
dy ≤ 2 u(ε) dy = u(ε) = o(1) as ε → 0,
−1 |y| −1
 OPTIMAL BOUNDARY REGULARITY FOR THE LOGARITHMIC LAPLACIAN 11

because u(0) = 0. Using that t 7→ |u0 (t)| is decreasing in (0, 41 ), it follows that
ε 1
< ε((1 − s)y + 1) < 2ε < for s ∈ (0, 1) and y ∈ (− 12 , 1),
2 4
and then
1 1 1
|u(ε) − u((y + 1)ε)|
Z Z Z
dy ≤ ε |u0 (ε((1 − s)y + 1))| ds dy
− 12 |y| − 21 0
3
≤ ε|u0 ( 14 ε)| = o(1)
2
as ε → 0.
Lemma 2.7. It holds that
Z 1 !
1 1 1
I := I(ε) := p −p dy = o(1)
ε y − ln (y + ε) + ln 2 − ln(y) + ln 2
as ε → 0.
Proof. Using the change of variables z = − ln y (y = e−z , dy = −e−z dz) we obtain
that
Z − ln ε
1 1
I= p −√ dz = o(1)
−z
− ln(e + ε) + ln 2 z + ln 2
0

as ε → 0. Indeed, consider
!
1 1
Fε (z) := p −√ χ{z<− ln ε} (z),
− ln(e−z + ε) + ln 2 z + ln 2
for z ∈ (0, ∞) and ε ∈ (0, 1). Observe that
1 1 1 1
√ − √ ≤ 3/2 (t − s), ln t − ln s ≤ (t − s) for all 0 < s ≤ t,
s t s s
by the mean value theorem. By applying these estimations to Fε we deduce that
−z −z
z + ln(e + ε) ln(e + ε) − ln (e−z )
Fε (z) ≤ 3/2
= 3/2
2 [− ln(e−z + ε) + ln 2] 2 [− ln(e−z + ε) + ln 2]
ez ε 1
≤ 3/2
≤ 3/2
for z ≤ − ln ε.
2 [− ln(e−z + ε) + ln 2] 2 [− ln(e−z + ε) + ln 2]
(27)
So let us define, for any ε ≥ 0,
1
Gε (z) := 3/2
, z ∈ R.
2 [− ln(e−z + ε) + ln 2]
Observe that Gε (z) &R ∞G0 (z) as εR → 0 for z ∈ R, so the monotone convergence
∞
theorem implies that 0 Gε (z) & 0 G0 (z) as ε → 0. On the other hand, by (27),
Fε (z) ≤ Gε (z) for z ∈ R and Fε (z) → F0 (z) = 0 as ε → 0 for zR ∈ R, so the
∞
generalized Lebesgue dominated convergence theorem implies that 0 Fε → 0 as
ε → 0, as claimed.
Lemma 2.8. It holds that
Z 1+ε
u(y) √ √
J2 = J2 (ε) := − dy = 2 ln 2 − 2 − ln ε + o(1) as ε → 0. (28)
2ε |ε − y|
12 V. HERNÁNDEZ-SANTAMARÍA, L. F. LÓPEZ RÍOS AND A. SALDAÑA

Proof. Let ε ∈ (0, ζ) with ζ as in (24). Observe that

Z 1 √
r
1 ε
y dy = 2 − ln − 2 ln 2. (29)
p
ε y − ln 2 2
Then, using (29),
Z 1+ε Z 1
u(y) u(y + ε)
J2 = − dy = − dy
2ε |ε − y| ε y
!
√ Z 1
r
ε 1 1
= 2 ln 2 − 2 − ln + − u(y + ε) dy
− ln y2
p
2 ε y
!
√ Z 1
r
ε 1 1 1
= 2 ln 2 − 2 − ln + p −p dy
2 ε y − ln(y) + ln 2 − ln (y + ε) + ln 2
√
r
ε
= 2 ln 2 − 2 − ln − I, (30)
2
with I as in Lemma 2.7. Therefore, by (30) and Lemma 2.7,
√ √
J2 = 2 ln 2 − 2 − ln ε + o(1) as ε → 0.

Theorem 2.9. The function u given by (26) belongs to C ∞ ((0, ∞)) and
Z ε+1
u(ε) − u(y) √
J = J(ε) := dy = 2 ln 2 + o(1) as ε → 0+ . (31)
ε−1 |y − ε|
Proof. That u ∈ C ∞ ((0, ∞)) is clear. Moreover, since u = 0 in (−∞, 0),
Z ε+1 Z 0
u(ε) − u(y)
J= dy + u(ε) |ε − y|−1 dy
0 |ε − y| ε−1
Z ε+1
u(ε) − u(y)
= dy + u(ε)(− ln ε) (32)
0 |ε − y|
for ε > 0 small. Let J1 and J2 as in Lemmas 2.6 and 2.8. Then,
Z 1+ε Z 2ε Z 1+ε
u(ε) − u(y) u(ε) − u(y) u(ε) − u(y)
dy = dy + dy
0 |ε − y| 0 |ε − y| 2ε |ε − y|
Z 1+ε
1
= J1 + J2 + u(ε) dy = J1 + J2 + u(ε)(− ln ε).
2ε |ε − y|
(33)
Now the claim (31) follows by (32), (33), Lemma 2.6, Lemma 2.8, and the fact that
√
2u(ε)(− ln(ε)) = 2 − ln ε + o(1) as ε → 0+ .

Lemma 2.10. The function u given by (26) belongs to H((0, 2)).

Proof. Let ζ be as in (24). For x ∈ U := (−3, 3)\(−ζ, ζ) and y ∈ (−1, 1), using a
Taylor expansion, we have that
|u(x + y) − u(x)| = |yu0 (x) + |y|h(y)| ≤ |y|(ku0 kL∞ (U ) + |h(y)|)
for some h ∈ L∞ ((−1, 1)) such that h(y) = o(1) as y → 0. Then,
Z Z 1
(u(y + x) − u(x))u(y + x)
A1 := dy dx
U −1 |y|
 OPTIMAL BOUNDARY REGULARITY FOR THE LOGARITHMIC LAPLACIAN 13

Z Z 1
≤ kuk∞ ku0 kL∞ (U ) + khkL∞ ((−1,1)) dy dx < ∞,
U −1
ζ 1 ζ
(u(y + x) − u(x))u(y + x)
Z Z Z
A2 := dy dx ≤ 2kuk2∞ − ln(x) dx < ∞.
0 x |y| 0
Moreover, using that u = 0 in (−∞, 0) and Lemma 2.6,
Z ζZ x
(u(y + x) − u(x))u(y + x)
A3 := dy dx
0 −1 |y|
Z ζZ x
|u(x) − u(y + x)|
≤ kuk∞ dy dx < ∞.
0 −x |y|
Furthermore, using that u = 0 in (−∞, 0),
Z 0 Z 1
(u(y + x) − u(x))u(y + x)
A4 := dy dx
−ζ −1 |y|
Z 0 Z 1
(u(y + x) − u(x))u(y + x)
= dy dx
−ζ −x |y|
Z 0
≤ 2kuk2∞ − ln(−x) dx < ∞.
−ζ

Then,
1
(u(y + x) − u(x))u(y + x)
Z Z
A := dy dx = A1 + A2 + A3 + A4 < ∞. (34)
R −1 |y|
Recall the definition of the norm k · k given in (16). Then,
Z Z x+1
2 (u(x) − u(y))(u(x) − u(y))
2kuk = dy dx
R x−1 |x − y|
Z Z x+1 Z Z x+1
(u(x) − u(y))u(x) (u(y) − u(x))u(y)
= dy dx + dy dx.
R x−1 |x − y| R x−1 |x − y|
Using Fubini’s theorem, a change of variables (z = x − y), and (34),
Z Z x+1 Z Z y+1
(u(x) − u(y))u(x) (u(x) − u(y))u(x)
dy dx = dx dy
R x−1 |x − y| R y−1 |x − y|
Z Z 1
(u(z + y) − u(y))u(z + y)
= dz dy = A < ∞.
R −1 |z|
Similarly,
Z Z x+1 Z Z 1
(u(y) − u(x))u(y) (u(z + x) − u(x))u(z + x)
dy dx = dz dx
R x−1 |x − y| R −1 |z|
= A < ∞.
Thus kuk < ∞. Since we also know that u ∈ L2 (R), it follows that u ∈ H((0, 2)) as
claimed.
We are ready to show Theorem 2.4.
Proof of Theorem 2.4. By Lemma 2.10 and by the definition of u it holds that u ∈
1
H((0, 2))∩C ∞ ((0, ∞))∩L 2 (RN ). By Theorem 2.9 and the fact that u ∈ C ∞ ((0, ∞))
we have that (22) holds (with Ω = (0, 2) and V = u). Then, by Lemma 2.2, we
14 V. HERNÁNDEZ-SANTAMARÍA, L. F. LÓPEZ RÍOS AND A. SALDAÑA
√
have that L∆ u ∈ L∞ ((0, 2)). Finally, by
√ Lemma 2.3 (with σ = ln 2), Lemma 2.5,
and Theorem 2.9 we have that L∆ u ≥ ln 2 in (0, δ) for some δ > 0.

2.2. The higher-dimensional case. In this section we extend the ideas of the
one-dimensional case to higher dimensions. Let N ≥ 2,
RN
+ := {x ∈ R
N
: x1 > 0},
and let ζ be such that
 √ r ! N1 
1 N ln 2 4
ζ< 1+ ln − 1 . (35)
2 4 3
1
Note that, since (1 + N x) N < 1 + x for x ≥ 0, inequality (35) implies (24); namely,
√ r
ln 2 4 1
0<ζ< ln < .
4 3 4
Let u and ϕ as in (25) and (26); namely, ϕ ∈ Cc∞ (R) is an even function such
that
ϕ = 1 in (−1 − ζ, 1 + ζ), ϕ = 0 in R\(−1 − 2ζ, 1 + 2ζ), 0 ≤ ϕ ≤ 1 in R
− 12
and u(x) := (− ln x2 ) ϕ(x)χ[0,∞) (x) for x ∈ R. Let
ϕ(x1 )
V ∈ L1loc (RN ) be given by V (x) := u(x1 ) = p χ[0,∞) (x1 ), (36)
− ln x21
see Figure 1 (right). For ε ∈ (0, 14 ), let
xε = (ε, 0, . . . , 0) ∈ RN and Uε := {y ∈ B1 (0) : y1 ∈ (−ε, ε)}.
For N ≥ 1, we use σN to denote the surface measure of the sphere in RN , namely,
N
σN = Γ(NNπ+1)
2
.
2
We show first some auxiliary lemmas.
Lemma 2.11.
V (xε ) − V (y + xε )
Z
J1 = J1 (ε) := dy = o(1) as ε → 0.
Uε |y|N
Proof. Let B := {y 0 ∈ RN −1 : |y 0 | < 1}. Using that u is increasing in (0, ε) for
ε > 0 small and that u(0) = 0,
Z Z − 2ε − N2
ε ε2

|u(ε) − u(y1 + ε)|
Z
0 0 2
J1,1 := N dy1 dy ≤ u(ε) + |y | dy 0
B −ε (y12 + |y 0 |2 ) 2 B 2 4
Z 1 2 − N2
ε ε
= u(ε)σN −1 + r2 rN −2 dr
2 0 4
Z ∞

− N
≤ u(ε)σN −1 1 + r2 2 rN −2 dr = o(1)
0

as ε → 0. Furthermore, using that t 7→ |u0 (t)| is decreasing in (0, 31 ),

Z Z ε
|u(ε) − u(y1 + ε)|
J1,2 := N dy1 dy 0
ε 2 0 2
(y1 + |y | )
B −2 2
 OPTIMAL BOUNDARY REGULARITY FOR THE LOGARITHMIC LAPLACIAN 15

Z Z ε Z 1
y1
≤ |u0 (ty1 + ε)| N dt dy1 dy 0
B − 2ε 0 (y12 + |y 0 |2 ) 2
Z Z ε
y1
≤ |u0 ( 2ε )| N dy1 dy 0
B − 2ε (y12 + |y 0 |2 ) 2
Z 1Z ε
y1
= σN −1 |u0 ( 2ε )| N rN −2 dy1 dr
0 − 2ε (y12 + r2 ) 2
∞ 1
y1 rN −2
Z Z
≤ σN −1 ε|u0 ( 2ε )| N dy1 dr
0 − 12 (y12 + r2 ) 2
∞
rN −2
Z
3
= σN −1 ε|u0 ( 2ε )| N dr = o(1)
2 0 (1 + r2 ) 2
as ε → 0+ , where we used that ε|u0 ( 2ε )| → 0 as ε → 0+ . Then,
|V (xε ) − V (y + xε )|
Z
dy ≤ J1,1 + J1,2 = o(1) as ε → 0 (37)
Uε |y|N
and this ends the proof.
h i
For ε ∈ (0, 14 ) and η ∈ √1 , 1
2
, let

Qε,η := {y = (y1 , y 0 ) ∈ RN : y1 ∈ (ε, η), |y 0 | < η}.

Note that Qε, √1 ⊂ B1 and {y ∈ B1 : y1 > ε} ⊂ Qε,1 .
2

Lemma 2.12. For ε ∈ (0, 14 ) and η ∈ [ √12 , 1], let

Z
V (y)
J21 (η) := N
dy > 0.
Qε,η |y|
√ √ √
Then J21 (η) = σN − ln ε + O(1) and J21 (η) < σN − ln ε − σN ln 2 − ln η + o(1)
as ε → 0.
Proof. For ε ∈ (0, 14 ) and η ∈ [ √12 , 1], let
Z 1 Z − ln η
N 1
R(η) := σN −1 |t2 + 1|− 2 √ dz dt > 0.
ε
η ln t−ln η ln 2 + z − ln t

Let Bη := {y 0 ∈ RN −1 : |y 0 | < η}. Using spherical coordinates, changes of

variables (t = ρ−1 y1 , dt = ρ−1 dy1 , z = − ln ρ, ρ = e−z , dz = −ρ−1 dρ), and
Fubini’s theorem (see Figure 4),
Z Z η 1
(ln 2 − ln y1 )− 2
J21 (η) = dy1 dy 0
Bη ε |y|N
Z ηZ η 1
(ln 2 − ln y1 )− 2 N −2
= σN −1 N ρ dy1 dρ
0 ε |y12 + ρ2 | 2
Z η Z ηρ−1 1
−1 (ln 2 − ln(tρ))− 2
= σN −1 ρ N dt dρ
0 ερ−1 |t2 + 1| 2
Z ∞ Z ηez
1
= σN −1 N √ dt dz
z 2
|t + 1| ln 2 − ln t + z
− ln η εe 2
16 V. HERNÁNDEZ-SANTAMARÍA, L. F. LÓPEZ RÍOS AND A. SALDAÑA

Z ∞ Z ln t−ln ε
2 −N 1
= σN −1 |t + 1| 2 √dz dt − R(η)
ε
η ln t−ln η ln 2 − ln t + z
Z ∞ Z − ln ε !
2 −N 1
= σN −1 |t + 1| 2 √ dz dt − R(η)
ε
η − ln η ln 2 + z
√ p Z ∞ N
= 2σN −1 ln 2 − ln ε − ln 2 − ln η |t2 + 1|− 2 dt − R(η)
η −1 ε
√ !
√ πΓ N 2−1


p
= ln 2 − ln ε − ln 2 − ln η σN −1 + o(1) − R(η)
Γ N2


√ p
= σN − ln ε − σN ln 2 − ln η − R(η) + o(1),

ηez

εez
-1

p
− ln η z

z ln t − ln ε
ln t − ln η

− ln η
ε
η
p p t
1

Figure 4. Domains of integration.

because
√ N −1

N N
πΓ π2 π2
σN −1 N

2 =2 N =N N = σN .
Γ 2 Γ( 2 ) Γ( 2 + 1)
 OPTIMAL BOUNDARY REGULARITY FOR THE LOGARITHMIC LAPLACIAN 17

The claim now follows because

Z 1 Z − ln η
N 1
0 < σN −1 |t2 + 1|− 2 √ dz dt
1
2 ln t−ln η ln 2 + z − ln t
Z 1 Z − ln η
N 1
< R(η) = σN −1 |t2 + 1|− 2 √ dz dt
ε
η ln t−ln η ln 2 + z − ln t
Z 1 Z − ln t−ln η
1
≤ σN −1 dz dt √
0 − ln η ln 2+z
Z 1p p
= 2σN −1 ln 2 − ln (ηt) − ln 2 − ln η dt = O(1)
0
as ε → 0.
Lemma 2.13. For ε ∈ (0, 14 ) and η ∈ [ √12 , 1],
V (y) − V (xε + y)
Z
J22 (η) := dy = o(1) as ε → 0.
Qε,η |y|N
Proof. Arguing as in Lemma 2.12, we have that
− J22 (η)
ηρ−1
!
Z η Z
−1 1 1 N
= σN −1 ρ p −p |t2 + 1|− 2 dt dρ
0 ερ−1 ln 2 − ln(tρ + ε) ln 2 − ln(tρ)
ηez
!
Z ∞ Z
1 1 N
= σN −1 p −√ |t2 + 1|− 2 dt dz.
− ln η εez (ln 2 − ln(te−z + ε) ln 2 + z − ln t
Arguing as in the proof of Lemma 2.7, this last term tends to zero as ε → 0.
Lemma 2.14. It holds that
√
Z
V (y + xε )
J2 = J2 (ε) := − N
dy = −σN − ln ε + O(1) as ε → 0, (38)
B1 \Uε |y|
and
√ √
J2 > −σN − ln ε + σN ln 2 + o(1) as ε → 0. (39)
Proof. By Lemmas 2.12 and 2.13,
Z
V (y + xε ) √ √
−J2 < N
dy = J21 (1) − J22 (1) < σN − ln ε − ω ln 2 + o(1)
Qε,1 |y|
as ε → 0, and (39) follows. On the other hand,
Z Z
V (y + xε ) V (y + xε )
J2 = − dy − dy. (40)
Qε, √1 |y|N B1 \(Uε ∪Qε, √1 ) |y|N
2 2

Note that, by Lemmas 2.12 and 2.13,

Z
V (y + xε ) √ √ √
0< N
dy = J21 (1/ 2) − J22 (1/ 2) = σN − ln ε + O(1) (41)
Qε, √1 |y|
2

as ε → 0, whereas
Z
V (y + xε ) N
0< dy ≤ 2 2 kV k∞ |B1 |. (42)
B1 \(Uε ∪Qε, √1 ) |y|N
2
18 V. HERNÁNDEZ-SANTAMARÍA, L. F. LÓPEZ RÍOS AND A. SALDAÑA

But then, (40), (41), and (42) imply (38).

Theorem 2.15. Let V be given by (36), then

Z
V (xε ) − V (y) √
J := N
dy > σN ln 2 + o(1) and J = O(1) as ε → 0.
B1 (xε ) |y − xε |
(43)
Proof. Let J1 and J2 be as in Lemmas 2.11 and 2.14. By Lemma 2.11,
V (xε ) − V (y)
Z Z
J := dy = J1 + J2 + V (xε ) |y|−N dy + o(1)
B1 (xε ) |y − xε |N B1 \Uε
− ln(ε) √ √
= J1 + J2 + σN p ε
= J2 + σN − ln ε + o(1) > σN ln 2 + o(1)
− ln 2
as ε → 0. The claim (43) now follows from Lemma 2.14.

Let Ve ∈ L1loc (RN ) be given by

ϕ(x1 )ϕ(|x|)
Ve (x) = V (x)ϕ(|x|) = p χ(0,∞) (x1 ), x ∈ RN , (44)
− ln x21

where ϕ is given in (25). Recall that B2+ is the half ball of radius 2 given in (21).

Lemma 2.16. Let Ve be given by (44), then L∆ Ve ∈ L∞ (B2+ ).

Proof. First observe that Theorem 2.15 together with the translation invariance of
V (V (x) = u(x1 )) and the fact that V ∈ C ∞ (RN+ ) imply that

V (z) − V (z + y)
Z
sup dy < C
z∈B + B1 |y|N
2

for some C > 0, as a simple argument by contradiction shows. From this estimate
and Lemma 2.2 we deduce that L∆ V ∈ L∞ (B2+ ). On the other hand, by the
Leibniz-type formula (78),

L∆ Ve (x) = L∆ V (x)ϕ(x) + V (x)L∆ ϕ(x) − I(V, ϕ)(x) for all x ∈ RN .

Thus, it suffices to show that I(V, ϕ) ∈ L∞ (B2+ ). Observe that, for x ∈ B2+ ,
|I(V, ϕ)(x)|
|V (x) − V (y)| |ϕ(x) − ϕ(y)|
Z
≤ cN dy
B1 (x) |x − y|N
|V (y)ϕ(y) − V (x)ϕ(y) − V (y)ϕ(x)|
Z
+ cN dy + ρN V (x)ϕ(x)
RN \B1 (x) |x − y|N
|V (x) − V (y)| |ϕ(x) − ϕ(y)| |V (y) − V (x)|
Z Z
≤ cN N
dy + cN ϕ(y)dy
B1 (x) |x − y| BR (x)\B1 (x) |x − y|N
Z
V (y)
+ cN ϕ(x) dy + ρN V (x)ϕ(x) =: A1 + A2 + A3 + A4 ,
N
R \B1 (x) |x − y|N
for R > 1 such that B1 (x) ∪ B1+2ζ ⊂ BR (x), see (25). Since V is bounded and ϕ is
smooth and bounded, it easily follows that A1 , A2 , and A4 are uniformly bounded.
 OPTIMAL BOUNDARY REGULARITY FOR THE LOGARITHMIC LAPLACIAN 19

Furthermore, recall that V has support on the strip {|y1 | < 1 + 2ζ} (see (36)). Since
x1 ∈ (0, 2), we have that {|x1 + y1 | < 1 + 2ζ} ⊂ {|y1 | < 4} and then
Z
V (x + y)
A3 = cN ϕ(x) dy
RN \B1 |y|N
Z
≤ cN kV k∞ kϕk∞ |y|−N dy
(RN \B1 )∩{|x1 +y1 |<1+2ζ}
Z
≤ cN kV k∞ kϕk∞ |y|−N dy
{|y 0 |<1}∩{|y1 |<4}∩{|y|>1}
Z Z !
0 −N 0
+ |y | dy dy1
{|y1 |<4} {|y 0 |>1}
 Z ∞ 
≤ cN kV k∞ kϕk∞ {|y 0 | < 1} ∩ {|y1 | < 4} + {|y1 | < 4} σN −1 ρ−2 dρ
1
< ∞,
where y = (y1 , y 0 ) ∈ R × RN −1 .
Lemma 2.17. It holds that
σN √
Z
Ve (y)
N
dy < ln 2 for ε ∈ (0, ζ).
RN \B1 (xε ) |y − xε | 4
Proof. By (35),
ϕ(y1 )χ(0,∞) (y1 )
Z Z
Ve (y) 1
N
dy = y1 dy
|y − xε |N
p
RN \B1 (xε ) |y − xε | B1+2ζ \B1 (xε ) − ln 2
Z
1 1 1
≤q dy ≤ q |B1+2ζ \B1 (xε )|
1+2ζ B1+2ζ \B1 (xε ) |y − xε |N
− ln 2 − ln 1+2ζ 2
1 1 σN
σN √
=q |B1+2ζ \B1 | ≤ q (1 + 2ζ)N − 1 < ln 2
− ln 1+2ζ 1+ 12 N 4
2 − ln 2

for ε ∈ (0, ζ).

Lemma 2.18. It holds that Ve ∈ H(B2+ ).

Proof. We argue as in Lemma 2.10. Let ζ > 0 be as in (35) and define
K := {x ∈ B3 : −ζ < x1 < ζ} and U := B3 \K.
For x ∈ U , we have that
|Ve (x + y) − Ve (x)| = |∇Ve (x) · y + o(y)| ≤ |y|(k∇Ve kL∞ (U ) + o(1)) as |y| → 0.
Then
(Ve (y + x) − Ve (x))Ve (y + x)
Z Z
A1 := dy dx
U B1 |y|N
Z Z
≤ kVe k∞ (k∇Ve kL∞ (U ) + o(1))|y|1−N dy dx < ∞,
U B1

(Ve (y + x) − Ve (x))Ve (y + x)
Z Z
A2 := dy dx
K∩{x1 >0} B1 ∩{y1 >x1 } |y|N
20 V. HERNÁNDEZ-SANTAMARÍA, L. F. LÓPEZ RÍOS AND A. SALDAÑA

Z Z 1 Z 1
N
≤ 2σN −1 kVe k2∞ |y12 + ρ2 |− 2 ρN −2 dy1 dρ dx
K∩{x1 >0} 0 x1
1 ∞
ρN −2
Z Z Z
≤ 2σN −1 kVe k2∞ y1−1 N dρ dy1 dx
K∩{x1 >0} x1 0 |1 + ρ2 | 2
√ N −1

Z
πΓ
= 2σN −1 kVe k2∞ N
2

− ln(x1 ) dx < ∞. (45)
2Γ 2 K∩{x1 >0}

Moreover,
(Ve (y + x) − Ve (x))Ve (y + x)
Z Z
A3 := dy dx
K∩{x1 >0} B1 ∩{y1 <x1 } |y|N
|Ve (x) − Ve (y + x)|
Z Z
≤ kVe k∞ dy dx < ∞,
K∩{x1 >0} B1 ∩{−x1 <y1 <x1 } |y|N
where the finiteness of A3 follows from (37) (note that (37) is stated for V instead
of Ve , but since Ve is given by (44), the bound easily extends to Ve ).
Furthermore, since Ve (x + y) = 0 if x1 + y1 < 0, arguing as in (45),

(Ve (y + x) − Ve (x))Ve (y + x)
Z Z
A4 := dy dx
K∩{x1 <0} B1 |y|N
(Ve (y + x) − Ve (x))Ve (y + x)
Z Z
= dy dx
K∩{x1 <0} B1 ∩{−x1 <y1 <1} |y|N
Z 1Z 1
ρN −2
Z
≤ 2σN −1 kVe k2∞ dx dρ dy1 dx
2 2 N
K∩{x1 >0} x1 0 |y1 + ρ | 2
Z 1 Z ∞
rN −2
Z
2 −1
≤ 2σN −1 kV k∞
e dx y1 dy1 dr
K∩{x1 >0} x1 0 |1 + r2 |N/2
Z
= CkVe k2∞ − ln(x1 )dx < ∞,
K∩{x1 >0}

where C > 0 is a constant only depending on N .

Then,
(Ve (y + x) − Ve (x))Ve (y + x)
Z Z
A := dy dx = A1 + A2 + A3 + A4 < ∞. (46)
RN B1 |y|N
Recall that
(Ve (x) − Ve (y))Ve (x)
Z Z
2 e 2
kV k = χB1 (x) (y)
cN RN RN |x − y|N
(Ve (y) − Ve (x))Ve (y)
+ χB1 (x) (y) dy dx.
|x − y|N
Therefore, using Fubini’s theorem, a change of variables, and (46),

(Ve (x) − Ve (y))Ve (x)

Z Z
χB1 (x) (y) dy dx
RN RN |x − y|N
(Ve (x) − Ve (y))Ve (x)
Z Z
= χB1 (x) (y) dx dy
RN RN |x − y|N
 OPTIMAL BOUNDARY REGULARITY FOR THE LOGARITHMIC LAPLACIAN 21

(Ve (z + y) − Ve (y))Ve (z + y)
Z Z
= dz dy = A < ∞.
RN B1 |z|N
Similarly,
(Ve (y) − Ve (x))Ve (y)
Z Z
χB1 (x) (y) dy dx
RN RN |x − y|N
(Ve (z + x) − Ve (x))Ve (z + x)
Z Z
= dz dx = A < ∞.
RN B1 |z|N
Then kVe k2 < ∞. Since we also know that Ve ∈ L2 (RN ), we obtain that Ve ∈ H(B2+ )
as claimed.
The following result implies Theorem 2.1 in the case N ≥ 2.
Theorem 2.19. The function Ve given by (36) belongs to H(B2+ ) ∩ C ∞ (RN +) ∩
1
L 2 (RN ) and there is δ ∈ (0, 2) such that
σN √
L∆ Ve ∈ L∞ (B2+ ) and L∆ Ve (x) > ln 2 for x ∈ Bδ ∩ RN
+.
2
Proof. By Lemma 2.16 we know that L∆ Ve ∈ L∞ (B2+ ). By Lemma 2.18 and by
1
construction we have that Ve ∈ H(B2+ ) ∩ C ∞ (RN N
+ ) ∩ L (R ). Finally, by Theo-
2

rem 2.15,
Z
V (xε ) − V (y) √
J := N
dy > σN ln 2 + o(1) as ε → 0.
B1 (xε ) |y − xε |
Since, by (25), it holds that ϕ(|x|) = 1 for x ∈ B1+ζ , we have that Ve = V in B1+ζ
(with V given in (36)) and therefore
Ve (x) − Ve (y) V (x) − V (y)
Z Z
N
dy = dy for all x ∈ Bζ .
B1 (x) |y − x| B1 (x) |y − x|N
In particular,
Z
Ve (yε ) − Ve (y) √
N
dy = J > σN ln 2 + o(1) as ε → 0 (47)
B1 (yε ) |y − yε |
for all yε = (ε, y 0 ) where y 0 ∈ R√
N −1
is such that yε ∈ Bζ . Then, by Lemma 2.17, (47)
and Lemma 2.3 (with σ = σN ln 2/2 and η = 3/4), there is δ ∈ (0, ζ) such that
σN √
L∆ Ve (yε ) > ln 2
2
for all yε = (ε, y 0 ) ∈ Bδ ∩ B2+ .
Proof of Theorem 2.1. The claim follows from Theorems 2.4 (for the case N = 1)
and Theorem 2.19 (for the case N ≥ 2).

3. A new barrier via the Kelvin transform. Let R > 0 and x0 ∈ RN . The
inversion of a point x ∈ RN with respect to the sphere SR (x0 ) is given by
x − x0
x∗ := x0 + R2 , x 6= x0 , (48)
|x − x0 |2
and the Kelvin transform of u is
u∗ (x∗ ) := |x∗ − x0 |−N u(x), x∗ 6= x0 . (49)
22 V. HERNÁNDEZ-SANTAMARÍA, L. F. LÓPEZ RÍOS AND A. SALDAÑA

For Ω ⊂ RN we define Ω∗ := {x∗ : x ∈ Ω}. Recall that

|x − z|
|x∗ − z ∗ | = for x, z ∈ RN \{0}, (50)
|x||z|
see, for instance, [5, Proposition A.3].
To prove Proposition 1.3, we show an auxiliary lemma first. Recall the definition
of hΩ : Ω → R (see (18)) given by
Z Z !
−N −N
hΩ (x) := cN |x − y| dy − |x − y| dy .
B1 (x)\Ω Ω\B1 (x)

Lemma 3.1. Let R, x0 , and x∗ as in (48). Let Ω ⊂ RN be an open bounded subset

such that BR (x0 ) ⊂ Ωc . The function x 7→ hΩ∗ (x∗ ) − hΩ (x) is bounded in Ω.
Proof. Without loss of generality we assume that x0 = 0 and R = 1; in particular
this implies that |x| > 1 for every x ∈ Ω. Given x ∈ Ω consider B1/2 (x) that, under
the inversion with respect to the unit sphere, is transformed into the ball Bρ (P ),
where
4x 2
P = 2
, ρ= 2
,
4|x| − 1 4|x| − 1
see [3, Section 2]. This ball contains another one centered at x∗ such that
Bσ (x∗ ) ⊂ Bρ (P ), (51)
where
2|x| − 1
σ= . (52)
|x|(4|x|2 − 1)
Indeed, if y ∈ Bσ (x∗ ), then
1
|y − P | ≤ |y − x∗ | + |x∗ − P | < σ + = ρ.
|x|(4|x|2 − 1)
Observe that 0 < σ < ρ < 1, as |x| ≥ 1.
Let A1 := [B1 (x) \ (B1/2 (x) ∪ Ω)] ∪ [Ω \ B1 (x)] and
Z
1
R1 (x) := cN N
dy.
A1 |y − x|

Note that R1 ∈ L∞ (Ω). Then, by a change of variables (y = z ∗ , see [5, Proposition

A.3]) and (50),
Z
1
hΩ (x) = cN N
dy + R1 (x)
B1/2 (x)\Ω |y − x|

|z|−2N
Z
= cN ∗ N
dz + R1 (x) (53)
Bρ (P )\Ω∗ |z − x|

|z|−N |x∗ |N
Z
= cN ∗ N
dz + R1 (x).
Bρ (P )\Ω∗ |z − x |

On the other hand, let A2 = [B1 (x∗ )\(Bσ (x∗ )∪Ω)]∪[Ω\B1 (x∗ )] (with σ as in (52))
and Z
∗ 1
R2 (x ) = cN dz.
A2 |z − x∗ |N
 OPTIMAL BOUNDARY REGULARITY FOR THE LOGARITHMIC LAPLACIAN 23

Note that R2 ∈ L∞ (Ω∗ ) and

Z
∗ 1
h Ω∗ (x ) = cN dz + R2 (x∗ ). (54)
Bσ (x∗ )\Ω∗ |z − x∗ |N
Let A3 := [Bρ (P ) \ (Bσ (x∗ ) ∪ Ω∗ )] and
|z|−N |x∗ |N
Z
R3 (x) := |R1 (x) − R2 (x)| + cN dz.
A3 |z − x∗ |N
Note that R3 ∈ L∞ (Ω) and, from (51), (53), and (54) we deduce that
|z|−N |x∗ |N − 1
Z
|hΩ (x) − hΩ∗ (x∗ )| ≤ dz + R3 (x)
Bσ (x∗ )\Ω∗ |z − x∗ |N
1 |z|N − |x∗ |N
Z
≤ dz + R3 (x)
Bσ (x∗ ) |z|
N |z − x∗ |N
N Z
|z|N − |x∗ |N
 2
4C − 1
≤ dz + R3 (x). (55)
2 Bσ (x∗ ) |z − x∗ |N
For the last inequality in (55), we used that Bσ (x∗ ) ⊂ RN \ B2/(4C 2 −1) with C :=
supx∈Ω |x|.
To conclude, we use polar coordinates to deduce that, up to a constant,
|z|N − |x∗ |N |x∗ + rθ|N − |x∗ |N
Z Z σZ
dz = dS(θ)dr, (56)
Bσ (x∗ ) |z − x∗ |N 0 S1 r
where S1 = ∂B1 . By a Taylor expansion,
|x∗ + rθ|N − |x∗ |N
≤ N |θ||x∗ |N −1 + R(r), (57)
r
where R is continuous and R(r) → 0 as r → 0+ . Then (55), (56), and (57) yield
that hΩ − hΩ∗ ∈ L∞ (Ω).
Proof of Proposition 1.3. Without loss of generality we assume that x0 = 0 and
R = 1.
To prove i) let u be a Dini continuous function at x and x 6= x0 . We argue
as in [30, Proposition A.1]; however the logarithmic case is more involved. In
particular, one cannot assume without loss of generality that u∗ (x∗ ) = 0. By [10,
Proposition 2.2],
u∗ (x∗ ) − u∗ (y)
Z
∗ ∗
L∆ u (x ) = cN dy + (hΩ∗ (x∗ ) + ρN )|x∗ |−N u(x), (58)
Ω∗ |x∗ − y|N
where hΩ∗ is given by (18). Using the change of variables y = z ∗ (see [5, Proposition
A.3]) and (58),
u∗ (x∗ ) − u∗ (y)
Z
cN dy
Ω∗ |x∗ − y|N
Z ∗ ∗
u (x ) − u∗ (z ∗ ) −2N
= cN |z| dz
Ω |x∗ − z ∗ |N
|x∗ |−N u(x) − |z ∗ |−N u(z) ∗ −N ∗ −N ∗ 2N
Z
= cN |x | |z | |z | dz
Ω |x − z|N
|z ∗ |N u(x) − |x∗ |N u(z)
Z
= cN |x∗ |−2N dz
Ω |x − z|N
24 V. HERNÁNDEZ-SANTAMARÍA, L. F. LÓPEZ RÍOS AND A. SALDAÑA

|z|−N − |x|−N u(x) − u(z)

Z Z
= cN |x∗ |−2N u(x) N
dz + cN |x ∗ −N
| N
dz
Ω |z − x| Ω |x − z|
|z|−N − |x|−N
Z
∗ −2N
= cN |x | u(x) dz + |x∗ |−N L∆ u(x)
Ω |z − x|N
− (hΩ (x) + ρN )|x∗ |−N u(x).

This equation and (58) imply (10).

To show ii), assume that u, L∆ u ∈ L∞ (Ω) and that
BR (x0 ) = B1 (0) ⊂ Ωc . (59)

By (59), it is clear that u∗ ∈ L∞ (Ω∗ ). Moreover, by Lemma 3.1, (59), and (10), it
also follows that L∆ u∗ ∈ L∞ (Ω∗ ), as claimed.
To prove iii), let u ∈ H(Ω) and recall that we have assumed that x0 = 0,
R = 1, and that (59) holds. Note that, in this setting, Ω∗ ⊂ B1 , B2 (y) ⊂ B3
for all y ∈ Ω∗ , and B2 (y) ⊂ RN \Ω∗ for all y ∈ RN \B3 . To ease notation, let
∗ ∗
(y))2
S = S(x, y) := (u (x)−u
|x−y|N
, then, using that u∗ = 0 in RN \Ω∗ ,
Z Z Z Z Z Z
S dxdy = S dxdy + S dxdy
RN B2 (y) Ω∗ B (y) RN \Ω∗ B2 (y)
Z Z 2 Z Z Z Z
= S dxdy + S dxdy + S dxdy.
Ω∗ Ω∗ Ω∗ B2 (y)\Ω∗ B3 \Ω∗ B2 (y)

Note that, by (59), there is δ > 0 such that Bδ ⊂ (Ω∗ )c and therefore there is
some M > 0 such that
(B3 \Ω∗ )∗ ⊂ A\Ω, A := BM \B 31 .

Then, by a change of variables (x = ξ ∗ , y = z ∗ ) and (50),

2
(u∗ (x) − u∗ (y))
Z Z
dxdy
RN B2 (y) |x − y|N
2
(u∗ (x) − u∗ (y)) u∗ (y)2
Z Z Z Z
≤ N
dxdy + 2 N
dxdy
Ω∗ Ω∗ |x − y| Ω∗ B3 \Ω∗ |x − y|

2
|ξ|N u(ξ) − |z|N u(z)
Z Z
≤ N
|ξ|−N |z|−N dξdz
Ω Ω |ξ − z|
|z|2N u(z)2 −N −N
Z Z
+2 N
|ξ| |z| dξdz. (60)
Ω A\Ω |ξ − z|

Using that Ω is bounded, (59), and that (|ξ|N u(ξ) − |z|N u(z))2 ≤ 2(|ξ|2N (u(ξ) −
u(z))2 + 2(|ξ|N − |z|N )2 u(z)2 , we have that

2
|ξ|N u(ξ) − |z|N u(z)
Z Z
|ξ|−N |z|−N dξdz
Ω Ω |ξ − z|N
(u(ξ) − u(z))2 N −N (|ξ|N − |z|N )2
Z Z
≤2 |ξ| |z| + u(z)2 |ξ|−N |z|−N dξdz
Ω Ω |ξ − z|N |ξ − z|N
(u(ξ) − u(z))2
Z Z Z
≤C dξdz + C 1 u(z)2 dz (61)
Ω Ω |ξ − z|N Ω
 OPTIMAL BOUNDARY REGULARITY FOR THE LOGARITHMIC LAPLACIAN 25

for some C1 = C1 (Ω) > 0, where we used that

 N 2
|a| − |b|N
Z
sup |a|−N |b|−N < ∞ and sup |ξ − a|2−N dξ < ∞.
a,b∈Ω |a − b| a∈Ω Ω

Moreover, using (59) and that Ω is bounded, we find some C2 = C2 (Ω) > 0 such
that
u(z)2 |z|N
Z Z Z Z
N |ξ|N
dξdz < C 2 u(z)2
|ξ − z|−N dξdz
Ω A\Ω |ξ − z| Ω A\Ω
Z Z
C2 2
= u(z) κΩ (z)dz + C2 |A| u(z)2 dz, (62)
cN Ω Ω

where
R κΩ is the so-called killing measure (see [10, eq. (3.5)]) given by κΩ (z) :=
cN B1 (z)\Ω |z − y|−N dy for z ∈ Ω. By the logarithmic boundary Hardy inequality
(see [10, Remark 4.3 and Corollary A.2]), we have that Ω u(z)2 κΩ (z)dz < ∞. Claim
R

iii) now follows from (60), (61), and (62).

Remark 3.2. The proof of the previous lemma can be slightly adapted to study
the properties of functions in the fractional Sobolev spaces
( )
s 2 N |u(x) − u(y)| 2 N N N
H0 (Ω) := u ∈ L (R ) : N ∈ L (R × R ) and u = 0 in R \ Ω ,
|x − y| 2 +s
s ∈ (0, 1), under their corresponding Kelvin transform
u∗ (x∗ ) = |x∗ − x0 |2s−N u(x),
see [30]. The following result is probably well-known, but we could not locate it in
the literature. We state it here for future reference.
Lemma 3.3. Let Ω be a bounded open set and BR (x0 ) ⊂ Ωc . If u ∈ H0s (Ω), then
u∗ ∈ H0s (Ω∗ ).
The Kelvin transform can now be used to produce a new barrier that will be
useful to show the optimal regularity of the torsion function, see Figure 2.
Proposition 3.4. Let N ≥ 2 and D := B 21 ( 12 e1 ) ∩ {x1 > 21 } = {x ∈ RN : |x − 21 | <
1
1 1
2 , x1 > 2 }. There are f ∈ L∞ (D) and v ∈ L 2 (RN ) ∩ C ∞ (D) ∩ H(D) such that
L∆ v = f pointwisely in D and there are c > 0 and δ > 0 such that
1 1
c ` 2 (dist(x, ∂D)) < v(x) < c−1 ` 2 (dist(x, ∂D)) for all x ∈ D ∩ Bδ (e1 ). (63)

Proof. Let w be given by Theorem 2.1 and Ω := B2 ∩ RN + . Let w(x)

e := w(2(x − e1 ))
and Ω := {x ∈ B1 (e1 ) : x1 > 1}. Then, by Theorem 2.1 and Lemmas 1.5 and A.4,
e
1
e ∈ L 2 (RN ) ∩ C ∞ (Ω)
we have that w e ∩ H(Ω)
e is a solution of L∆ w
e = fe in Ω
e for some
∞
fe ∈ L (Ω)
e and there are c1 > 0 and δ > 0 such that
1 1
c1 ` 2 (dist(x, ∂ Ω))
e < w(x)
e < c−1
1 ` (dist(x, ∂ Ω))
2 e for all x ∈ Ω
e ∩ Bδ (e1 ).

Let κ : RN → RN be the Kelvin transform κ(x) := |x|x 2 , that is, the point inversion
with respect to the sphere S1 . For x 6= 0, let v(x) := w(κ(x))
e and let D be as in
the statement. It is not hard to see that κ(Ω)
e = D and that
e ∩ {x ∈ RN : x1 = 1}) = ∂D ∩ ∂B 1 ( 1 e1 ),
κ(∂ Ω 2 2
26 V. HERNÁNDEZ-SANTAMARÍA, L. F. LÓPEZ RÍOS AND A. SALDAÑA

namely, that the flat part of the boundary of Ω e goes to the curved part of the
boundary of D through κ (see Figure 2). Using Proposition 1.3, it follows that
v ∈ C ∞ (D) ∩ H(D) is a solution of L∆ v = f in D for some f ∈ L∞ (D). A
1
simple calculation using Lemma 1.5 and (50) yields that v ∈ L 2 (RN ) and that (63)
holds.

We close this subsection with the following result on weak solutions under the
Kelvin transform.
Lemma 3.5. Let Ω ⊂ RN be as in Proposition 1.3 ii). Let u be a weak solution of
L∆ u = f in Ω, u=0 in RN \ Ω,
Then u∗ is a weak solution of
L∆ u∗ = f¯ in Ω∗ , u∗ = 0 in RN \ Ω∗ ,
where
|z|−N − |y ∗ |−N
Z
f¯(y) := f (y ∗ )|y|−N + cN u(y ∗ )|y|−2N dz
Ω |z − y ∗ |N
+ u(y ∗ )(hΩ∗ (y) − hΩ (y ∗ ))|y|−N , y ∈ Ω∗ .

Proof. Let x0 ∈ RN be as in the statement of Proposition 1.3 and without loss of

generality assume that x0 = 0. Let ψ ∈ Cc∞ (Ω∗ ) and let ϕ ∈ Cc∞ (Ω) be such that
ϕ∗ (y) = |y|−N ϕ(y ∗ ) = ψ(y) for y ∈ Ω∗ . By definition of weak solution and (19), we
have that
Z Z
∗ ∗
E(u , ψ) = u (y)L∆ ψ(y) dy = u∗ (y)L∆ ϕ∗ (y) dy. (64)
Ω∗ Ω∗

Then, by Proposition 1.3,

|z|−N − |y ∗ |−N
Z
L∆ ϕ∗ (y) = |y|−N L∆ ϕ(y ∗ ) + cN |y|−2N ϕ(y ∗ ) dz
Ω |z − y ∗ |N
+ (hΩ∗ (y) − hΩ (y ∗ ))|y|−N ϕ(y ∗ )

for y ∈ Ω∗ . Thus, using that u∗ (y) = |y|−N u(y ∗ ) and ψ(y) = |y|−N ϕ(y ∗ ),
|z|−N − |y ∗ |−N
Z Z
E(u∗ , ψ) = u∗ (y)|y|−N L∆ ϕ(y ∗ ) + cN u∗ (y)|y|−2N ϕ(y ∗ ) dz
Ω∗ Ω |z − y ∗ |N
+ u∗ (y)(hΩ∗ (y) − hΩ (y ∗ ))|y|−N ϕ(y ∗ ) dy
|z|−N − |y ∗ |−N
Z Z
= u(y ∗ )|y|−2N L∆ ϕ(y ∗ ) + cN u(y ∗ )|y|−2N ψ(y) dz
Ω∗ Ω |z − y ∗ |N
+ u(y ∗ )(hΩ∗ (y) − hΩ (y ∗ ))|y|−N ψ(y ∗ ) dy.
Using that u is a weak solution and the change of variables y = x∗ , we obtain that
Z Z
u(y ∗ )|y|−2N L∆ ϕ(y ∗ ) dy = u(x)L∆ ϕ(x) dx
Ω∗ Ω
f (y ∗ )
Z Z
= E(u, ϕ) = f (x)ϕ(x) dx = N
ψ(y) dy.
Ω Ω∗ |y|

The claim now follows.

 OPTIMAL BOUNDARY REGULARITY FOR THE LOGARITHMIC LAPLACIAN 27

4. Optimal boundary bounds for solutions. Recall that Ψ is the digamma

function, γ is the Euler-Mascheroni constant, and, for a bounded open set Ω, the
space V(Ω) is defined in (11). We use the following result shown in [10, Corollary
1.9].
Theorem 4.1 (Weak maximum principle for the logarithmic Laplacian). Let Ω ⊂
RN be an open bounded domain such that |Ω| < 2N e 2 (Ψ( 2 )−γ ) |B1 | and let u ∈
N N

V(Ω) be such that

EL (u, ϕ) ≥ 0, u≥0 in RN \Ω,
for all nonnegative ϕ ∈ Cc∞ (Ω). Then u ≥ 0 a.e. in RN .
Recall that Bε is the ball in RN centered at 0 of radius ε and Bε+ := {x ∈ Bε :
x1 > 0}. We are ready to show Theorem 1.1.
Proof of Theorem 1.1. Let w, c, and δ as in Theorem 2.1 and let u be as in the
statement of the Theorem. First observe that, by Lemma A.3 and Lemma 1.5, it is
enough to prove (6) for Ω ⊂ Bδ ; in particular L∆ w > c in Ω ∩ RN
+ . Moreover, by
making δ smaller, if necessary, we may assume that
|Ω| < 2N e 2 (Ψ( 2 )−γ ) |B1 |.
N N

Let x0 ∈ ∂Ω. We can assume x0 = 0 and the tangent space to ∂Ω in x0 to be ∂RN +.

Let η be as in (5). By making η smaller, if necessary, choose y0 = (η, 0, . . . , 0) ∈ RN
+,
η < δ/2 such that Bη (y0 ) ∩ Ω = {x0 }.
We now consider the inversion with respect to the sphere Sη (y0 ) and the Kelvin
transform of u,
u∗ (x∗ ) = |x∗ − y0 |−N u(x), x∗ ∈ Ω∗ ,
see (48), (49) (with R = η and x0 = y0 ). Observe that Ω∗ ⊂ Bδ+ and y0 6∈ Ω∗ .
Let v := M w − u∗ with M = c−1 kf kL∞ (Ω) and f as in Lemma 3.5. Since
u ∈ C(RN ) ∩ H(Ω∗ ), by Proposition 1.3 and [10, Theorem 1.11], and w ≥ 0 in RN ,
∗

we have that v ≥ 0 in RN \Ω∗ . Moreover, for every ϕ ∈ Cc∞ (Ω∗ ) nonnegative,

Z
EL (v, ϕ) ≥ (M c − kf kL∞ (Ω) )ϕ dx ≥ 0,
Ω∗
R R
where we have used that EL (w, ϕ) = Ω∗ L∆ w(x)ϕ(x) dx ≥ c Ω∗ ϕ(x) dx. Therefore,
by Theorem 4.1, v ≥ 0 in RN ; in particular,
1
u∗ (x∗ ) ≤ M w(x∗ ) ≤ C` 2 (x∗1 )
for some C > 0 and for all x∗ in the segment joining x0 and y0 , by Theorem 2.1.
From this inequality we deduce that for every x ∈ Ω that belongs to the segment
joining −y0 and x0 ,
1
u(x) = r0−N (r0 − x∗1 )N u∗ (x∗1 ) ≤ C` 2 (|x1 |)
p (65)
= C `(dist(x, ∂Ω)),
as x∗1 ≤ |x1 |.
We conclude that (65) is true for all x in the η-neighborhood of ∂Ω. A similar
argument can now be done using −u instead of u, which yields that (65) holds for
−u too. This implies (6), as claimed.
28 V. HERNÁNDEZ-SANTAMARÍA, L. F. LÓPEZ RÍOS AND A. SALDAÑA

5. Applications.
5.1. Optimal regularity for the torsion function in a small ball. In this
section, we show Theorem 1.2. We begin with some existence and qualitative prop-
erties for the torsion function.
N N
Proposition 5.1. Let N ≥ 1, r > 0 be such that |Br | < 2N e 2 (Ψ( 2 )−γ) |B1 |, and
consider the following problem
L∆ τ = 1 in Br , τ =0 in RN \Br . (66)
There is a unique classical positive solution τ to (66). In particular, τ is radially
symmetric and continuous in RN .
Proof. Under the assumption on r, by Theorem 4.1, we have that L∆ satisfies the
maximum principle in Br . By [6, Theorem 1.1], there is a classical solution of (66),
namely, (66) holds pointwisely. Since L∆ satisfies the maximum principle in Br ,
we have that τ ≥ 0 in Br . Moreover, τ > 0 in Br , because, if τ (y0 ) = 0 for some
y0 ∈ Ω, then we would have that
Z
τ (y)
1 = L∆ τ (y0 ) = −cN N
dy < 0,
B |y 0 − y|
which is absurd. Finally, τ is radially symmetric by the uniqueness of the
solution.
5.1.1. 1D case. We show the one-dimensional case first.
1 1
Theorem 5.2. Let r > 0 be such that r < 2e 2 (Ψ( 2 )−γ) ≈ 0.561459 and let τ be the
torsion function of the interval (0, r), namely, the unique weak solution of
L∆ τ = 1 in (0, r), τ =0 in R \ (0, r).
Then there is c ∈ (0, 1) such that
1
` 2 (δ(x)) 1
≥ τ (x) ≥ c ` 2 (δ(x)) for x ∈ (0, r),
c
where δ(x) = dist(x, ∂(0, r)) = dist(x, {0, r}) = min{x, r − x}.
Proof. The upper bound follows from Theorem 1.1. For the lower bound, let u be
given by (26). Then L∆ u ∈ L∞ (Ω) with Ω = (0, 2), by Theorem 2.4. Let r > 0 be
as in the statement, ε := 41 r, Ω
e := (0, 2ε) ⊂ (0, r), and let u e → R be given by
e:Ω
x ∞ e
u
e(x) := u( ε ). By Lemma A.3, L∆ u e ∈ L (Ω).
By Proposition 5.1, τ > 0 in [0, r) and τ ∈ C(R). Let U := τ − kL∆1uek∞ u e, then
L∆ U ≥ 0 in Ω and U ≥ 0 on R\Ω. By Theorem 4.1, we obtain that U ≥ 0 in R,
e e
namely
C 1
x 1

τ (x) ≥ C u
e(x) = p x
= C` 2 ≥ C1 ` 2 (x) for x ∈ (0, 10 )
− ln 2 2
for some constants C, C1 > 0, by Lemma 1.5. One can argue similarly in a neigh-
1 
borhood of r to obtain that τ (x) ≥ C2 ` 2 (r − x) for x ∈ (r − 10 , r) and for some
 
constant C2 > 0. Let I(ε) := ( 10 , r − 10 ). Since τ > 0 in (0, r), then
1 inf I(ε) τ
τ ≥ C3 ` 2 (δ(x)) for x ∈ I(ε), C3 := 1 .
supI(ε) ` 2 (δ(x))
The lower bound now follows with c := min{C1 , C2 , C3 }. This ends the proof.
 OPTIMAL BOUNDARY REGULARITY FOR THE LOGARITHMIC LAPLACIAN 29

5.1.2. The general case. We are ready to show Theorem 1.2.

Proof of Theorem 1.2. The case N = 1 is shown in Theorem 5.2. Assume N ≥ 2,

let r be as in the statement, and let v be as in Proposition 3.4. Let D := {x ∈
RN : |x − 12 | < 12 , x1 > 12 } ⊂ B1 , ve(x) := v( xr ), and D
e := rD ⊂ Br . Then, by
1
Lemma A.3 and Proposition 3.4, we have that ve ∈ L 2 (RN ) ∩ C ∞ (D) e ∩ H(D)e is a
∞ e
weak solution of L∆ ve = f in D for some f ∈ L (D)\{0}. Moreover, there is c > 0
e
and δ > 0 such that
1
ve(x) > c ` 2 (dist(x, ∂ D))
e for all x ∈ D
e ∩ r(Bδ (e1 )). (67)

By Proposition 5.1, τ ∈ C(RN ) and τ > 0 in Br . Let U := τ − ke v with

N e
k := 1/kf kL∞ (D)
e . Then L∆ U ≥ 0 (weakly) in D and U = τ ≥ 0 on R \D. By
e
Theorem 4.1, we obtain that U ≥ 0 in RN , namely, τ (x) ≥ ke
v (x) for x ∈ D.
e Then,
by (67),
τ (re1 − te1 )
lim inf 1
t→0 +
` 2 (t)
1
ve(re1 − te1 ) c ` 2 (dist(re1 − te1 , ∂ D))
e
≥ k lim inf 1 ≥ k lim inf 1
+
t→0 ` 2 (t) +
t→0 ` 2 (t)
1 1
c ` 2 (dist(re1 − te1 , re1 )) c ` 2 (t)
= k lim inf 1 = k lim inf 1 = kc > 0. (68)
t→0+ ` (t)
2 t→0+ ` (t)
2

Since τ is radially symmetric, then the lower bound in (8) follows. The lower
estimate in (9) now follows from (68) (with a proof by contradiction, for instance).
On the other hand, the upper bound in (8) and in (9) holds by Theorem 1.1.

5.2. A Hopf-type lemma for the logarithmic Laplacian. For a bounded open
set Ω, the space V(Ω) is defined in (11). We say that a function v ∈ V(Ω) solves
weakly that L∆ v ≥ 0 in Ω if
EL (v, ϕ) ≥ 0 for all ϕ ∈ Cc∞ (Ω) with ϕ ≥ 0 in Ω.
We remark that EL (v, ϕ) < ∞ for v ∈ V(Ω) and ϕ ∈ Cc∞ (Ω) by [10, Lemma 4.4].
Recall that η denotes the outer unit normal vector field along ∂Ω. We are ready to
show Theorem 1.4.

Proof of Theorem 1.4. Let x0 ∈ ∂Ω and v be as in the statement. By continuity

and because v 6= 0, there are δ > 0, an open set V ⊂ {x ∈ Ω : v(x) > δ}, and
r > 0 such that v solves weakly that L∆ v ≥ 0 in B := Br (x0 − rη(x0 )) ⊂ Ω and
dist(B, V ) > 0. Note that x0 ∈ ∂B.
By Theorem 4.1, we can consider, if necessary, r smaller so that L∆ satisfies the
weak maximum principle in B. By Proposition 5.1, there is a classical solution τ of
L∆ τ = 1 in B, τ =0 in RN \ B.
Moreover, τ > 0 in B. Now we argue as in [16]. Let χV denote the characteristic
function of V and note that, for x ∈ B, χV (x) = 0 and, therefore,
Z Z
χV (y) 1
L∆ χV (x) = −cN N
dy = −c N N
dy ≤ −cN |V | diam(Ω)−N .
RN |x − y| V |x − y|
30 V. HERNÁNDEZ-SANTAMARÍA, L. F. LÓPEZ RÍOS AND A. SALDAÑA

Let K := cN |V | diam(Ω)−N and

 
K 1
ϕ := k τ + χV with k := K
.
2 2 kτ kL (B)
∞ +1
Then, L∆ ϕ ≤ k(K/2 − K) ≤ 0 in B. Moreover, since v > δ in V , we have that
v − δϕ solves weakly that
L∆ (v − δϕ) ≥ 0 in B and v − δϕ ≥ 0 in RN \ B.
Then, by Theorem 4.1, v ≥ δϕ ≥ k K
2 δτ > 0 in B. But then, for t ∈ (0, r),
v(x0 − tη(x0 )) K τ (x0 − tη(x0 ))
1 ≥k δ 1 ≥c>0
` (t)
2 2 ` 2 (t)
for some c > 0, by Theorem 1.2, as claimed.
Corollary 5.3. Let Ω ⊂ RN be an open bounded set of class C 2 and let v ∈ C(RN )
be such that v = 0 on RN \Ω, v > 0 in Ω and there is c ∈ (0, 1) with
v(x0 − tη(x0 )) v(x0 − tη(x0 ))
c−1 > lim sup 1 ≥ lim inf+ 1 > c, (69)
t→0+ ` (t)
2 t→0 ` 2 (t)
where η(x0 ) is a unitary exterior normal vector at x0 . Then there is C > 1 such
that
1 1
C −1 ` 2 (δ(x)) < v(x) < C` 2 (δ(x)) for all x ∈ Ω,
where δ(x) := dist(x, ∂Ω).
Proof. This is a standard consequence of Hopf-type lemmas. We include a proof
for completeness. We show first that
1
v(x) > c ` 2 (δ(x)) for all x ∈ Ω (70)
for some c > 0. Indeed, assume by contradiction that there are (xn )n∈N ⊂ Ω with
1 1
v(xn ) < ` 2 (δ(xn )) for all x ∈ Ω. (71)
n
Since v > 0 in Ω, then (up to a subsequence) xn → x∗ with x∗ ∈ ∂Ω as n → ∞.
Let yn ∈ ∂Ω be such that xn = yn − δ(xn )η(yn ). By (69),
v(xn )
lim inf 1 ≥ c > 0;
n→∞ ` 2 (δ(xn ))
1
but this contradicts (71) and thus (70) follows. The fact that v(x) < C ` 2 (δ(x)) for
all x ∈ Ω follows from Theorem 1.1.
5.3. Uniqueness of positive solutions of logarithmic sublinear problems.
Let Ω be a bounded set of class C 2 ; in particular Ω satisfies uniform exterior and
interior sphere conditions. For µ > 0, consider the problem
L∆ u = −µ ln(|u|)u in Ω, v=0 in RN \Ω, (72)
We say that u ∈ H(Ω) is a weak solution of (72) if
Z
EL (u, v) = −µ uv ln(|u|) dx for all v ∈ H(Ω).
Ω
In [2, Theorem 1.1], the following result is shown. Recall that
1
`(r) = | ln(min{r, 10 })|−1 .
 OPTIMAL BOUNDARY REGULARITY FOR THE LOGARITHMIC LAPLACIAN 31

Theorem 5.4. For every µ > 0 there is a unique (up to a sign) least-energy weak
solution u ∈ H(Ω) of (72) which is a global minimizer of the energy functional
Z
1 µ
v 2 ln(v 2 ) − 1 dx. (73)


J0 : H(Ω) → R, J0 (v) := EL (v, v) + I(v), I(v) :=
2 4 Ω
1 1
Moreover, 0 < |u(x)| ≤ (R2 e 2 −ρN ) µ for x ∈ Ω, where R := 2 diam(Ω) and ρN is an
explicit constant given in (14). Furthermore, u ∈ C(RN ), and there are α ∈ (0, 1)
and C > 0 such that
|u(x) − u(y)|
sup < C. (74)
x,y∈R N `α (|x − y|)
x6=y

In this theorem, the uniqueness (up to a sign) of the least energy solution is
shown using a convexity-by-paths argument as in [4, Section 6]. In particular, the
following is shown: Given u and v in H(Ω) such that u2 6= v 2 , let
1
θ(t, u, v) := ((1 − t)u2 + tv 2 ) 2 , (75)
then
the function t 7→ J0 (θ(t, u, v)) is strictly convex in [0, 1], (76)
Since a strictly convex function cannot have two global minimizers, (76) immediately
yields the uniqueness of least-energy solutions.
Now we use (76) and Theorem 1.4 to yield the uniqueness of positive solutions
(which has to be the positive least-energy solution obtained in Theorem 5.4).
Theorem 5.5. For every µ > 0 there is only one positive and bounded solution
of (72).
Proof. Let A be the set of positive and bounded critical points of J0 . By The-
orem 5.4 we know that A is nonempty. Let u, v ∈ A. Since u, v ∈ L∞ (Ω), it
follows that ln |u|u, ln |v|v ∈ L∞ (Ω), and, by [10, Theorem 1.11], we have that
u, v ∈ C(Ω). Then there is ε > 0 such that −µ ln |u|u > 0 and −µ ln |v|v > 0 in
Ωε := {x ∈ Ω : dist(x, ∂Ω) < ε}. Since u and v are continuous weak solutions
of (72), we have in particular that u and v solve weakly
L∆ u ≥ 0, L∆ v ≥ 0 in Ωε , u ≥ 0, v ≥ 0 in RN \Ωε .
By Theorem 1.4, there is c ∈ (0, 1) such that
u(x0 − tη(x0 )) u(x0 − tη(x0 ))
c−1 > lim sup 1 ≥ lim inf 1 > c,
t→0+ ` 2 (t) t→0 +
` 2 (t)
v(x0 − tη(x0 )) v(x0 − tη(x0 ))
c−1 > lim sup 1 ≥ lim inf 1 > c,
` 2 (t)
t→0+ t→0+ ` 2 (t)
for all x0 ∈ ∂Ω. Then, by Corollary 5.3, we obtain that u and v are comparable,
namely, that there is M > 1 such that
v(x)
M> > M −1 for x ∈ Ω. (77)
u(x)
v
Let θ be as in (75). We now argue as in [4, Theorem 6.1]. Let w := u χΩ ,
2
ξ −1
z(ξ, t) := 1 , and let χΩ denote the characteristic function of Ω, then
(1−t+tξ 2 ) 2 +1

θ(t) − θ(0) θ(t)2 − u2 (1 − t)u2 + tv 2 − u2

= =
t t(θ(t) + u) t(θ(t) + u)
32 V. HERNÁNDEZ-SANTAMARÍA, L. F. LÓPEZ RÍOS AND A. SALDAÑA

v 2 − u2 w2 − 1
= =u 1 = uz(w, t).
θ(t) + u (1 − t + tw2 ) 2 + 1
By (77), for x, y ∈ Ω,
u(x)z(w(x), t) − u(y)z(w(y), t)
= (u(x) − u(y))z(w(x), t) − u(y)(z(w(y), t) − z(w(x), t))
and, by the Mean-Value Theorem,
v(x)
u(y)|z(w(x), t) − z(w(y), t)| ≤ C1 u(y)|w(x) − w(y)| = C1 u(y) − v(y)
u(x)
v(x)
= C1 v(x) − v(y) + (u(y) − u(x))
u(x)
≤ C2 (|v(x) − v(y)| + |u(y) − u(x)|),
where C1 := sup(k,t)∈[0,M ]×[0,1] |∂ξ z(k, t)| < ∞ and C2 := C1 + M .
On the other hand, if x ∈ Ω and y ∈ RN \Ω, then u(y) = 0 and
|u(x)z(w(x), t) − u(y)z(w(y), t)| ≤ C1 M |u(x)| = C1 M |u(x) − u(y)|.
Then there is C3 > 0 such that
2
θ(t) − θ(0)
t
   
Z Z | θ(t)−θ(0)
t (x) − θ(t)−θ(0)
t (y)|2
= dy dx
RN B1 (x) |x − y|N
!
|u(x) − u(y)|2 |v(x) − v(y)|2
Z Z Z Z
≤ C3 dy dx + dy dx
RN B1 (x) |x − y|N RN B1 (x) |x − y|N
= C3 (kuk2 + kvk2 ).
This, together with (76), guarantees that all the assumptions of [4, Theorem 1.1]
are satisfied. Then, this result implies that A has at most one element, and this
ends the proof.

Appendix A. Some auxiliary results.

A.1. Scaling properties of the logarithmic Laplacian. In this section, for
completeness, we show some easy scaling properties for the logarithmic Laplacian
(see also [29, Lemma 2.5]). For λ > 0 and a function ϕ : RN → R, let
ϕλ (x) := ϕ(λ−1 x).
Let Ω be an open bounded Lipschitz subset of RN . We set Ωλ := λΩ = {λx :
x ∈ Ω}.
Lemma A.1 (On smooth functions). Let u ∈ Cc∞ (Ω) be a solution of L∆ u = f in
Ω, u = 0 on RN \Ω for some f ∈ L∞ (Ω). Then, for any λ > 0, uλ is a solution of
L∆ uλ = feλ in Ωλ , uλ = 0 on RN \Ωλ , with feλ := fλ + ln(λ−2 )uλ ∈ L∞ (Ωλ ).
Proof. Let x ∈ Ωλ = λΩ, then (−∆)s uλ (x) = λ−2s (−∆)s u(λ−1 x) and then
L∆ uλ = ∂s |s=0 (−∆)s uλ (x) = ∂s |s=0 λ−2s (−∆)s u(λ−1 x)
= ln(λ−2 )u(λ−1 x) + L∆ u(λ−1 x),
 OPTIMAL BOUNDARY REGULARITY FOR THE LOGARITHMIC LAPLACIAN 33

where we used [10, Theorem 1.1] to justify the first and last equalities.
The next result is an integration by parts formula under slightly weaker assump-
tions than those in [10, equation (3.11)].
Lemma A.2 (Integration by parts). Let Ω ⊂ RN be a bounded Lipschitz set and
let w ∈ H(Ω) be such that L∆ w ∈ L∞ (Ω). Then,
Z
[L∆ w] v dx = EL (w, v) for all v ∈ H(Ω).
RN

Proof. A similar result for uniformly Dini continuous functions can be found in [10,
equation (3.11)], and the same arguments can be extended for w as in the statement.
We give a proof for completeness. Let k : RN \{0} → R and j : RN → R be given
by
k(z) := cN 1B1 (z)|z|−N and j(z) := cN 1RN \B1 (z)|z|−N .
Then, for x ∈ Ω,
Z
L∆ w(x) = (w(x) − w(y))k(x − y)dy − [j ∗ w](x) + ρN w(x)
RN
and, by a standard argument using a change of variables,
Z
[L∆ w] v dx
RN
Z Z 
= (w(x) − w(y))k(x − y)dy − [j ∗ w](x) + ρN w(x) v(x) dx
RN N
Z ZR Z
1
= (w(x) − w(y))(v(x) − v(y))k(x − y) dx dy − (j ∗ w − ρN w) v dx
2 RN RN RN
= EL (w, v)
for all v ∈ H(Ω), as claimed.
Lemma A.3 (On weak solutions). Let u ∈ H(Ω) ∩ L∞ (Ω) be a weak solution of
L∆ u = f in Ω, u = 0 on RN \Ω for some f ∈ L∞ (Ω). Then, for any λ > 0, uλ is a
weak solution of L∆ uλ = feλ in λΩ, uλ = 0 on RN \λΩ, with feλ := fλ + ln(λ−2 )uλ ∈
L∞ (Ω).
Proof. Let ϕ ∈ Cc∞ (Ω). Using Lemmas A.1 and A.2,
Z
EL (uλ , ϕλ ) = uλ L∆ ϕλ dx
N
ZR
= u(λ−1 x)L∆ ϕ(λ−1 x) + ln(λ−2 )u(λ−1 x)ϕ(λ−1 x) dx
R N
Z
= λN uL∆ ϕ + ln(λ−2 )uϕ dx
RN
 Z 
N −2
=λ EL (u, ϕ) + ln(λ ) uϕ dx
RN
Z
= (fλ + ln(λ−2 )uλ )ϕλ dx.
RN

Now, a scaling property for pointwise solutions easily follows from the previous
result.
34 V. HERNÁNDEZ-SANTAMARÍA, L. F. LÓPEZ RÍOS AND A. SALDAÑA

Lemma A.4 (On pointwise solutions). Let w and Ω be as in Lemma A.2. Moreover,
assume that w ∈ L∞ (Ω) and let f (x) := L∆ w(x) for x ∈ Ω. Then, for any
λ > 0, wλ is a solution of L∆ wλ = feλ in Ωλ , wλ = 0 on RN \Ωλ , with feλ :=
fλ + ln(λ−2 )wλ ∈ L∞ (Ωλ ).
Proof. Let λ > 0, w, and Ω as in the statement, ϕ ∈ Cc∞ (Ωλ ), and let ψ(x) :=
ϕ(λx). Then, by Lemmas A.1, A.2, and a change of variables,
Z Z Z Z
λ−N fλ ϕ dx = f ψ dx = L∆ wψ dx = EL (w, ψ) = wL∆ ψ dx
Ωλ Ω Ω Ω
Z
= w(x)(L∆ ϕ(λx) + ln(λ2 )ϕ(λx)) dx
Ω
Z
−N
=λ wλ (x)(L∆ ϕ(x) + ln(λ2 )ϕ(x)) dx
Ωλ
Z
−N
=λ (L∆ wλ (x) + ln(λ2 )wλ (x))ϕ(x) dx.
Ωλ

(L∆ wλ − (fλ + ln(λ−2 )wλ ))ϕ dx = 0. Since ϕ is arbitrary, the

R
As a consequence, Ωλ
claim follows.

A.2. A Leibniz-type formula for the logarithmic Laplacian. Let E be a

bounded measurable set of RN and u : E → R be a measurable function. Recall
that the modulus of continuity of u at a point x ∈ E is defined as
ωu,x,E : (0, ∞) → [0, ∞), ωu,x,E (r) = sup |u(y) − u(x)|,
y∈E,
|y−x|≤r

R1 ωu,x,E (r)
and u is said to be Dini continuous at x if 0 r dr < ∞. Let
 
|u(x)|
Z
L10 (RN ) := u ∈ L1loc (RN ) : dx < ∞ .
RN (1 + |x|)N
Lemma A.5. Let u, v ∈ L10 (RN ) be such that u and v are Dini continuous functions
at some x ∈ RN and v ∈ L∞ (RN ). Then L∆ [uv](x) is well defined (in the sense of
formula (2)) and
L∆ [uv](x) = u(x)L∆ v(x) + v(x)L∆ u(x) − I(u, v)(x), (78)
where
I(u, v)(x)
(u(x) − u(y))(v(x) − v(y))
Z
= cN dy
B1 (x) |x − y|N
u(y)v(y) − u(x)v(y) − u(y)v(x)
Z
+ cN dy + ρN u(x)v(x). (79)
RN \B1 (x) |x − y|N
Proof. Let x ∈ Ω be as in the statement. First we show that the right hand side
of (78) is well defined. The first two terms, u(x)L∆ v(x) and v(x)L∆ u(x) are well
defined by our assumptions on u and v, see [10, Proposition 2.2]. On the other
hand, since v ∈ L∞ (RN ) and u is Dini continuous at x,
(u(x) − u(y))(v(x) − v(y)) (u(x) − u(z + x))(v(x) − v(z + x))
Z Z
N
dy = dz
B1 (x) |x − y| B1 |z|N
 OPTIMAL BOUNDARY REGULARITY FOR THE LOGARITHMIC LAPLACIAN 35

1
|u(x) − u(x + z)|
Z Z
ωu,x,B1 (r)
≤ 2kvk L∞ dz ≤ 2kvk∞ |SN −1 | dr < ∞.
B1 |z|N 0 r
∞ N
Furthermore, since v ∈ L (R ) and u ∈ L10 (R),
there is a constant Cx > 0 only
depending on x such that
|u(y)|
Z Z
u(y)v(y)
N
dy ≤ kvk L ∞ dy
N
R \B1 (x) |x − y| N
R \B1 (x) |x − y|N
|u(y)|
Z
≤ Cx kvkL ∞ dy < ∞.
RN (1 + |y|)N
Similarly, RN \B1 (x) u(x)v(y) dy and RN \B1 (x) u(y)v(x)
R R
|x−y|N |x−y|N
dy are also finite. Then, since
L∆ [uv](x)
u(x)v(x) − u(y)v(y)
Z Z
u(y)v(y)
= cN dy − cN dy + ρN u(x)v(x),
B1 (x) |x − y|N RN \B1 (x) |x − y|N
identity (78) follows by noting that
u(x)v(x) − u(y)v(y)
Z
cN dy
B1 (x) |x − y|N
v(x) − v(y) (u(x) − u(y))v(y)
Z Z
= u(x) cN N
dy + cN dy
B1 (x) |x − y| B1 (x) |x − y|N
v(x) − v(y) u(x) − u(y)
Z Z
= u(x) cN N
dy + v(x) cN N
dy
B1 (x) |x − y| B1 (x) |x − y|
(u(x) − u(y))(v(x) − v(y))
Z
− cN dy.
B1 (x) |x − y|N

Acknowledgments. We thank Héctor Chang-Lara for discussions on this topic

and we thank the anonymous referees for their helpful comments and suggestions
that substantially improved the quality of our paper.
The work of V. Hernández-Santamarı́a is supported by the program “Estancias
Posdoctorales por México para la Formación y Consolidación de las y los Investi-
gadores por México” of CONAHCYT (Mexico). He also received support from
Projects A1-S-17475 and A1-S-10457 of CONAHCYT and by UNAM-DGAPA-
PAPIIT grants IN109522, IN104922, and IA100324. L.F. López Rı́os is supported
by CONAHCYT grant CF-2023-G-122. A. Saldaña is supported by CONAHCYT
grants CBF2023-2024-116 and A1-S-10457 and by UNAM-DGAPA-PAPIIT grant
IA100923.

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Received February 2024; revised June 2024; early access June 2024.