ERRATUM to A Critical Nonlocal Double Phase Problem in the Heisenberg Group With a Modified Hardy Potential

Abstract

Mukherjee A. & Tiwari S. (2025). A critical nonlocal double phase problem in the heisenberg group with a modified hardy potential. Asymptotic Analysis. 0(0). https://doi.org/10.1177/09217134251386204

In the published version of the article, the section labelled “The Heisenberg Group ℍN and the Space HWs,℘(ℍ N )” was incomplete until the next section “Lemma 2.1.”. Below is the corrected version of the section:

2. The Heisenberg Group $H^{N}$ and the Space $H W^{s, ℘} (H^{N})$

In this section, we first present a few basic properties of the Heisenberg group $(H^{N}, \circ)$ . For more details we refer to Garofalo and Lanconelli (1990) and Garofalo and Nhieu (1996).

Let $H^{N}$ be the Heisenberg group of topological dimension $2 N + 1$ , which is a nilpotent Lie group having $R^{2 N + 1}$ as a background manifold equipped with a non-Abelian group law

ξ \circ ξ^{'} = (z + z^{'}, t + t^{'} + 2 \sum_{i = 1}^{N} (x_{i}^{'} y_{i} - x_{i} y_{i}^{'}))

for

\begin{aligned} ξ & = (z, t) = (x_{1}, \dots, x_{N}, y_{1}, \dots, y_{N}, t) \in H^{N} \\ ξ^{'} & = (z^{'}, t^{'}) = (x_{1}^{'}, \dots, x_{N}^{'}, y_{1}^{'}, \dots, y_{N}^{'}, t^{'}) \in H^{N} \end{aligned}

Throughout the article, we will denote

\tilde{0} = (0, 0) \in R^{2 N} \times R

as the natural origin of

H^{N}

. The

2 N + 1

left-invariant vector fields on

H^{N}

given by

X_{j} = \frac{\partial}{\partial x_{j}} + 2 y_{j} \frac{\partial}{\partial t}; Y_{j} = \frac{\partial}{\partial y_{j}} - 2 x_{j} \frac{\partial}{\partial t}; T = \frac{\partial}{\partial t}

for

j = 1, \dots, N

form a basis for the real Lie algebra of

H^{N}

of left-invariant vector fields. These bases satisfy the following canonical commutator relations

[X_{j}, Y_{k}] = - 4 δ_{j k} T; [Y_{j}, Y_{k}] = [X_{j}, X_{k}] = [Y_{j}, T] = [X_{j}, T] = 0

for all

j, k = 1, \dots, N

. Since the commutators of length more than two vanish,

H^{N}

is a nilpotent graded stratified group of step two. A left-invariant vector field belonging to

span {X_{j}, Y_{j}}_{j = 1}^{N}

is called horizontal. For each real

R > 0

, the dilation

δ_{R} : H^{N} \to H^{N}

defined by

δ_{R} (ξ) = (R z, R^{2} t) for all ξ = (z, t) \in H^{N}

It is easily verified that Jacobian of dilatations

δ_{R}

has constant determinant and equals

R^{2 N + 2}

, hence

Q = 2 N + 2

is the homogeneous dimension of

H^{N}

. The Korányi norm of any

ξ \in H^{N}

is defined by

r (ξ) = r (z, t) = (| z |^{4} + t^{2})^{\frac{1}{4}}

(2.1)

for

ξ = (z, t) = (x, y, t) \in R^{N} \times R^{N} \times R

and

| z |

being the Euclidean norm in

R^{2 N}

z

. The Korányi norm of

δ_{R}

r (δ_{R} (ξ)) = r (R z, R^{2} t) = (| R z |^{4} + R^{4} t^{2})^{\frac{1}{4}} = R r (ξ)

for all

ξ = (z, t) \in H^{N}

. The corresponding distance in

H^{N}

, called Korányi distance, is

d_{K} (ξ, ξ^{'}) = r (ξ^{- 1} \circ ξ^{'}) for all (ξ, ξ^{'}) \in H^{N} \times H^{N}

Throughout the article, we denote by

B_{H} (ξ_{0}, R) = {ξ \in H^{N} : d_{K} (ξ, ξ_{0}) < R}

the Korányi open ball of radius

R

centered at

ξ_{0}

. For simplicity we put

B_{H} (R) = B_{H} (\tilde{0}, R)

. The Lebesgue measure on

R^{2 N + 1}

is invariant under left translations of

H^{N}

. Since Haar measures on Lie groups are unique up to constant multipliers, we denote by

d ξ

the Haar measure on

H^{N}

with

(2 N + 1)

-dimensional Lebesgue measure, i.e.

d ξ = d x d y d t

R^{N} \times R^{N} \times R

and

| U |

denotes the same of any measurable

U \subseteq H^{N}

. Furthermore

| δ_{R} (U) | = R^{Q} | U |; d (δ_{R} ξ) = R^{Q} d ξ

We define the horizontal gradient of some

u \in C^{1} (H^{N})

\nabla_{H} u = \sum_{j = 1}^{N} [(X_{j} u) X_{j} + (Y_{j} u) Y_{j}]

We consider the natural inner product

⟨ X, Y ⟩_{H} = \sum_{j = 1}^{N} (x^{j} y^{j} + {\bar{x}}^{j} {\bar{y}}^{j})

for vector fields

X = {x^{j} X_{j} + {\bar{x}}^{j} Y_{j}}_{j = 1}^{N}

and

Y = {y^{j} X_{j} + {\bar{y}}^{j} Y_{j}}_{j = 1}^{N}

, here repeated indices follow Einstein summation convention. Then the Hilbertian norm induced by the above inner product is

| X |_{H} = \sqrt{⟨ X, X ⟩_{H}}

for any horizontal vector field

X

. The horizontal divergence of

X

is then defined by

{div}_{H} X = \nabla_{H} \cdot X = \sum_{j = 1}^{N} [X_{j} (x^{j}) + Y_{j} ({\bar{x}}^{j})]

We then have the Kohn–Spencer Laplacian as

Δ_{H} u = \sum_{j = 1}^{N} (X_{j}^{2} + Y_{j}^{2}) u

and the generalized horizontal

℘

-Laplacian for

℘ \in (1, \infty)

is defined by

Δ_{H, ℘} u = {div}_{H} (| \nabla_{H} u |^{℘ - 2} \nabla_{H} u)

for all

u \in C_{c}^{\infty} (H^{N})

. We now define the horizontal fractional Sobolev spaces

H W^{s, ℘} (H^{N})

. Interested readers may follow Adimurthi and Mallick (2018), Ghosh et al. (2024), Zhou et al. (2022) for more details. Let

℘ \in (1, \infty)

and

s \in (0, 1)

. Let

u : H^{N} \to R

be a Haar-measurable function. Then one defines

H W^{s, ℘} (H^{N}) = {u \in L^{℘} (H^{N}) : \frac{| u (ξ) - u (η) |}{r (η^{- 1} \circ ξ)^{s + \frac{Q}{℘}}} \in L^{℘} (H^{N} \times H^{N})}

endowed with the natural norm

| | u | |_{W^{s, ℘} (H^{N})} = {(\int_{H^{N}} | u |^{℘} d ξ + \underset{=: [u]_{H W^{s, ℘} (H^{N})}^{℘}}{\underset{⏟}{\iint_{H^{N} \times H^{N}} \frac{| u (ξ) - u (η) |^{℘}}{r (η^{- 1} \circ ξ)^{Q + s ℘}} d ξ d η}})}^{\frac{1}{℘}}

for

u \in H W^{s, ℘} (H^{N})

. In a similar fashion, given some domain

Ω \subset H^{N}

, one defines

H W_{0}^{s, ℘} (Ω)

as the closure of

C_{0}^{\infty} (Ω)

H W^{s, ℘} (H^{N})

. We note

H W_{0}^{s, ℘} (H^{N}) = H W^{s, ℘} (H^{N})

. The fractional

(s, ℘)

-horizontal gradient of any

u \in H W^{s, ℘} (H^{N})

is denoted by

\begin{aligned} | (\nabla_{H})^{s} u |^{℘} (ξ) = \int_{H^{N}} \frac{| u (ξ) - u (η) |^{℘}}{r (η^{- 1} \circ ξ)^{Q + s ℘}} d η = \int_{H^{N}} \frac{| u (ξ \circ h) - u (ξ) |^{℘}}{r {(h)}^{Q + s ℘}} d h \end{aligned}

There are some classical embeddings in case of

H W^{s, ℘} (H^{N})

as seen in

R^{N}

. We state a few of them. The following embedding is Theorem

2

from Kassymov and Suragan (2020b), see also Theorem

1

in Ghosh et al. (2024).

The article has been updated online.

ERRATUM to A Critical Nonlocal Double Phase Problem in the Heisenberg Group With a Modified Hardy Potential

Abstract

2. The Heisenberg Group H N and the Space H W s , ℘ ( H N )

2. The Heisenberg Group $H^{N}$ and the Space $H W^{s, ℘} (H^{N})$