New characterizations of magnetic Sobolev spaces

TL;DR

This paper introduces two novel characterizations of magnetic Sobolev spaces using nonlocal functionals, extending classical results and analyzing convergence properties.

Contribution

It provides new nonlocal characterizations of magnetic Sobolev spaces for Lipschitz magnetic fields, linking to BBM formulas and classical Sobolev space work.

Findings

New characterizations of magnetic Sobolev spaces established

Convergence properties in almost everywhere and L^1 sense analyzed

Connections made to BBM formula and classical Sobolev spaces

Abstract

We establish two new characterizations of magnetic Sobolev spaces for Lipschitz magnetic fields in terms of nonlocal functionals. The first one is related to the BBM formula, due to Bourgain, Brezis, and Mironescu. The second one is related to the work of the first author on the classical Sobolev spaces. We also study the convergence almost everywhere and the convergence in $L^1$ appearing naturally in these contexts.

Equations

$$H_A^1({\mathbb{R}}^N)\ni u\mapsto {\int }_{{\mathbb{R}}^N}|\nabla u-\mathrm{i}A(x)u{|}^{2}dx,$$

$$\|u\|_{H^{1}_{A}(\Omega)}:=\Big{(}\|u\|_{L^{2}(\Omega)}^{2}+[u]_{H^{1}_{A}(\Omega)}^{2}\Big{)}^{1/2},\quad[u]_{H^{1}_{A}(\Omega)}:=\Big{(}\int_{\Omega}|\nabla u-{\rm i}A(x)u|^{2}dx\Big{)}^{1/2}.$$

$$H_A^s({\mathbb{R}}^N)\ni u\mapsto (1-s){\iint }_{{\mathbb{R}}^{2N}}\frac{|u(x)-e^{\mathrm{i}(x-y)\cdot A\left(\frac{x+y}{2}\right)}u(y){|}^2}{|x-y{|}^{N+2s}}dxdy,$$

$$\underset{s\nearrow 1}{\lim }(1-s){\int }_{\Omega }{\int }_{\Omega }\frac{|u(x)-e^{\mathrm{i}(x-y)\cdot A\left(\frac{x+y}{2}\right)}u(y){|}^2}{|x-y{|}^{N+2s}}dxdy=Q_N{\int }_{\Omega }|\nabla u-\mathrm{i}A(x)u{|}^{2}dx,$$

$$Q_N:=\frac{1}{2}{\int }_{{\mathbb{S}}^{N-1}}|\mathbf{\omega }\cdot \sigma {|}^{2}d\sigma$$

$$\rho_{n}(r):=\left\{\begin{array}[]{cl}2(1-s_{n})\mbox{diam}(\Omega)^{2s_{n}-2}r^{2-2s_{n}-N}&\mbox{ for }0<r\leq\mbox{diam}(\Omega),\\[4.0pt] 0&\mbox{ for }r>\mbox{diam}(\Omega),\end{array}\right.$$

$$\underset{n\to +\infty }{\lim }{\int }_{\delta }^{\infty }{\rho }_n(r)r^{N-1}dr=0.$$

$${\Psi }_u(x,y):=e^{\mathrm{i}(x-y)\cdot A\left(\frac{x+y}{2}\right)}u(y),x,y\in \Omega .$$

$$\underset{n\to +\infty }{\lim }{\int }_{\Omega }{\int }_{\Omega }\frac{|{\Psi }_u(x,y)-{\Psi }_u(x,x){|}^2}{|x-y{|}^2}{\rho }_n(|x-y|)dxdy=2Q_N{\int }_{\Omega }|\nabla u-\mathrm{i}A(x)u{|}^{2}dx.$$

$$\int_{0}^{\infty}\rho_{n}(r)r^{N-1}dr=1\quad\mbox{ and }\quad\lim_{n\to+\infty}\int_{\delta}^{\infty}\rho_{n}(r)r^{N-1}dr=0,\,\,\,\mbox{for all $\delta>0$}.$$

$$\underset{n\in \mathbb{N}}{\sup }{\iint }_{{\mathbb{R}}^{2N}}\frac{|{\Psi }_u(x,y)-{\Psi }_u(x,x){|}^2}{|x-y{|}^2}{\rho }_n(|x-y|)dxdy<+\infty .$$

$$\underset{n\to +\infty }{\lim }{\iint }_{{\mathbb{R}}^{2N}}\frac{|{\Psi }_u(x,y)-{\Psi }_u(x,x){|}^2}{|x-y{|}^2}{\rho }_n(|x-y|)dxdy=2Q_N{\int }_{{\mathbb{R}}^N}|\nabla u-\mathrm{i}A(x)u{|}^{2}dx,$$

$$\iint_{{\mathbb{R}}^{2N}}\frac{|\Psi_{u}(x,y)-\Psi_{u}(x,x)|^{2}}{|x-y|^{2}}\rho_{n}(|x-y|)\,dx\,dy\\ \leq 2|\mathbb{S}^{N-1}|\int_{{\mathbb{R}}^{N}}|\nabla u-{\rm i}A(x)u|^{2}\,dx+2|\mathbb{S}^{N-1}|\big{(}2+\|\nabla A\|_{L^{\infty}({\mathbb{R}}^{N})}^{2}\big{)}\int_{{\mathbb{R}}^{N}}|u|^{2}\,dx.$$

$$J_{\delta}(u):=\iint_{\{|\Psi_{u}(x,y)-\Psi_{u}(x,x)|>\delta\}}\frac{\delta^{2}}{|x-y|^{N+2}}\,dxdy,\quad\mbox{for $u\in L^{1}_{{\rm loc}}({\mathbb{R}}^{N})$}.$$

$$I_{\delta }(u):={\iint }_{\{|u(y)-u(x)|>\delta \}}\frac{{\delta }^2}{|x-y{|}^{N+2}}dxdy,\text{mbox}foru\in L_{\mathrm{loc}}^1({\mathbb{R}}^N).$$

$$\lim_{\delta\searrow 0}I_{\delta}(u)=Q_{N}\int_{\Omega}|\nabla u|^{2}dx,\quad\mbox{for $u\in H^{1}({\mathbb{R}}^{N})$}.$$

$$\underset{0<\delta <1}{\sup }{J}_{\delta }(u)<+\infty .$$

$$\underset{\delta \searrow 0}{\lim }{J}_{\delta }(u)=Q_N{\int }_{{\mathbb{R}}^N}|\nabla u-\mathrm{i}A(x)u{|}^{2}dx$$

$$\sup_{\delta>0}J_{\delta}(u)\leq C_{N}\left(\int_{{\mathbb{R}}^{N}}|\nabla u-{\rm i}A(x)u|^{2}\,dx+\big{(}\|\nabla A\|_{L^{\infty}({\mathbb{R}}^{N})}^{2}+1\big{)}\int_{{\mathbb{R}}^{N}}|u|^{2}\,dx\right).$$

$$A(x,y,z)=\frac{1}{2}(-y,x,0),(x,y,z)\in {\mathbb{R}}^3,$$

$$\underset{n\to +\infty }{\lim }{\iint }_{{\mathbb{R}}^{2N}}\frac{|{\Psi }_u(x,y)-{\Psi }_u(x,x){|}^2}{|x-y{|}^2}{\rho }_n(|x-y|)dxdy=0$$

$$\underset{\delta \searrow 0}{\lim }{J}_{\delta }(u)=0,$$

$$\{\begin{array}{ll}\nabla \Re u=-A\Im u,& \\ \nabla \Im u=A\Re u,& \end{array}$$

$$\iint_{{\mathbb{R}}^{2N}}\frac{|\Psi_{u}(x,y)-\Psi_{u}(x,x)|^{2}}{|x-y|^{2}}\rho_{n}(|x-y|)\,dx\,dy\\ \leq 2|\mathbb{S}^{N-1}|\int_{{\mathbb{R}}^{N}}|\nabla u-{\rm i}A(x)u|^{2}\,dx+2|\mathbb{S}^{N-1}|\big{(}2+\|\nabla A\|_{L^{\infty}({\mathbb{R}}^{N})}^{2}\big{)}\int_{{\mathbb{R}}^{N}}|u|^{2}\,dx.$$

$${\int }_{{\mathbb{R}}^N}{\rho }_n(|z|)dz=|{\mathbb{S}}^{N-1}|{\int }_0^{\infty }{\rho }_n(r)r^{N-1}dr=|{\mathbb{S}}^{N-1}|.$$

$$\mathop{\iint_{{\mathbb{R}}^{2N}}}_{\{|x-y|\geq 1\}}\frac{|\Psi_{u}(x,y)-\Psi_{u}(x,x)|^{2}}{|x-y|^{2}}\rho_{n}(|x-y|)\,dx\,dy\\[6.0pt] \leq 2\iint_{{\mathbb{R}}^{2N}}\big{(}|u(y)|^{2}+|u(x)|^{2}\big{)}\rho_{n}(|x-y|)\,dx\,dy\leq 4|\mathbb{S}^{N-1}|\int_{{\mathbb{R}}^{N}}|u|^{2}\,dx,$$

$$\begin{gathered} {{\iint }_{{\mathbb{R}}^{2N}}}_{\{|x-y|<1\}}\frac{|{\Psi }_u(x,y)-{\Psi }_u(x,x){|}^2}{|x-y{|}^2}{\rho }_n(|x-y|)dxdy \\ \le 2|{\mathbb{S}}^{N-1}|\left({\int }_{{\mathbb{R}}^N}|\nabla u-\mathrm{i}A(x)u{|}^{2}dx+\Vert \nabla A{\Vert }_{L^{\infty }({\mathbb{R}}^N)}^2{\int }_{{\mathbb{R}}^N}|u{|}^{2}dx\right). \end{gathered}$$

$$\frac{\partial {\Psi }_u(x,y)}{\partial y}$$

$$\times e^{\mathrm{i}(x-y)\cdot A\left(\frac{x+y}{2}\right)}u(y).$$

$$\Big{|}\frac{\partial\Psi_{u}(x,y)}{\partial y}\Big{|}\leq|\nabla u(y)-{\rm i}A(y)u(y)|+\Big{|}A\Big{(}\frac{x+y}{2}\Big{)}-A(y)\Big{|}|u(y)|+\frac{1}{2}|y-x|\Big{|}\nabla A\Big{(}\frac{x+y}{2}\Big{)}\Big{|}|u(y)|.$$

$$\Big{|}\frac{\partial\Psi_{u}(x,y)}{\partial y}\Big{|}\leq|\nabla u(y)-{\rm i}A(y)u(y)|+\|\nabla A\|_{L^{\infty}({\mathbb{R}}^{N})}|x-y||u(y)|,$$

Full text

A.P., M.S. and E.V. are members of Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). E.V. receives funding from the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme FP7/2007-2013/ under REA grant agreement No. 607643 (ERC Grant MaNET ‘Metric Analysis for Emergent Technologies’).

1. Introduction

In electromagnetism, a relevant role in the study of particles which interact with a magnetic field $B=\nabla\times A$, $A:{\mathbb{R}}^{3}\to{\mathbb{R}}^{3}$, is played by the magnetic Laplacian $(\nabla-{\rm i}A)^{2}$ [2, 26, 16]. This yields to nonlinear Schrödinger equations of the type $-(\nabla-{\rm i}A)^{2}u+u=f(u),$ which have been extensively studied (see e.g. [13, 1, 17, 15] and the references therein). The linear operator $-(\nabla-{\rm i}A)^{2}u$ is defined weakly as the differential of the energy functional

[TABLE]

over complex-valued functions $u$ on ${\mathbb{R}}^{N}$. Here ${\rm i}$ denotes the imaginary unit and $|\cdot|$ the standard Euclidean norm of ${\mathbb{C}}^{N}$. Given a measurable function $A:{\mathbb{R}}^{N}\to{\mathbb{R}}^{N}$ and given an open subset $\Omega$ of ${\mathbb{R}}^{N}$, one defines $H^{1}_{A}(\Omega)$ as the space of complex-valued functions $u\in L^{2}(\Omega)$ such that $\|u\|_{H^{1}_{A}(\Omega)}<\infty$ for the norm

[TABLE]

In [14], some physically motivated nonlocal versions of the local magnetic energy were introduced. In particular the operator $(-\Delta)^{s}_{A}$ is defined as the gradient of the nonlocal energy functional

[TABLE]

where $s\in(0,1)$. Recently, the existence of ground stated of $(-\Delta)^{s}_{A}u+u=f(u)$ was investigated in [12] via Lions concentration compactness arguments. In [28] a connection between the local and nonlocal notions was obtained on bounded domains, precisely, if $\Omega\subset{\mathbb{R}}^{N}$ is a bounded Lipschitz domain and $A\in C^{2}({\mathbb{R}}^{N})$, then for every $u\in H^{1}_{A}(\Omega)$ it holds

[TABLE]

where

[TABLE]

being ${\mathbb{S}}^{N-1}$ the unit sphere in ${\mathbb{R}}^{N}$ and ${\boldsymbol{\omega}}$ an arbitrary unit vector of ${\mathbb{R}}^{N}$. See also [23] for the general case of the $p$-norm with $1\leq p<+\infty$ as well as [24] where the limit as $s\searrow 0$ is covered. This provides a new characterization of the $H^{1}_{A}$ norm in terms of nonlocal functionals extending the results by Bourgain, Brezis and Mironescu [3, 4] (see also [11, 25]) to the magnetic setting. Let $\{s_{n}\}_{n\in{\mathbb{N}}}$ be a sequence of positive numbers converging to $1$ and less than $1$ and set

[TABLE]

where $\mbox{diam}(\Omega)$ denotes the diameter of $\Omega$. We have $\int_{0}^{\infty}\rho_{n}(r)r^{N-1}dr=1$ and, for all $\delta>0,$

[TABLE]

Given $u:\Omega\to{\mathbb{C}}$ a measurable complex-valued function, we denote

[TABLE]

The function $\Psi_{u}(\cdot,\cdot)$ also depends on $A$ but for notational ease, we ignore it. Assertion (1.1) can be then written as

[TABLE]

This paper is concerned with the whole space setting. Our first goal is to obtain formula (1.3) for $\Omega={\mathbb{R}}^{N}$ and to provide a characterization of $H^{1}_{A}({\mathbb{R}}^{N})$ in terms of the LHS of (1.3) in the spirit of the work of Bourgain, Brezis and Mironescu.

Here and in what follows, a sequence of nonnegative radial functions $\{\rho_{n}\}_{n\in{\mathbb{N}}}$ is called a sequence of mollifiers if it satisfies the conditions

[TABLE]

In this direction, we have

Theorem 1.1**.**

Let $A:{\mathbb{R}}^{N}\to{\mathbb{R}}^{N}$ be Lipschitz and let $\{\rho_{n}\}_{n\in{\mathbb{N}}}$ be a sequence of nonnegative radial mollifiers. Then $u\in H^{1}_{A}({\mathbb{R}}^{N})$ if and only if $u\in L^{2}({\mathbb{R}}^{N})$ and

[TABLE]

Moreover, for $u\in H^{1}_{A}({\mathbb{R}}^{N})$, we have

[TABLE]

and

[TABLE]

In this paper, $|\mathbb{S}^{N-1}|$ denotes the $(N-1)$-Hausdorff measure of the unit sphere $\mathbb{S}^{N-1}$ in ${\mathbb{R}}^{N}$.

The proof of Theorem 1.1 is given in Section 2.

Remark 1.1**.**

Similar results as in Theorem 1.1 hold for more general mollifiers $\{\rho_{n}\}_{n\in{\mathbb{N}}}$ with slight changes in the constants. See Remark 2.1 for details.

The second goal of this paper is to characterize $H^{1}_{A}({\mathbb{R}}^{N})$ in term of $J_{\delta}(\cdot)$ where, for $\delta>0$,

[TABLE]

This is motivated by the characterization of the Sobolev space $H^{1}({\mathbb{R}}^{N})$ provided in [5, 18] (see also [6, 7, 8, 9, 10, 19, 20, 21, 22]) in terms of the family of nonlocal functionals $I_{\delta}$ which is defined by, for $\delta>0$,

[TABLE]

It was showed in [5, 18] that if $u\in L^{2}({\mathbb{R}}^{N})$, then $u\in H^{1}({\mathbb{R}}^{N})$ if and only if $\sup_{0<\delta<1}I_{\delta}(u)<\infty$; moreover,

[TABLE]

Concerning this direction, we establish

Theorem 1.2**.**

Let $A:{\mathbb{R}}^{N}\to{\mathbb{R}}^{N}$ be Lipschitz. Then $u\in H^{1}_{A}({\mathbb{R}}^{N})$ if and only if $u\in L^{2}({\mathbb{R}}^{N})$ and

[TABLE]

Moreover, we have, for $u\in H^{1}_{A}({\mathbb{R}}^{N})$,

[TABLE]

and

[TABLE]

Throughout the paper, we shall denote by $C_{N}$ a generic positive constant depending only on $N$ and possibly changing from line to line.

The proof of Theorem 1.2 is given in Section 3.

As pointed out in [13], a physically meaning example of magnetic potential in the space is

[TABLE]

which in fact fulfills the requirement of Theorems 1.1 and 1.2 that $A$ is Lipschitz. Furthermore, in the spirit of [6], as a byproduct of Theorems 1.1 and 1.2, for $u\in L^{2}({\mathbb{R}}^{N})$, if we have

[TABLE]

or

[TABLE]

then

[TABLE]

namely the direction of $\nabla\Re u,\nabla\Im u$ is that of the magnetic potential $A$. In the particular case $A=0$, this implies that $u$ is a constant function.

The $L^{p}$ versions of the above mentioned results are given in Sections 2 and 3. In addition to these results, we also discuss the convergence almost everywhere and the convergence in $L^{1}$ of the quantities appearing in Theorems 1.1 and 1.2 in Section 4.

The paper is organized as follows. The proof of Theorems 1.1 and 1.2 are given in Sections 2 and  3 respectively. The convergence almost everywhere and the convergence in $L^{1}$ are investigated in Section 4.

2. Proof of Theorem 1.1 and its $L^{p}$ version

The proof of Theorem 1.1 can be derived from a few lemmas which we present below. The first one is on (1.7).

Lemma 2.1** (Upper bound).**

Let $A:{\mathbb{R}}^{N}\to{\mathbb{R}}^{N}$ be Lipschitz and let $\{\rho_{n}\}_{n\in{\mathbb{N}}}$ be a sequence of nonnegative radial mollifiers. We have, for all $u\in H^{1}_{A}({\mathbb{R}}^{N})$,

[TABLE]

Proof.

Since $C^{\infty}_{c}({\mathbb{R}}^{N})$ is dense in $H^{1}_{A}({\mathbb{R}}^{N})$ (cf. [16, Theorem 7.22]), using Fatou’s lemma, without loss of generality, one might assume that $u\in C^{1}_{c}({\mathbb{R}}^{N})$. Recall that

[TABLE]

Since

[TABLE]

it suffices to prove that

[TABLE]

For a.e. $x,y\in{\mathbb{R}}^{N}$, we have

[TABLE]

It follows that

[TABLE]

This implies

[TABLE]

which yields, for $x,y\in{\mathbb{R}}^{N}$ with $|x-y|<1$,

[TABLE]

Since, for $f\in L^{2}({\mathbb{R}}^{N})$, in light of (1.4) and (2.1), we get

[TABLE]

we then derive from (2) that

[TABLE]

which is (2.2). ∎

We next establish the following result which is used in the proof of (1.6) and in the proof of Theorem 1.2.

Lemma 2.2**.**

Let $u\in C^{2}({\mathbb{R}}^{N})$, $A:{\mathbb{R}}^{N}\to{\mathbb{R}}^{N}$ be Lipschitz, and let $\{\rho_{n}\}_{n\in{\mathbb{N}}}$ be a sequence of nonnegative radial mollifiers. Then

[TABLE]

Moreover, for any $(\varepsilon_{n})\searrow 0$, there holds

[TABLE]

Throughout this paper, for $R>0$, let $B_{R}$ denote the open ball in ${\mathbb{R}}^{N}$ centered at the origin and of radius $R$.

Proof.

Fix $R>1$ (arbitrary). Using the fact

[TABLE]

we have, for $x,y\in B_{R}$,

[TABLE]

Here and in what follows, $C$ denotes a positive constant. On the other hand, we obtain, for $x,y\in B_{R}$,

[TABLE]

It follows that

[TABLE]

Since

[TABLE]

it follows from (2.7) that

[TABLE]

We have, by the definition of $Q_{N}$,

[TABLE]

By the arbitrariness of $R>1$ we get

[TABLE]

which implies (2.5).

Assertion (2.6) can be derived as follows. We have, by Hölder’s inequality,

[TABLE]

Since, for every $R>0$, there holds

[TABLE]

we get (2.6) from (2.9) and the arbitrariness of $R>1$. ∎

We are ready to prove (1.6).

Lemma 2.3** (Limit formula).**

Let $A:{\mathbb{R}}^{N}\to{\mathbb{R}}^{N}$ be Lipschitz and let $\{\rho_{n}\}_{n\in{\mathbb{N}}}$ be a sequence of nonnegative radial mollifiers. Then, for $u\in H^{1}_{A}({\mathbb{R}}^{N})$,

[TABLE]

Proof.

By Lemma 2.1 and the density of $C^{\infty}_{c}({\mathbb{R}}^{N})$ in $H^{1}_{A}({\mathbb{R}}^{N})$, one might assume that $u\in C^{2}_{c}({\mathbb{R}}^{N})$. From Lemma 2.2, it suffices to prove that, for $u\in C^{2}_{c}({\mathbb{R}}^{N})$,

[TABLE]

Fix $R>4$ such that $\operatorname{supp}u\subset B_{R/2}$. Using (2.7) and (2.8), one derives that

[TABLE]

which yields

[TABLE]

On the other hand, we have

[TABLE]

and the fact that

[TABLE]

by the choice of $R$. Combining (2.11), (2.12), and (2.13) yields (2.10). ∎

The following result is about uniform bounds for the integrals in (1.5).

Lemma 2.4**.**

Let $A:{\mathbb{R}}^{N}\to{\mathbb{R}}^{N}$ be Lipschitz and let $\{\rho_{n}\}_{n\in{\mathbb{N}}}$ be a sequence of nonnegative radial mollifiers. Then $u\in H^{1}_{A}({\mathbb{R}}^{N})$ if $u\in L^{2}({\mathbb{R}}^{N})$ and

[TABLE]

Proof.

Let $\{\tau_{m}\}$ be a sequence of nonnegative mollifiers with $\operatorname{supp}\tau_{m}\subset B_{1}$ which is normalized by the condition $\int_{{\mathbb{R}}^{N}}\tau_{m}(x)\,dx=1$. Set

[TABLE]

We estimate

[TABLE]

We have

[TABLE]

By the change of variables $y^{\prime}=y-z$ and $x^{\prime}=x-z$ and using the inequality $|a+b|^{2}\leq 2(|a|^{2}+|b|^{2})$ for all $a,b\in{\mathbb{C}}$ and applying Jensen’s inequality, we deduce that

[TABLE]

Since, for $t\in{\mathbb{R}}$,

[TABLE]

it follows that, for all $x,\,y,\,z\in{\mathbb{R}}^{N}$,

[TABLE]

Here and in what follows in this proof, $C$ denotes some positive constant independent of $m$ and $n$. Taking into account the fact that $\operatorname{supp}\tau_{m}\subset B_{1}$, we obtain

[TABLE]

Combining (2.14), (2), (2.16) yields

[TABLE]

On the other hand, by Lemma 2.2 we have

[TABLE]

The conclusion now immediately follows from (2.17) and (2.18) after letting $m\to+\infty$. ∎

We are ready to give the proof of Theorem 1.1.

Proof of Theorem 1.1. Theorem 1.1 is a direct consequence of Lemmas 2.1, 2.3 and 2.4. ∎

Remark 2.1**.**

Let $\{\rho_{n}\}_{n\in{\mathbb{N}}}$ be a sequence of non-negative radial functions such that

[TABLE]

and

[TABLE]

Theorem 1.1 then holds for such a sequence $\{\rho_{n}\}_{n\in{\mathbb{N}}}$ provided that the constant $2$ in (1.7) is replaced by an appropriate positive constant $C$ independent of $u$. This follows by taking into account the fact that, for $u\in L^{2}({\mathbb{R}}^{N})$,

[TABLE]

For example, this applies to the radial sequence

[TABLE]

which provides a characterization of $H^{1}_{A}({\mathbb{R}}^{N})$ and yields

[TABLE]

Consider now the space $(\mathbb{C}^{n},|\cdot|_{p})$ ($n\geq 1$), endowed with the norm

[TABLE]

where $|\cdot|$ is the Euclidean norm of ${\mathbb{R}}^{n}$ and $\Re a$, $\Im a$ denote the real and imaginary parts of $a\in\mathbb{C}$ respectively. We emphasize that this is not related to the $p$-norm in ${\mathbb{R}}^{n}$. In what follows, we use this notation with $n=N$ and $n=1$. Notice that $|z|_{p}=|z|$ whenever $z\in{\mathbb{R}}^{n}$, which makes our next statements consistent with the case $A=0$ and $u$ being a real valued function. Also $|\cdot|_{2}=|\cdot|,$ consistently with the previous definition. Define, for some ${\boldsymbol{\omega}}\in\mathbb{S}^{N-1}$,

[TABLE]

We have, for $z\in\mathbb{C}^{N}$, (see [3, 23]),

[TABLE]

Using the same approach and technique, one can prove the following $L^{p}$ version of Theorem 1.1.

Theorem 2.1**.**

Let $p\in(1,+\infty)$, $A:{\mathbb{R}}^{N}\to{\mathbb{R}}^{N}$ be Lipschitz, and let $\{\rho_{n}\}_{n\in{\mathbb{N}}}$ be a sequence of nonnegative radial mollifiers. Then $u\in W^{1,p}_{A}({\mathbb{R}}^{N})$ if and only if $u\in L^{p}({\mathbb{R}}^{N})$ and

[TABLE]

Moreover, for $u\in W^{1,p}_{A}({\mathbb{R}}^{N})$, we have

[TABLE]

and

[TABLE]

for some positive constant $C_{N,p}$ depending only on $N$ and $p$.

Remark 2.2**.**

Assume that $C$ is a positive constant such that, for all $a,b\in{\mathbb{C}}$,

[TABLE]

Then assertion (2.21) of Theorem 2.1 holds with $C_{N,p}=|\mathbb{S}^{N-1}|C$.

3. Proof of Theorem 1.2 and its $L^{p}$ version

Let us set, for $\sigma\in\mathbb{S}^{N-1}$,

[TABLE]

and denote $\mathscr{M}_{e_{N}}$ by $\mathscr{M}_{N}$, $e_{N}:=(0,\ldots,0,1)$. We have the following result which is a direct consequence of the theory of maximal functions, see e.g., [29, Theorem 1, page 5].

Lemma 3.1** (Maximal function estimate).**

There exists a universal constant $C>0$ such that, for all $\sigma\in\mathbb{S}^{N-1}$,

[TABLE]

The following lemma yields an upper bound of $J_{\delta}(u)$ in terms of the norm of $u$ in $H^{1}_{A}({\mathbb{R}}^{N})$.

Lemma 3.2** (Uniform upper bound).**

Let $A:{\mathbb{R}}^{N}\to{\mathbb{R}}^{N}$ be Lipschitz and $u\in H^{1}_{A}({\mathbb{R}}^{N})$. We have

[TABLE]

Proof.

By density of $C^{\infty}_{c}({\mathbb{R}}^{N})$ in $H^{1}_{A}({\mathbb{R}}^{N})$, using Fatou’s lemma, we can assume that $u\in C^{1}_{c}({\mathbb{R}}^{N})$. For each $\delta>0$, let us define

[TABLE]

and

[TABLE]

We have

[TABLE]

Since $|\Psi_{u}(x,y)-\Psi_{u}(x,x)|\leq|u(x)|+|u(y)|$ and

[TABLE]

it follows that

[TABLE]

We are therefore interested in estimating the integral

[TABLE]

Let us now define

[TABLE]

Performing the change of variables $y=x+h\sigma$, for $h\in(0,1)$ and $\sigma\in\mathbb{S}^{N-1}$, yields

[TABLE]

where ${\mathcal{C}}_{\sigma}$ denotes the set

[TABLE]

Without loss of generality it suffices to prove that, for $\sigma=e_{N}=(0,\ldots,0,1)\in\mathbb{S}^{N-1}$,

[TABLE]

We have, by virtue of (2.3),

[TABLE]

Using the fact that if $a+b>\delta$ then either $a>\delta/2$ or $b>\delta/2$, we derive that

[TABLE]

where the last inequality follows recalling that since $(x,h)\in{\mathcal{C}}_{e_{N}}$ then $h\in(0,1)$. As usual, by using the theory of maximal functions stated in Lemma 3.1, we have

[TABLE]

and

[TABLE]

Assertion (3.1) follows from (3.3) and (3.4). The proof is complete. ∎

We next establish

Lemma 3.3** (Limit formula).**

Let $A:{\mathbb{R}}^{N}\to{\mathbb{R}}^{N}$ be Lipschitz and $u\in H^{1}_{A}({\mathbb{R}}^{N})$. Then

[TABLE]

where $Q_{N}$ is the constant defined in (1.2).

Proof.

By virtue of Lemma 3.2, for every $\delta>0$ and all $w\in H^{1}_{A}({\mathbb{R}}^{N})$, we have

[TABLE]

Since

[TABLE]

it follows that, for every $\varepsilon\in(0,1)$,

[TABLE]

This implies, for $\varepsilon\in(0,1)$ and $u,\,v\in H^{1}_{A}({\mathbb{R}}^{N})$,

[TABLE]

From (3.5) and (3.6), we derive that, for $u,u_{n}\in H^{1}_{A}({\mathbb{R}}^{N})$ and $\varepsilon\in(0,1)$,

[TABLE]

and

[TABLE]

Since $C^{1}_{c}({\mathbb{R}}^{N})$ is dense in $H^{1}_{A}({\mathbb{R}}^{N})$, from (3.7) and (3.8), it suffices to prove the assertion for $u\in C^{1}_{c}({\mathbb{R}}^{N})$. This fact is assumed from now on.

Let $R>0$ be such that $\operatorname{supp}u\subset B_{R/2}$. We claim that, for every $\sigma\in\mathbb{S}^{N-1}$, there holds

[TABLE]

Without loss of generality, we can assume $\sigma=e_{N}\in\mathbb{S}^{N-1}$. Then, we aim to prove that

[TABLE]

where $A_{N}$ denotes the $N$-th component of $A$. To this end, we consider the sets

[TABLE]

Therefore, we obtain $\chi_{\mathcal{C}_{e_{N}}(x^{\prime},\delta)}(x_{N},h)\leq\chi_{\mathcal{F}(x^{\prime})}(x_{N},h)$ for a.e. $(x,h)\in B_{R}\times(0,\infty)$ (by (3.2) in the proof of Lemma 3.2) and

[TABLE]

where we have set

[TABLE]

and we have denoted $\chi$ the characteristic function. We have, by the theory of maximal functions,

[TABLE]

and, by a straightforward computation,

[TABLE]

The validity of Claim (3.9) with $\sigma=e_{N}$ now follows from Dominated Convergence theorem since

[TABLE]

and, by a direct computation,

[TABLE]

Now, performing a change of variables we get

[TABLE]

where

[TABLE]

Exploiting (3.9), we obtain

[TABLE]

On the other hand, since $\operatorname{supp}u\subset B_{R/2}$, we have

[TABLE]

Combining (3.10) and (3.11) yields

[TABLE]

In order to conclude, we notice the following, see (2.20),

[TABLE]

where $Q_{N}$ is the constant defined in (1.2). ∎

We next deal with (1.8).

Lemma 3.4**.**

Let $u\in L^{2}({\mathbb{R}}^{N})$ and let $A:{\mathbb{R}}^{N}\to{\mathbb{R}}^{N}$ be Lipschitz. Then $u\in H^{1}_{A}({\mathbb{R}}^{N})$ if

[TABLE]

Proof.

The proof is divided into two steps.

Step 1. We assume that $u\in L^{2}({\mathbb{R}}^{N})\cap L^{\infty}({\mathbb{R}}^{N})$. Set

[TABLE]

In light of (3.12), we obtain

[TABLE]

for some positive constant $C$ independent of $\varepsilon\in(0,1)$. By Fubini’s theorem and by the definition of $L$, we have

[TABLE]

It follows that

[TABLE]

By virtue of inequality (2.6) of Lemma 2.4, we have

[TABLE]

which implies $u\in H^{1}_{A}({\mathbb{R}}^{N})$.

Step 2. We consider the general case. For $M>1$, define ${\mathcal{T}}_{M}:{\mathbb{C}}\to{\mathbb{C}}$ by setting

[TABLE]

and denote

[TABLE]

Then, we have

[TABLE]

It follows that

[TABLE]

Hence we obtain

[TABLE]

Applying the result in Step 1, we have $u_{M}\in H^{1}_{A}({\mathbb{R}}^{N})$ and hence by Lemma 3.3,

[TABLE]

Combining (3.13) and (3.14) and letting $M\to+\infty$, we derive that $u\in H^{1}_{A}({\mathbb{R}}^{N})$. The proof is complete. ∎

Remark 3.1**.**

Similar approach used for $H^{1}({\mathbb{R}}^{N})$ is given in [18].

Proof of Theorem 1.2. The limit formula stated in Theorem 1.2 follows by Lemma 3.3. Now, if $u\in H^{1}_{A}({\mathbb{R}}^{N})$, then (1.9) follows from Lemma 3.2. On the contrary, if $u\in L^{2}({\mathbb{R}}^{N})$ and (1.8) holds, it follows from Lemma 3.4 that $u\in H^{1}_{A}({\mathbb{R}}^{N})$. ∎

Given $u$ a measurable complex-valued function, define, for $1<p<+\infty$,

[TABLE]

We have the following $L^{p}$-version of Theorem 1.2.

Theorem 3.1**.**

Let $p\in(1,+\infty)$ and let $A:{\mathbb{R}}^{N}\to{\mathbb{R}}^{N}$ be Lipschitz. Then $u\in W^{1,p}_{A}({\mathbb{R}}^{N})$ if and only if $u\in L^{p}({\mathbb{R}}^{N})$ and

[TABLE]

Moreover, we have, for $u\in W^{1,p}_{A}({\mathbb{R}}^{N})$,

[TABLE]

and

[TABLE]

for some positive constant $C_{N,p}$ depending only on $N$ and $p$.

Recall that $Q_{N,p}$ is defined by (2.19).

Proof.

We have the maximal function estimates in the form

[TABLE]

for all $\sigma\in\mathbb{S}^{N-1}$ and $g\in L^{p}({\mathbb{R}}^{N})$, either complex or real valued. It is readily checked (repeat the proof of [16, Theorem 7.22] with straightforward adaptations) that $C^{\infty}_{c}({\mathbb{R}}^{N})$ is dense in $W^{1,p}_{A}({\mathbb{R}}^{N})$. Lemma 3.2 holds in the modified form

[TABLE]

for all $u\in W^{1,p}_{A}({\mathbb{R}}^{N})$ and $\delta>0$. To achieve this conclusion, it is sufficient to observe that, see (3.2),

[TABLE]

The rest of the proof follows verbatim. Lemma 3.3 holds in the form

[TABLE]

for every $u\in W^{1,p}_{A}({\mathbb{R}}^{N})$. In fact, mimicking the proof of Lemma 3.3, one obtains

[TABLE]

The final conclusion follows from (2.20). Lemma 3.4 can be modified accordingly with minor modifications, replacing $|\cdot|$ with $|\cdot|_{p}$. ∎

4. Convergence almost everywhere and convergence in $L^{1}$

Motivated by the work in [9] (see also [27]), we are interested in other modes of convergence in the context of Theorems 1.1 and 1.2. We only consider the case $p=2$. Similar results hold for $p\in(1,+\infty)$ with similar proofs. We begin with the corresponding results related to Theorem 1.1. For $u\in L^{1}_{{\rm loc}}({\mathbb{R}}^{N})$, set

[TABLE]

We have

Proposition 4.1**.**

Let $A:{\mathbb{R}}^{N}\to{\mathbb{R}}^{N}$ be Lipschitz, $u\in H^{1}_{A}({\mathbb{R}}^{N})$, and let $(\rho_{n})$ be a sequence of radial mollifiers such that

[TABLE]

We have

[TABLE]

and

[TABLE]

Before giving the proof of Proposition 4.1, we recall the following result established in [10, Lemma 1] (see also[9, Lemma 2] for a more general version).

Lemma 4.1**.**

Let $r>0$, $x\in{\mathbb{R}}^{N}$ and $f\in L^{1}_{{\rm loc}}({\mathbb{R}}^{N})$. We have

[TABLE]

Here and in what follows, for $x\in{\mathbb{R}}^{N}$ and $r>0$, let $B_{x}(r)$ denote the open ball in ${\mathbb{R}}^{N}$ centered at $x$ and of radius $r$. Moreover, $M(f)$ denotes the maximal function of $f$,

[TABLE]

As a consequence of Lemma 4.1, we have

Corollary 4.1**.**

Let $f\in L^{1}_{{\rm loc}}({\mathbb{R}}^{N})$ and $\rho$ be a nonnegative radial function such that

[TABLE]

Then, for a.e. $x\in{\mathbb{R}}^{N}$,

[TABLE]

Proof.

Using polar coordinates, we have

[TABLE]

Applying Lemma 4.1, we obtain, for a.e. $x\in{\mathbb{R}}^{N}$

[TABLE]

It follows from (4.1) that, for a.e. $x\in{\mathbb{R}}^{N}$,

[TABLE]

which is the conclusion. ∎

We are ready to give the proof of Proposition 4.1.

Proof of Proposition 4.1.

We first establish that, for a.e. $x\in{\mathbb{R}}^{N}$,

[TABLE]

where

[TABLE]

Here and in what follows in this proof, $C$ denotes a positive constant independent of $x$. Indeed, we have, as in (2), for a.e. $x,y\in{\mathbb{R}}^{N}$ with $|y-x|<1$,

[TABLE]

This implies, for a.e. $x\in{\mathbb{R}}^{N}$,

[TABLE]

Applying Corollary 4.1, we have, for a.e. $x\in{\mathbb{R}}^{N}$,

[TABLE]

On the other hand, we get

[TABLE]

A combination of (4.4) and (4.5) yields (4.2). Set, for $v\in H^{1}_{A}({\mathbb{R}}^{N})$ and $\varepsilon\geq 0$,

[TABLE]

By (2.7), one has, for $v\in C^{2}_{c}({\mathbb{R}}^{N})$ and $\varepsilon\geq 0$,

[TABLE]

Using the theory of maximal functions, see e.g., [29, Theorem 1 on page 5], we derive from (4.2) that, for any $\varepsilon>0$ and for any $w\in H^{1}_{A}({\mathbb{R}}^{N})$ with $m\int_{{\mathbb{R}}^{N}}|w(y)|^{2}\,dy\leq\varepsilon/2$,

[TABLE]

Fix $\varepsilon>0$ and let $v\in C^{2}_{c}({\mathbb{R}}^{N})$ with $\max\{1,m\}\|v-u\|_{H^{1}_{A}({\mathbb{R}}^{N})}\leq\varepsilon/2$. We derive from (4.6) that

[TABLE]

Since $\varepsilon>0$ is arbitrary, one reaches the conclusion that $|\Omega_{0}(u)|=0$. The proof is complete. ∎

We next discuss the corresponding results related to Theorem 1.2. Given $u\in L^{1}_{{\rm loc}}({\mathbb{R}}^{N})$, set, for $x\in{\mathbb{R}}^{N}$,

[TABLE]

We have

Proposition 4.2**.**

Let $A:{\mathbb{R}}^{N}\to{\mathbb{R}}^{N}$ be Lipschitz and let $u\in H^{1}_{A}({\mathbb{R}}^{N})$. We have

[TABLE]

and

[TABLE]

Proof.

For $v\in H^{1}_{A}({\mathbb{R}}^{N})$, set

[TABLE]

and denote

[TABLE]

We first establish a variant of (4.7) and (4.8) in which $J_{\delta}$ is replaced by $\hat{J}_{\delta}$. Using (3.2), as in the proof of Lemma 3.2, we have, for any $v\in H^{1}_{A}({\mathbb{R}}^{N})$,

[TABLE]

We derive that, for $u,u_{n}\in H^{1}_{A}({\mathbb{R}}^{N})$, and $\varepsilon\in(0,1)$,

[TABLE]

and

[TABLE]

On the other hand, one can check that, as in the proof of Lemma 3.3, for $u_{n}\in C^{2}_{c}({\mathbb{R}}^{N})$,

[TABLE]

We derive from (4.9), (4.10), and (4.11) that, for $u\in H^{1}_{A}({\mathbb{R}}^{N})$,

[TABLE]

and, we hence obtain, by the Dominate convergence theorem,

[TABLE]

since $\mathscr{M}(u,x)\in L^{1}({\mathbb{R}}^{N})$. A straightforward computation yields

[TABLE]

It follows that

[TABLE]

We also have, for $w\in C^{2}_{c}({\mathbb{R}}^{N})$,

[TABLE]

where $R>0$ is such that $\operatorname{supp}w\subset B_{R}$. Using Lemma 3.2 and the density of $C^{2}_{c}({\mathbb{R}}^{N})$ in $H^{1}_{A}({\mathbb{R}}^{N})$, we derive that,

[TABLE]

The conclusion now follows from (4.12), (4.13), (4.14) and (4.15). ∎