NEW CHARACTERIZATIONS OF MAGNETIC SOBOLEV SPACES

Transcription

1 NEW CHARACTERIZATIONS OF MAGNETIC SOBOLEV SPACES HOAI-MINH NGUYEN, ANDREA PINAMONTI, MARCO SQUASSINA, AND EUGENIO VECCHI 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, 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 the convergence in L 1 appearing naturally in these contexts. 1. Introduction In electromagnetism, a relevant role in the study of particles which interact with a magnetic field B = A, A : R 3 R 3, is played by the magnetic Laplacian ( ia) [, 16, 6]. This yields to nonlinear Schrödinger equations of the type ( ia) u + u = f(u), which have been extensively studied (see e.g. [1, 13, 15, 17] the references therein). The linear operator ( ia) u is defined weakly as the differential of the energy functional HA(R 1 N ) u u ia(x)u dx, over complex-valued functions u on. Here i denotes the imaginary unit the stard Euclidean norm of C N. Given a measurable function A : given an open subset Ω of, one defines HA 1 (Ω) as the space of complex-valued functions u L (Ω) such that u H 1 A (Ω) < for the norm u H 1 A (Ω) := ( u L (Ω) + [u] H 1 A (Ω) ) 1/, ( 1/. [u]h 1 A (Ω) := u ia(x)u dx) Ω In [14], some physically motivated nonlocal versions of the local magnetic energy were introduced. In particular the operator ( ) s A is defined as the gradient of the nonlocal energy functional HA(R s N u(x) e i(x y) A( x+y ) u(y) ) u (1 s) x y N+s dx dy, where s (, 1). Recently, the existence of ground stated of ( ) s Au + u = f(u) was investigated in [1] via Lions concentration compactness arguments. In [8] a connection between the local nonlocal notions was obtained on bounded domains, precisely, if Ω is a bounded Lipschitz domain A C ( ), then for every u HA 1 (Ω) it holds x+y u(x) e i(x y) A( ) u(y) (1.1) (1 s) s 1 x y N+s dx dy = Q N u ia(x)u dx, Ω Ω 1 Mathematics Subject Classification. 49A5, 6A33, 8D99. Key words phrases. Magnetic Sobolev spaces, new characterization, nonlocal functionals. A.P., M.S. 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 FP7/7-13/ under REA grant agreement No (ERC Grant MaNET Metric Analysis for Emergent Technologies ). 1 Ω

2 H.-M. NGUYEN, A. PINAMONTI, M. SQUASSINA, AND E. VECCHI where (1.) Q N := 1 S N 1 ω σ dσ being S N 1 the unit sphere in ω an arbitrary unit vector of. See also [3] for the general case of the p-norm with 1 p < + as well as [4] where the it as s is covered. This provides a new characterization of the HA 1 norm in terms of nonlocal functionals extending the results by Bourgain, Brezis Mironescu [3, 4] (see also [11, 5]) to the magnetic setting. Let s n } n N be a sequence of positive numbers converging to 1 less than 1 set (1 sn )diam(ω) sn r sn N for < r diam(ω), ρ n (r) := for r > diam(ω), where diam(ω) denotes the diameter of Ω. We have ρ n (r)r N 1 dr = 1, for all δ >, δ ρ n (r)r N 1 dr =. Given u : Ω C a measurable complex-valued function, we denote x+y Ψ u (x, y) := e i(x y) A( ) u(y), x, y Ω. The function Ψ u (, ) also depends on A but for notational ease, we ignore it. Assertion (1.1) can be then written as (1.3) = Q N u ia(x)u dx. Ω Ω This paper is concerned with the whole space setting. Our first goal is to obtain formula (1.3) for Ω = to provide a characterization of H 1 A (RN ) in terms of the LHS of (1.3) in the spirit of the work of Bourgain, Brezis Mironescu. Here in what follows, a sequence of nonnegative radial functions ρ n } n N is called a sequence of mollifiers if it satisfies the conditions (1.4) In this direction, we have ρ n (r)r N 1 dr = 1 δ ρ n (r)r N 1 dr =, for all δ >. Theorem 1.1. Let A : be Lipschitz let ρ n } n N be a sequence of nonnegative radial mollifiers. Then u HA 1 (RN ) if only if u L ( ) RN (1.5) sup n N < +. Moreover, for u H 1 A (RN ), we have RN (1.6) = Q N (1.7) Ω u ia(x)u dx, RN S N 1 u ia(x)u dx + S N 1 ( + A L (R )) u dx. N

3 NEW CHARACTERIZATIONS OF MAGNETIC SOBOLEV SPACES 3 In this paper, S N 1 denotes the (N 1)-Hausdorff measure of the unit sphere S N 1 in. The proof of Theorem 1.1 is given in Section. Remark 1.1. Similar results as in Theorem 1.1 hold for more general mollifiers ρ n } n N with slight changes in the constants. See Remark.1 for details. The second goal of this paper is to characterize HA 1 (RN ) in term of J δ ( ) where, for δ >, J δ (u) := x y N+ dxdy, for u L1 loc (RN ). Ψ u(x,y) Ψ u(x,x) >δ} This is motivated by the characterization of the Sobolev space H 1 ( ) provided in [5, 18] (see also [6 1,19 ]) in terms of the family of nonlocal functionals I δ which is defined by, for δ >, I δ (u) := x y N+ dx dy, for u L1 loc (RN ). u(y) u(x) >δ} It was showed in [5, 18] that if u L ( ), then u H 1 ( ) if only if sup <δ<1 I δ (u) < ; moreover, I δ(u) = Q N u dx, for u H 1 ( ). δ Ω Concerning this direction, we establish Theorem 1.. Let A : be Lipschitz. Then u H 1 A (RN ) if only if u L ( ) (1.8) sup J δ (u) < +. <δ<1 Moreover, we have, for u HA 1 (RN ), J δ(u) = Q N u ia(x)u dx δ ( (1.9) sup J δ (u) C N u ia(x)u dx + ( ) A L ( ) + 1) u dx. δ> Throughout the paper, we shall denote by C N a generic positive constant depending only on N possibly changing from line to line. The proof of Theorem 1. is given in Section 3. As pointed out in [13], a physically meaning example of magnetic potential in the space is A(x, y, z) = 1 ( y, x, ), (x, y, z) R3, which in fact fulfills the requirement of Theorems that A is Lipschitz. Furthermore, in the spirit of [6], as a byproduct of Theorems , for u L ( ), if we have or RN = J δ(u) =, δ

4 4 H.-M. NGUYEN, A. PINAMONTI, M. SQUASSINA, AND E. VECCHI then Ru = AIu, Iu = ARu, namely the direction of Ru, Iu is that of the magnetic potential A. In the particular case A =, this implies that u is a constant function. The L p versions of the above mentioned results are given in Sections 3. In addition to these results, we also discuss the convergence almost everywhere the convergence in L 1 of the quantities appearing in Theorems in Section 4. The paper is organized as follows. The proof of Theorems are given in Sections 3 respectively. The convergence almost everywhere the convergence in L 1 are investigated in Section 4.. Proof of Theorem 1.1 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.1 (Upper bound). Let A : be Lipschitz let ρ n } n N be a sequence of nonnegative radial mollifiers. We have, for all u H 1 A (RN ), RN S N 1 u ia(x)u dx + S N 1 ( + A L (R )) u dx. N Proof. Since Cc ( ) is dense in HA 1 (RN ) (cf. [16, Theorem 7.]), using Fatou s lemma, without loss of generality, one might assume that u Cc 1 ( ). Recall that (.1) ρ n ( z )dz = S N 1 ρ n (r)r N 1 dr = S N 1. Since x y 1} ( u(y) + u(x) ) ρ n ( x y ) dx dy 4 S N 1 u dx, it suffices to prove that (.) For a.e. x, y, we have Ψ u (x, y) y ( ) S N 1 u ia(x)u dx + A L ( ) u dx. ( x+y = e i(x y) A( ) x + y u(y) i A x+y e i(x y) A( ) u(y). ) + 1 ( x + y ) } (y x) A

5 NEW CHARACTERIZATIONS OF MAGNETIC SOBOLEV SPACES 5 It follows that (.3) Ψ u(x, y) ( x + y ) u(y) ia(y)u(y) + A A(y) u(y) + 1 ( A x + y ) u(y). y y x This implies Ψ u(x, y) u(y) ia(y)u(y) + A y L ( ) x y u(y), which yields, for x, y with x y < 1, (.4) x y 1 ( ) ( ) ( ) u ty + (1 t)x ia ty + (1 t)x u ty + (1 t)x dt + A L ( ) 1 Since, for f L ( ), in light of (1.4) (.1), we get 1 f ( ty + (1 t)x ) ρ n ( x y ) dt dx dy we then derive from (.4) that which is (.). ( ) u ty + (1 t)x dt. = f(x) dx ρ n ( z ) dz = S N 1 f(x) dx, S N 1 u(y) ia(y)u(y) dy + S N 1 A L ( ) u(y) dy, We next establish the following result which is used in the proof of (1.6) in the proof of Theorem 1.. Lemma.. Let u C ( ), A : be Lipschitz, let ρ n } n N be a sequence of nonnegative radial mollifiers. Then RN (.5) inf Q N u ia(x)u dx. Moreover, for any (ε n ), there holds +εn (.6) inf ρ n ( x y ) dx dy Q N u ia(x)u dx. x y +εn Throughout this paper, for R >, let B R denote the open ball in centered at the origin of radius R. Proof. Fix R > 1 (arbitrary). Using the fact e it (1 + it) Ct, for t R,

6 6 H.-M. NGUYEN, A. PINAMONTI, M. SQUASSINA, AND E. VECCHI we have, for x, y B R, ( ) ( ( Ψ u (x, y) 1 + i(x y) A(y) u(y) x + y ) ) Ψ u(x, y) 1 + i(x y) A u(y) ( x + y ) + x y A A(y) u(y) C u C (B R )(1 + A W 1, (B R )) x y. Here in what follows, C denotes a positive constant. On the other h, we obtain, for x, y B R, u(x) u(y) u(y) (x y) C u C (B R ) x y. It follows that [ ] (.7) Ψ u (x, y) Ψ u (x, x) Since (.8) it follows from (.7) that ( ) u(y) ia(y)u(y) B R B R (y x) C u C (B R )( 1 + A W 1, (B R )) x y. =, inf B R B R We have, by the definition of Q N, ( u(y) ia(y)u(y) ) (x y) inf B R B R. ( u(y) ia(y)u(y) ) (x y) (.9) inf B R B R Q N B R 1 u(y) ia(y)u(y) dy. By the arbitrariness of R > 1 we get inf which implies (.5). Q N u ia(x)u dx,

7 NEW CHARACTERIZATIONS OF MAGNETIC SOBOLEV SPACES 7 Assertion (.6) can be derived as follows. We have, by Hölder s inequality, B R B R B R B R x y ρ n ( x y )dx dy +εn ρ n ( x y )dx dy x y +εn +εn B R B R ρ n ( x y ) dx dy εn +εn. Since, for every R >, there holds B R B R x y 1} ρ n ( x y ) dx dy εn +εn = 1, we get (.6) from (.9) the arbitrariness of R > 1. We are ready to prove (1.6). Lemma.3 (Limit formula). Let A : be Lipschitz let ρ n } n N be a sequence of nonnegative radial mollifiers. Then, for u HA 1 (RN ), RN = Q N u ia(x)u dx. Proof. By Lemma.1 the density of Cc ( ) in HA 1 (RN ), one might assume that u Cc ( ). From Lemma., it suffices to prove that, for u Cc ( ), RN (.1) sup Q N u ia(x)u dx. Fix R > 4 such that supp u B R/. Using (.7) (.8), one derives that sup B R B R which yields ( u(y) ia(y)u(y) ) (x y) sup B R B R, (.11) sup B R B R Q N u(y) ia(y)u(y) dy.

8 8 H.-M. NGUYEN, A. PINAMONTI, M. SQUASSINA, AND E. VECCHI On the other h, we have (.1) sup the fact that x y 1} sup x y 1} ( u(x) + u(y) ) ρ n ( x y ) dx dy =, (.13) if (x, y) B R B R x y < 1 then Ψ u (x, y) Ψ u (x, x) =, by the choice of R. Combining (.11), (.1), (.13) yields (.1). The following result is about uniform bounds for the integrals in (1.5). Lemma.4. Let A : be Lipschitz let ρ n } n N be a sequence of nonnegative radial mollifiers. Then u H 1 A (RN ) if u L ( ) RN (.14) sup n N < +. Proof. Let τ m } be a sequence of nonnegative mollifiers with supp τ m B 1 which is normalized by the condition τ m (x) dx = 1. Set We estimate We have e u m = u τ m. RN Ψ um (x, y) Ψ um (x, x). i(x y) A( x+y ) um (y) u m (x) = ( x+y R e i(x y) A( N ) u(y z) u(x z) )τ m (z) dz. By the change of variables y = y z x = x z using the inequality a+b ( a + b ) for all a, b C applying Jensen s inequality, we deduce that (.15) RN Ψ um (x, y) Ψ um (x, x) Since, for t R, RN x+y e i(x y) A( +z) x+y e i(x y) A( ) u(y) + x y τ m (z)ρ n ( x y ) dz dx dy. e it 1 C t,

9 NEW CHARACTERIZATIONS OF MAGNETIC SOBOLEV SPACES 9 it follows that, for all x, y, z, x+y e i(x y) A( +z) x+y e i(x y) A( ) e = i(x y) (A( x+y x+y +z) A( )) 1 C A L ( ) x y z C x y z. Here in what follows in this proof, C denotes some positive constant independent of m n. Taking into account the fact that supp τ m B 1, we obtain x+y e i(x y) A( +z) x+y e i(x y) A( ) u(y) (.16) x y τ m (z)ρ n ( x y ) dz dx dy Combining (.14), (.15), (.16) yields (.17) RN Ψ um (x, y) Ψ um (x, x) C u(y) τ m (z)ρ n ( x y ) dz dx dy C. C. On the other h, by Lemma. we have RN Ψ um (x, y) Ψ um (x, x) (.18) inf n x y ρ n ( x y ) dx dy Q N u m ia(x)u m dx. The conclusion now immediately follows from (.17) (.18) after letting m +. 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.1,.3.4. Remark.1. Let ρ n } n N be a sequence of non-negative radial functions such that 1 ρ n (r)r N 1 dr = 1, 1 δ 1 ρ n (r)r N 1 dr =, for every δ >, ρ n (r)r N 3 dr =. Theorem 1.1 then holds for such a sequence ρ n } n N provided that the constant 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 L ( ), sup x y 1} sup x y 1} For example, this applies to the radial sequence ( u(x) + u(y) ) ρ n ( x y ) x y dx dy =. ρ n (r) = (1 s n )r sn N, for r >, which provides a characterization of HA 1 (RN ) yields (1 s u(x) e i(x y) A( x+y ) u(y) n) dxdy = Q N u ia(x)u dx. x y N+sn

10 1 H.-M. NGUYEN, A. PINAMONTI, M. SQUASSINA, AND E. VECCHI Consider now the space (C n, p ) (n 1), endowed with the norm z p := ( (Rz 1,..., Rz n ) p + (Iz 1,..., Iz n ) p) 1/p, where is the Euclidean norm of R n Ra, Ia denote the real imaginary parts of a C respectively. We emphasize that this is not related to the p-norm in R n. In what follows, we use this notation with n = N n = 1. Notice that z p = z whenever z R n, which makes our next statements consistent with the case A = u being a real valued function. Also =, consistently with the previous definition. Define, for some ω S N 1, (.19) Q N,p := 1 ω σ p p dσ. p S N 1 We have, for z C N, (see [3, 3]), (.) z σ p p dσ = S N 1 Rz σ p dσ + S N 1 Iz σ p dσ = Rz p pq N,p + Iz p pq N,p = z p ppq N,p. S N 1 Using the same approach technique, one can prove the following L p version of Theorem 1.1. Theorem.1. Let p (1, + ), A : be Lipschitz, let ρ n } n N be a sequence of nonnegative radial mollifiers. Then u W 1,p A (RN ) if only if u L p ( ) p Ψ u (x, y) Ψ u (x, x) p x y p ρ n ( x y ) dx dy < +. sup n N Moreover, for u W 1,p A (RN ), we have (.1) p Ψ u (x, y) Ψ u (x, x) p x y p ρ n ( x y ) dx dy = pq N,p u ia(x)u p p dx p Ψ u (x, y) Ψ u (x, x) p x y p ρ n ( x y ) dx dy C N,p u ia(x)u p ( p dx + C N,p + A p L (R )) u p N p dx, for some positive constant C N,p depending only on N p. Remark.. Assume that C is a positive constant such that, for all a, b C, a + b p p C( a p p + b p p). Then assertion (.1) of Theorem.1 holds with C N,p = S N 1 C. Let us set, for σ S N 1, 3. Proof of Theorem 1. its L p version 1 M σ (g, x) := sup t> t t g(x + sσ) ds. denote M en by M N, e N := (,...,, 1). We have the following result which is a direct consequence of the theory of maximal functions, see e.g., [9, Theorem 1, page 5].

11 NEW CHARACTERIZATIONS OF MAGNETIC SOBOLEV SPACES 11 Lemma 3.1 (Maximal function estimate). There exists a universal constant C > such that, for all σ S N 1, M σ (g, x) dx C g dx, for all g L ( ). The following lemma yields an upper bound of J δ (u) in terms of the norm of u in H 1 A (RN ). Lemma 3. (Uniform upper bound). Let A : be Lipschitz u HA 1 (RN ). We have ( sup J δ (u) C N u ia(x)u dx + ( ) A L ( ) + 1) u dx. δ> Proof. By density of Cc ( ) in HA 1 (RN ), using Fatou s lemma, we can assume that u Cc 1 ( ). For each δ >, let us define } A δ := (x, y) : Ψ u (x, y) Ψ u (x, x) > δ, x y < 1 We have B δ := RN } (x, y) : Ψ u (x, y) Ψ u (x, x) > δ, x y 1. x y N+ 1 B δ dx dy Since Ψ u (x, y) Ψ u (x, x) u(x) + u(y) it follows that x y 1} RN x y N+ 1 x y 1} dx dy. u(x) x y N+ dx dy C N u(x) dx, x y N+ 1 B δ dx dy C N u(x) dx. We are therefore interested in estimating the integral δ A dx dy. x y N+ Let us now define X δ := } (x, h, σ) (, 1) S N 1 : Ψ u (x, x + hσ) Ψ u (x, x) > δ. Performing the change of variables y = x + hσ, for h (, 1) σ S N 1, yields δ A δ dx dy = x y N+ X δ dh dx dσ = h3 Cσ dh dx dσ, h3 where C σ denotes the set } C σ := (x, h) (, 1) : Ψ u (x, x + hσ) Ψ u (x, x) > δ, σ S N 1. Without loss of generality it suffices to prove that, for σ = e N = (,...,, 1) S N 1, δ (3.1) CeN ( ) h 3 dhdx C N u ia(x)u dx + A L ( ) u dx. We have, by virtue of (.3), (3.) Ψ(x, x + he N ) Ψ(x, x) hm N ( u iau, x) + h A L ( )M N ( u, x). S N 1

12 1 H.-M. NGUYEN, A. PINAMONTI, M. SQUASSINA, AND E. VECCHI Using the fact that if a + b > δ then either a > δ/ or b > δ/, we derive that δ CeN h 3 dhdx dh dx + hm N ( u iau,x)>δ/} h3 dh dx + h3 hm N ( u iau,x)>δ/} h A L ( ) M N ( u,x)>δ/} h A L ( ) M N ( u,x)>δ/} dh dx h3 dh dx, h3 where the last inequality follows recalling that since (x, h) C en then h (, 1). As usual, by using the theory of maximal functions stated in Lemma 3.1, we have (3.3) hm N ( u iau,x)>δ/} h 3 dhdx C N u ia(x)u dx (3.4) h A L ( ) M N ( u,x)>δ/} h 3 dhdx C N A L u dx. Assertion (3.1) follows from (3.3) (3.4). The proof is complete. We next establish Lemma 3.3 (Limit formula). Let A : be Lipschitz u HA 1 (RN ). Then J δ(u) = Q N u ia(x)u dx, δ where Q N is the constant defined in (1.). Proof. By virtue of Lemma 3., for every δ > all w HA 1 (RN ), we have ( (3.5) J δ (w) C N w ia(x)w dx + ( ) A L ( ) + 1) w dx. Since Ψ u (x, y) Ψ u (x, x) Ψ v (x, y) Ψ v (x, x) + Ψ u v (x, y) Ψ u v (x, x), it follows that, for every ε (, 1), J δ (u) Ψ v(x,y) Ψ v(x,x) >(1 ε)δ} dx dy x y N+ This implies, for ε (, 1) u, v H 1 A (RN ), (3.6) J δ (u) (1 ε) J (1 ε)δ (v) + ε J εδ (u v). + dx dy. Ψ u v (x,y) Ψ u v (x,x) >εδ} x y N+ From (3.5) (3.6), we derive that, for u, u n H 1 A (RN ) ε (, 1), (3.7) J δ (u) (1 ε) J (1 ε)δ (u n ) ( ε C N (u u n ) ia(x)(u u n ) dx + ( ) A L ( ) + 1) u u n dx

13 NEW CHARACTERIZATIONS OF MAGNETIC SOBOLEV SPACES 13 (3.8) (1 ε) J δ/(1 ε) (u n ) J δ (u) ( ε C N (u u n ) ia(x)(u u n ) dx + ( ) A L ( ) + 1) u u n dx. Since Cc 1 ( ) is dense in HA 1 (RN ), from (3.7) (3.8), it suffices to prove the assertion for u Cc 1 ( ). This fact is assumed from now on. Let R > be such that supp u B R/. We claim that, for every σ S N 1, there holds (3.9) 1 δ (x,h) B R (, ): h>1} h 3 dhdx = 1 ( u iau) σ dx. Ψu(x,x+δhσ) Ψu(x,x) δh Without loss of generality, we can assume σ = e N S N 1. Then, we aim to prove that 1 } δ (x,h) B R (, ): h>1 h 3 dhdx = 1 u (x) ia N (x)u(x) y N Ψ u(x,x+δhe N ) Ψ u(x,x) δh where A N denotes the N-th component of A. To this end, we consider the sets C en (x, δ) := (x N, h) R (, ) : Ψ u (x, x + δhe N ) Ψ u (x, x) δh }, h > 1 E(x ) := (x N, h) R (, ) : Ψ u (x, x) y N }, h > 1 } F(x ) := (x N, h) R (, ) : hm N ( u iau, x) + h A L ( )M N ( u, x) > 1. Therefore, we obtain χ CeN (x,δ)(x N, h) χ F(x )(x N, h) for a.e. (x, h) B R (, ) (by (3.) in the proof of Lemma 3.) 1 h 3 χ F(x )(x N, h) dh dx I 1 + I, where we have set I 1 := I := B R (x,h) B R (, ): M N ( u iau,x)h>1/ } 1 dh dx, h3 } 1 dh dx, (x,h) B R (, ): h A L ( ) M N ( u,x)>1/ h3 we have denoted χ the characteristic function. We have, by the theory of maximal functions, I 1 C u ia(x)u dx,, by a straightforward computation, I C A L ( ) u L ( ) B R. The validity of Claim (3.9) with σ = e N now follows from Dominated Convergence theorem since δ χ C en (x,δ)(x N, h) = χ E(x )(x N, h),, by a direct computation, χ E(x )(x N, h) 1 h 3 dhdx = 1 B R B R for a.e. (x, h) B R (, ), u (x) ia N (x)u(x) y N dx. dx,

14 14 H.-M. NGUYEN, A. PINAMONTI, M. SQUASSINA, AND E. VECCHI Now, performing a change of variables we get dx dy = x y N+ where Ψ u(x,y) Ψ u(x,x) >δ, x B R } C σ (δ) := Exploiting (3.9), we obtain (3.1) δ (x, h) B R (, ) : Ψ u(x,y) Ψ u(x,x) >δ, x B R } On the other h, since supp u B R/, we have (3.11) dx dy δ Ψ u(x,y) Ψ u(x,x) >δ, x \B R } x y N+ = δ Combining (3.1) (3.11) yields δ Ψ u(x,y) Ψ u(x,x) >δ} B R S N 1 Ψ u (x, x + δhσ) Ψ u (x, x) δh χ Cσ(δ)(x, h) 1 dh dσ dx, h3 }. h > 1 x y N+ dx dy = 1 ( u iau) σ dx dσ. S N 1 B R x y N+ dx dy = 1 In order to conclude, we notice the following, see (.), x \B R, y B R/ } S N 1 S N 1 V σ dσ = Q N V, for any V C N, where Q N is the constant defined in (1.). We next deal with (1.8). dx dy =. x y N+ ( u iau) σ dx dσ. Lemma 3.4. Let u L ( ) let A : be Lipschitz. Then u H 1 A (RN ) if (3.1) sup J δ (u) < +. δ (,1) Proof. The proof is divided into two steps. Step 1. We assume that u L ( ) L ( ). Set In light of (3.1), we obtain L := sup Ψ u (x, y) Ψ u (x, x). x,y L εδ ε 1 J δ (u) dδ C, for some positive constant C independent of ε (, 1). By Fubini s theorem by the definition of L, we have L εδ ε 1 1 Ψu(x,y) Ψu(x,x) J δ (u) dδ = x y N+ εδ ε+1 dδ dx dy. It follows that 1 RN +ε ε + ε x y +ε dx dy C. x y N ε

15 NEW CHARACTERIZATIONS OF MAGNETIC SOBOLEV SPACES 15 By virtue of inequality (.6) of Lemma.4, we have inf ε RN +ε x y +ε ε x y N ε dx dy Q N which implies u H 1 A (RN ). u ia(x)u dx, Step. We consider the general case. For M > 1, define T M : C C by setting z if z M, T M (z) := M z/ z otherwise, denote Then, we have It follows that Hence we obtain u M := T M (u). T M (z 1 ) T M (z ) z 1 z, for all z 1, z C. Ψ um (x, y) Ψ um (x, x) Ψ u (x, y) Ψ u (x, x), for all x, y. (3.13) J δ (u M ) J δ (u). Applying the result in Step 1, we have u M HA 1 (RN ) hence by Lemma 3.3, (3.14) J δ (u M ) = Q N u M (x) ia(x)u M (x) dx. δ Combining (3.13) (3.14) letting M +, we derive that u HA 1 (RN ). The proof is complete. Remark 3.1. Similar approach used for H 1 ( ) is given in [18]. Proof of Theorem 1.. The it formula stated in Theorem 1. follows by Lemma 3.3. Now, if u HA 1 (RN ), then (1.9) follows from Lemma 3.. On the contrary, if u L ( ) (1.8) holds, it follows from Lemma 3.4 that u HA 1 (RN ). Given u a measurable complex-valued function, define, for 1 < p < +, δ p J δ,p (u) := dxdy, for δ >. x y N+p Ψ u(x,y) Ψ u(x,x) p>δ} We have the following L p -version of Theorem 1.. Theorem 3.1. Let p (1, + ) let A : be Lipschitz. Then u W 1,p A (RN ) if only if u L p ( ) sup J δ,p (u) <. <δ<1 Moreover, we have, for u W 1,p A (RN ), J δ,p(u) = Q N,p u ia(x)u p p dx δ ( J δ,p (u) C N,p u ia(x)u p p dx + ( ) A p L ( ) + 1) u p p dx, for some positive constant C N,p depending only on N p. Recall that Q N,p is defined by (.19).

16 16 H.-M. NGUYEN, A. PINAMONTI, M. SQUASSINA, AND E. VECCHI Proof. We have the maximal function estimates in the form M σ (g, x) p pdx C p g p pdx, for all g L p ( ). for all σ S N 1 g L p ( ), either complex or real valued. It is readily checked (repeat the proof of [16, Theorem 7.] with straightforward adaptations) that Cc ( ) is dense in W 1,p A (RN ). Lemma 3. holds in the modified form ( J δ,p (u) C N,p u ia(x)u p p dx + ( ) A p L ( ) + 1) u p p dx, for all u W 1,p A (RN ) δ >. To achieve this conclusion, it is sufficient to observe that, see (3.), Ψ(x, x + he N ) Ψ(x, x) p hm N ( u iau p, x) + h A L ( )M N ( u p, x). The rest of the proof follows verbatim. Lemma 3.3 holds in the form J δ,p(u) = Q N,p u ia(x)u p p dx, δ for every u W 1,p A (RN ). In fact, mimicking the proof of Lemma 3.3, one obtains δ p δ Ψ u(x,y) Ψ u(x,x) p>δ} x y N+p dx dy = 1 ( u iau) σ p dx dσ. p S N 1 The final conclusion follows from (.). Lemma 3.4 can be modified accordingly with minor modifications, replacing with p. 4. Convergence almost everywhere convergence in L 1 Motivated by the work in [9] (see also [7]), we are interested in other modes of convergence in the context of Theorems We only consider the case p =. Similar results hold for p (1, + ) with similar proofs. We begin with the corresponding results related to Theorem 1.1. For u L 1 loc (RN ), set We have D n (u, x) := RN sup t>1 x y ρ n ( x y ) dy, for x. Proposition 4.1. Let A : be Lipschitz, u HA 1 (RN ), let (ρ n ) be a sequence of radial mollifiers such that sup t ρ n (t) < +. n We have D n(u, x) = Q N u(x) ia(x)u(x), for a.e. x, D n(u, ) = Q N u( ) ia( )u( ), in L 1 ( ). Before giving the proof of Proposition 4.1, we recall the following result established in [1, Lemma 1] (see also [9, Lemma ] for a more general version). Lemma 4.1. Let r >, x f L 1 loc (RN ). We have r f(x + sσ) ds dσ C N rm(f)(x). S N 1

17 NEW CHARACTERIZATIONS OF MAGNETIC SOBOLEV SPACES 17 Here in what follows, for x r >, let B x (r) denote the open ball in centered at x of radius r. Moreover, M(f) denotes the maximal function of f, 1 M(f)(x) := sup f(y) dy, x. r> B x (r) As a consequence of Lemma 4.1, we have B x(r) Corollary 4.1. Let f L 1 loc (RN ) ρ be a nonnegative radial function such that (4.1) Then, for a.e. x, B x(r) 1 Proof. Using polar coordinates, we have 1 f ( t(y x) + x ) ρ( y x ) dt dy = B x(r) ρ(r)r N 1 dr = 1. f ( t(y x) + x ) ρ( y x ) dt dy C N M(f)(x). r S N 1 1 f ( x + tsσ ) s N 1 ρ(s) dt dσ ds. Applying Lemma 4.1, we obtain, for a.e. x 1 S N 1 f ( x + tsσ ) dt dσ C N M(f)(x). It follows from (4.1) that, for a.e. x, 1 f ( t(y x) + x ) ρ( y x ) dt dy C N M(f)(x), B x(r) 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, ( ) (4.) D n (u, x) C M( u iau )(x) + M( u )(x) + m where m := sup t>1 sup t ρ n (t). n \B x(1) u(y) dy, Here in what follows in this proof, C denotes a positive constant independent of x. Indeed, we have, as in (.4), for a.e. x, y with y x < 1, (4.3) x y 1 u ( t(y x) + x ) ia ( t(y x) + x ) u ( t(y x) + x ) dt + A L ( ) 1 u ( t(y x) + x ) dt.

18 18 H.-M. NGUYEN, A. PINAMONTI, M. SQUASSINA, AND E. VECCHI This implies, for a.e. x, x y ρ n ( y x ) dy B x(1) B x(1) 1 + A L ( ) u ( t(y x) + x ) ia ( t(y x) + x ) u ( t(y x) + x ) ρ n ( y x ) dt dy B x(1) 1 u ( t(y x) + x ) ρ n ( y x ) dt dy. Applying Corollary 4.1, we have, for a.e. x, (4.4) x y ρ n ( y x ) dy CM( u iau )(x) + CM( u )(x). B x(1) On the other h, we get (4.5) x y ρ n ( y x ) dy \B x(1) u(x) + \B x(1) u(y) ρ n ( y x ) x y dy u(x) + m u(y) dy. \B x(1) A combination of (4.4) (4.5) yields (4.). Set, for v HA 1 (RN ) ε, Ω ε (v) := x : sup D n (v, x) Q N v(x) ia(x)v(x) > ε }. By (.7), one has, for v C c ( ) ε, Ω ε (v) =. Using the theory of maximal functions, see e.g., [9, Theorem 1 on page 5], we derive from (4.) that, for any ε > for any w HA 1 (RN ) with m R w(y) dy ε/, N (4.6) Ω ε (w) C ( ) w(x) ia(x)w(x) + w(x) dx. ε Fix ε > let v Cc ( ) with max1, m} v u H 1 A ( ) ε/. We derive from (4.6) that Ω ε (u) Ω ε (u v) C ε v u H 1 A (RN ) Cε. Since ε > is arbitrary, one reaches the conclusion that Ω (u) =. The proof is complete. We next discuss the corresponding results related to Theorem 1.. Given u L 1 loc (RN ), set, for x, J δ (u, x) = dy. x y N+ We have Ψ u(x,y) Ψ u(x,x) >δ}

19 NEW CHARACTERIZATIONS OF MAGNETIC SOBOLEV SPACES 19 Proposition 4.. Let A : be Lipschitz let u H 1 A (RN ). We have (4.7) δ J δ (u, x) = Q N u(x) ia(x)u(x), for a.e. x (4.8) δ J δ (u, ) = Q N u( ) ia( )u( ), in L 1 ( ). Proof. For v HA 1 (RN ), set ( M (v, x) = M σ ( v iav, x) + A L ( ) M σ( v, x) ) dσ, for x, denote S N 1 Ĵ δ (u, x) = Ψ u(x,y) Ψ u(x,x) >δ, y x <1} x y N+ dy, for x RN. We first establish a variant of (4.7) (4.8) in which J δ is replaced by Ĵδ. Using (3.), as in the proof of Lemma 3., we have, for any v H 1 A (RN ), Ĵ δ (v, x) C N M (v, x) for all δ >. We derive that, for u, u n H 1 A (RN ), ε (, 1), (4.9) Ĵ δ (u, x) (1 ε) Ĵ (1 ε)δ (u n, x) ε C N M (u u n, x), (4.1) (1 ε) Ĵ δ/(1 ε) (u n, x) Ĵδ(u, x) ε C N M (u u n, x). On the other h, one can check that, as in the proof of Lemma 3.3, for u n C c ( ), (4.11) δ Ĵ δ (u n, x) = Q N u n (x) ia(x)u n (x), for x. We derive from (4.9), (4.1), (4.11) that, for u H 1 A (RN ), (4.1) δ Ĵ δ (u, x) = Q N u(x) ia(x)u(x), for a.e. x., we hence obtain, by the Dominate convergence theorem, (4.13) δ Ĵ δ (u, ) = Q N u( ) ia( )u( ), in L 1 ( ), since M (u, x) L 1 ( ). A straightforward computation yields dy =. δ x y N+ It follows that y x 1} (4.14) δ [Ĵδ(u, x) J δ (u, x)] =, for a.e. x. We also have, for w Cc ( ), δ δ Ψ w(x,y) Ψ w(x,x) >δ, y x 1} (B R ) ( B R ), y x 1} dx dy x y N+ dx dy =, x y N+

20 H.-M. NGUYEN, A. PINAMONTI, M. SQUASSINA, AND E. VECCHI where R > is such that supp w B R. Using Lemma 3. the density of C c ( ) in H 1 A (RN ), we derive that, (4.15) δ [Ĵδ(u, ) J δ (u, )] = in L 1 ( ). The conclusion now follows from (4.1), (4.13), (4.14) (4.15). References [1] G. Arioli, A. Szulkin, A semilinear Schrödinger equation in the presence of a magnetic field, Arch. Ration. Mech. Anal. 17 (3), [] J. Avron, I. Herbst, B. Simon, Schrödinger operators with magnetic fields. I. General interactions, Duke Math. J. 45 (1978), [3] J. Bourgain, H. Brezis, P. Mironescu, Another look at Sobolev spaces, in Optimal Control Partial Differential Equations. A Volume in Honor of Professor Alain Bensoussan s 6th Birthday (eds. J. L. Menaldi, E. Rofman A. Sulem), IOS Press, Amsterdam, 1, , 1 [4] J. Bourgain, H. Brezis, P. Mironescu, Limiting embedding theorems for W s,p when s 1 applications, J. Anal. Math. 87 (), [5] J. Bourgain, H.-M. Nguyen, A new characterization of Sobolev spaces, C. R. Acad. Sci. Paris 343 (6), [6] H. Brezis, How to recognize constant functions. Connections with Sobolev spaces, Russian Mathematical Surveys 57 (), [7] H. Brezis, New approximations of the total variation filters in imaging, Rend Accad. Lincei 6 (15), [8] H. Brezis, H.-M. Nguyen, Non-local functionals related to the total variation connections with image processing, preprint. 3 [9] H. Brezis, H.-M. Nguyen, The BBM formula revisited, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 7 (16) , 16 [1] H. Brezis, H.-M. Nguyen, Two subtle convex nonlocal approximations of the BV-norm, Nonlinear Anal. 137 (16), 45. 3, 16 [11] J. Davila, On an open question about functions of bounded variation, Calc. Var. Partial Differential Equations 15 (), [1] P. d Avenia, M. Squassina, Ground states for fractional magnetic operators, ESAIM COCV, to appear doi. org/1.151/cocv/ [13] M. Esteban, P.-L. Lions, Stationary solutions of nonlinear Schrödinger equations with an external magnetic field. Partial differential equations the calculus of variations, Vol. I, , Progr. Nonlinear Differential Equations Appl., 1, Birkhäuser Boston, Boston, MA, , 3 [14] T. Ichinose, Magnetic relativistic Schrödinger operators imaginary-time path integrals, Mathematical physics, spectral theory stochastic analysis, 47 97, Oper. Theory Adv. Appl. 3, Birkhäuser/Springer, Basel, [15] L.D. Lau, E.M. Lifshitz, Quantum mechanics. Pergamon Press, (1977). 1 [16] E. Lieb, M. Loss, Analysis, Graduate studies in Mathematics 14, 1. 1, 4, 16 [17] D.L. Mills, Nonlinear optics, Springer-Verlag, (1998). 1 [18] H.-M. Nguyen, Some new characterizations of Sobolev spaces, J. Funct. Anal. 37 (6), , 15 [19] H.-M. Nguyen, Further characterizations of Sobolev spaces, J. Eur. Math. Soc. 1 (8), [] H.-M. Nguyen, Γ-convergence, Sobolev norms, BV functions, Duke Math. J. 157 (11), [1] H.-M. Nguyen, Some inequalities related to Sobolev norms, Calc. Var. Partial Differential Equations 41 (11), [] H.-M. Nguyen, Estimates for the topological degree related topics, J. Fixed Point Theory 15 (14), [3] A. Pinamonti, M. Squassina, E. Vecchi, Magnetic BV functions the Bourgain-Brezis-Mironescu formula, preprint, 1 [4] A. Pinamonti, M. Squassina, E. Vecchi, The Maz ya-shaposhnikova it in the magnetic setting, J. Math. Anal. Appl. 449 (17), [5] A. Ponce, A new approach to Sobolev spaces connections to Γ-convergence, Calc. Var. Partial Differential Equations 19 (4), 9 55.

21 NEW CHARACTERIZATIONS OF MAGNETIC SOBOLEV SPACES 1 [6] M. Reed, B. Simon, Methods of modern mathematical physics, I, Functional analysis, Academic Press, Inc., New York, [7] A. Ponce, D. Spector, On formulae decoupling the total variation of BV functions, Nonlinear Anal. 154 (17), [8] M. Squassina, B. Volzone, Bourgain-Brezis-Mironescu formula for magnetic operators, C. R. Math. Acad. Sci. Paris 354 (16), [9] E. M. Stein, Singular integrals differentiability properties of functions. Princeton University Press, Princeton, N.J., , 18 (H.-M. Nguyen) Department of Mathematics EPFL SB CAMA Station 8 CH-115 Lausanne, Switzerl address: hoai-minh.nguyen@epfl.ch (A. Pinamonti) Dipartimento di Matematica Università di Trento Via Sommarive 14, 385 Povo (Trento), Italy address: rea.pinamonti@unitn.it (M. Squassina) Dipartimento di Matematica e Fisica Università Cattolica del Sacro Cuore Via dei Musei 41, I-511 Brescia, Italy address: marco.squassina@unicatt.it (E. Vecchi) Dipartimento di Matematica Università di Bologna Piazza di Porta S. Donato 5, 416, Bologna, Italy address: eugenio.vecchi@unibo.it