is devoted to a collection of results on nonlinear interpolation inequalities ith Schrödinger operators involving Aharonov-Bohm magnetic potentials, consequences. As symmetry plays an important role for establishing optimale shall consider various cases corresponding to a circle, a two-dimensional two-dimensional torus, and also the Euclidean spaces of dimensions 2 and he results are new and we put the emphasis on the methods, as very liton symmetry, rigidity and optimality in the presence of a magnetic field. ectacular applications are new magnetic Hardy inequalities in dimensions 2 haronov-Bohm magnetic potential; radial symmetry; cylindrical symmetry; reaking; magnetic Hardy inequality; magnetic interpolation inequality; optits; magnetic Schrödinger operator; magnetic Keller-Lieb-Thirring inequalc rings.

ion oblems involving magnetic fields play a peculiar role in the calculus of s fair to say that there are no simple physical intuitions that may serve one has to extract information through exact computations, such as he Landau Hamiltonian. A case in point are the symmetry properties where in the absence of magnetic fields one has fairly robust methods The isoperimetric inequality is one of the main sources of intuition; s are essentially different versions of the isoperimetric problem. This t the case in the presence of magnetic fields and there are very few direction. le is the work of Avron, Herbst and Simon [2] who proved that the of the hydrogenic atom in a constant magnetic field has cylindri-The proof is quite involved. Another result that comes to mind is who proved the equivalent of the Faber-Krahn inequality for the erator with a constant magnetic field and with a Dirichlet boundary domain. The disk yields the smallest ground state energy among ual area. Again, the proof is quite involved and some arguments are inear setting. In this connection one should mention the recent result ys and van Schaftingen [8] who showed perturbatively that in some ational problem involving a small constant magnetic field the minithe symmetry of the problem. Besides the constant magnetic field class of physically relevant variational problems involve Aharonovic fields and the purpose of this paper is to give an up-to-date account ge. nov-Bohm effect states that the wave function of a charged quanassing by a thin magnetic solenoid experiences a phase shift. This, here is no apparent interaction with the solenoid except through the the particle with the 'unphysical' vector potential. This prediction inally by Ehrenberg and Siday in 1949 (see [18]) and then again in nov-Bohm (see [1]) and we will stay with the custom of calling it the hm effect. It cannot be explained in terms of classical mechanics, but ss experimentally verified (see [4]). It counts as one of the important anical effects. estion one may pose is the influence of the Aharonov-Bohm potenergies of systems, say, of a particle in a potential interacting with t is relatively straightforward to write the Hamiltonian for this situmay ask for the effect of the Aharonov-Bohm field on the spectrum onian. One fruitful approach is to relate the ground state energy of echanical particle in an external potential to the minimization of a l variational problem. This also works in the presence of magnetic hich are the dual versions of the Keller-Lieb-Thirring spectral estius symmetry settings, we are interested in getting as much insight out the best constants in the inequalities and also about the qualiies of their extremal functions. Indeed, in many cases studying the perties of those extremal functions allows us to get very accurate, even sharp, estimates for the best constants in the inequalities (see ]). clidean space R d , the magnetic Laplacian is defined via a magnetic

he case of dimensions d = 2 and d = 3. The magnetic field is B = adratic form associated with -Δ A is given by R d |∇ A ψ| 2 and well functions in the space

netic gradient takes the form

-Bohm magnetic field can be considered as a singular measure supset x 1 = x 2 = 0, where (x i ) d i=1 is a system of cartesian coordinates. potential is defined as follows.

s consider polar coordinates (r, θ) such that

real constant and {e r , e θ }, with e r = x r , denotes the orthogonal basis ith our polar coordinates. The magnetic gradient and the magnetic e explicitly given by

s consider cylindrical coordinates (ρ, θ, z) where

ic gradient and the magnetic Laplacian are explicitly given by

onsider Aharonov-Bohm type magnetic potentials on compact manion the circle, the sphere and the torus. The expression of the magnetic be given case by case. is intended to provide a general overview of the mathematical results concerning various functional inequalities involving Aharonov-Bohm s: ound state energy estimates. agnetic interpolation inequalities. ults for optimal functions. eller-Lieb-Thirring inequalities. ardy inequalities.

rdy inequalities, all of the above inequalities will be considered on on the two-dimensional sphere S 2 , on the two-dimensional flat torus on R 3 , with consequences on Hardy inequalities on the Euclidean R 3 . It is crucial to consider precise geometric settings as we are ptimal inequalities, which rely on non-trivial symmetry results. A ear interpolation inequality is

n u in the appropriate H 1 A space, for any λ > 0, and for any p > 2, mpact manifold X in order to fix ideas. Assuming that vol(X ) = 1, e optimal inequality is to the determine the largest value of λ > 0 for μ A (λ) = λ + C for some constant C which is computed in terms of ds on X . Equality is then realized by the constants. It is usually not ove that the equality is achieved in the inequality if μ A (λ) denotes nstant, for any λ > 0. If we consider the Euler-Lagrange equation, formulated as the slightly more general rigidity question. For which we know that any solution is actually a constant? To obtain rigidity, to establish symmetry properties, which is usually the most difficult oof. In the non-compact case, optimal functions are not constants, ditional difficulty, but the problem can also be reduced to a symmetry other interpretation of the rigidity issue in terms of a phase transition. t manifold case, the optimal function for (1.3) is always a constant if and one can prove in many cases that there is a bifurcation from a se (solutions are constant) to a non-symmetric phase for a threshold esponding to the optimal inequality. -Thirring (KLT) inequalities are estimates of the ground state ] for the magnetic Schrödinger operator -Δ A -V in terms of re obtained by duality from (1.3) with q = p/(p-2). KLT inequalities pletely equivalent to (1.3), including for optimality issues and related ons, and essential for proving various magnetic Hardy inequalities, of the highlights and the main motivations of this paper. However, the fact that the accurate spectral information is carried by the KLT tually consider not only the superquadratic case p > 2, but also the ase p < 2 in which the role of the L 2 and L p norms are exchanged. ding nonlinear interpolation inequality is

(1.4)

, 2) and we can also draw a whole series of consequences (rigidity, as in the superquadratic case. In particular, we are able to prove d interesting results on optimality and rigidity. collects many results on functional inequalities with magnetic fields metric settings. Therefore it is difficult to pick particularly significant ver we believe that the interest of the paper lies as much in the s as in the results because very little is known on optimal inequalities Aharonov-Bohm magnetic fields and on the symmetry properties onding optimal functions. The most visible outcome of our work nequalities, which are important tools in functional analysis. The agnetic field is a key feature, for instance in dimension d = 2. Let us he attention of the reader to some results that are prominent in this deals with nonlinear magnetic interpolation inequalities, optimal d rigidity results on S 1 in the subquadratic case. This is a new alities which complements the theory on magnetic rings in the tic case studied in [14]. It was natural to study it in view of the carré chnique by Bakry and Emery in [3], but as far as we know, it is an result in the presence of a magnetic potential when p < 2. is the counterpart of Theorem 3.1 in the case of the torus T 2 ≈ s remarkable that we achieve an optimality result here as symmetry oducts of manifolds are known to be difficult. uadratic interpolation inequalities for proving KLT and then to consider also what happens on S 2 : see Proposition 2.2 and Corolresults in the superquadratic case. magnetic Hardy inequalities of Theorem 4.2 are a new and striking of the nonlinear Hardy-Sobolev interpolation inequalities of [7] on an space R 2 . .1 and 5.2 are two examples of application of the nonlinear magnetic n inequalities to magnetic Hardy inequalities on R 3 , which signifiove upon the results in [19,22]. clude this introduction by some mathematical observations and some erences. The overall question is to determine the functional spaces pted to magnetic Schrödinger operators in the spirit of [13]. Magnetic inequalities (without optimal constants) are usually not an issue as educed from the non-magnetic interpolation inequalities by the diauality: see for instance [27]. However we are interested in retaining out the magnetic field and characterizing optimality cases, which is difficult target. As a convention, we shall speak of Hardy-Sobolev en a term |x| -2 |u| 2 dx is subtracted from the kinetic energy and of n-Nirenberg inequalities when various pure power weights are taken n R d , Δ A has the same scaling properties as the non-magnetic Laplan Aharonov-Bohm magnetic potential and its spectrum is explicit. ies are spectral estimates but it has to be emphasized that they differ ssical estimates as they inherit nonlinear properties of the interpoities. Finally, it is a remarkable fact that, in presence of a magnetic ardy inequality can be established in the two-dimensional case (see ] for related papers). r is organized as follows. Section 2 is devoted to some preliminary circle S 1 and on the two-dimensional sphere S 2 . New interpolation e established on the sphere S 2 , with an optimality result. Section 3 the study of a class of subquadratic magnetic interpolation inequald on the flat torus T 2 . We are able to identify a sharp condition of deduce several Hardy inequalities in dimensions d = 2 and d = 3. start by recalling earlier, non-optimal but numerically almost sharp, rpolation inequalities on R 2 in the presence of an Aharonov-Bohm in order to establish some Hardy inequalities on R 2 . Section 5 is rdy inequalities on R 3 with singularities which are either spherically y symmetric.

devoted to various results on the sphere S d without magnetic field Inequalities with Aharonov-Bohm magnetic potentials nly for completeness but also as introductory material for Secs. 2.3 gnetic interpolation inequalities on S d S d , we consider the uniform probability measure dσ, which is the ed by the Lebesgue measure in R d+1 , duly normalized and denote by orresponding L q norm. Here we state known results for later use.

e know from [12] that there exists a concave monotone increasing μ 0,p (λ) on (0, +∞) such that μ 0,p (λ) is the optimal constant in the

In this range, equality is achieved is a constant function: this is a symmetry range. On the opposite, ), the optimal function is not constant and we shall say that there is king. ≤ p < 2 is similar: there exists a concave monotone increasing λ 0,p (μ) on (0, +∞) such that λ 0,p (μ) is the optimal constant in the

In this symmetry range, constants l functions, while there is symmetry breaking if μ > d/(2-p): optimal on-constant. etry range, positive constants are actually the only positive solutions agrange equation

1 is the sign of (p -2), while there are multiple solutions in the king range. The limit case p = 2 can be obtained by taking the limit al.

ted Poincaré inequality for the ultra-spherical operator

otes the Laplace-Beltrami operator on S d-1 and L d is the ultraator. In other words, L d is the Laplace-Beltrami operator on S d unctions which depend only on z. The operator L d has a basis of G ,d , the Gegenbauer polynomials, associated with the eigenvalues r any ∈ N (see [28]). Here d is not necessarily an integer. sider the eigenvalue problem

nes the eigenvalues λ = λ ,a given by

te by g ,a (z) = G ,2 (2 a+1) (z) the associated eigenfunctions and define z 2 ) a g ,a (z). By considering the lowest positive eigenvalue, we obtain incaré inequality.

For any a ∈ R and any function f

Inequalities with Aharonov-Bohm magnetic potentials t-hand side is the Dirichlet form associated with the operator

ic rings: Superquadratic inequalities on S 1 , we review a series of results which have been obtained in [14] in the case p > 2, in preparation for an extension to the subquadratic case will be studied in Sec. 3. c interpolation inequalities and consequences r the superquadratic case p > 2 in dimension d = 1. We recall that θ where θ ∈ [0, 2π) ≈ S 1 . As in [14] we consider the space H 1 (S 1 ) of c functions u ∈ C 0,1/2 (S 1 ), such that u ∈ L 2 (S 1 ). Inequality (2.2) en as

We also have the inequality

as shown that the inequality (for complex-valued functions)

) fter eliminating the phase, to the inequality

). ce is relatively easy to prove if ψ does not vanish, but some care erwise: see [14] for details. Here we denote by μ a,p (λ) the optimal 8). Using (2.8) and then (2.6), we obtain that

, which provides an estimate of al.

2) + λ (p -2) ≤ 1, then μ a,p (λ) = a 2 + λ and equality in (2.8) is nly by the constants.

2) + λ (p -2) > 1, then μ a,p (λ) < a 2 + λ and equality in (2.8) is not y the constants.

ion a ∈ [0, 1/2] is not a restriction. First, replacing ψ by e iks ψ(s) for ws that μ a+k,p (μ) = μ a,p (μ) so that we can assume that a ∈ [0, 1]. dering χ(s) = e -is ψ(s), we find that

ic Hardy inequalities on S 1 and R 2 can draw an easy consequence of Proposition 2.1 on a Keller-Liebinequality. By Hölder's inequality applied with q = p/(p -2), we

ith λ = 0 and μ such that μ -1 φ L q (S 1 ) = μ a,p (0), we know that side is non-negative. See [14] for more details. Altogether we obtain magnetic Hardy inequality on S 1 : for any a ∈ R, any p > 2 and if φ is a non-trivial potential in L q (S 1 ), then

(2.9) ial case of the more general interpolation inequality

(2.10) q (S 1 ) , where we denote by λ a,p (μ) the inverse function of λ → μ a,p (λ), roposition 2.1. See [13] for details. rd non-magnetic Hardy inequality on R d , i.e.

[24], under the condition that q ≥ 1 + 1 2 (d -2) 2 /(d -1), again with asure on S d-1 . Magnetic and non-radial improvements have been 4]. Let us give a statement in preparation for similar extensions to ension d = 3.

and is a non-negative function in L q (S 1 ). With the above notations, the

nstant τ > 0 which is the unique solution of the equation λ a,p (τ ϕ L q (S 1 ) ) = 0.

ic interpolation inequalities on S 2 f Secs. 2.1 and 2.2, we state some new results concerning the twohere with main results in Proposition 2.2 and Corollary 2.2. etic ground state estimate r the magnetic Laplacian on S 2 and the associated Dirichlet form where dσ is the uniform probability measure on S 2 . Using cylindrical , z) ∈ [0, 2π) × [-1, 1], we can write that dσ = 1 4 π dz dθ and assume etic gradient takes the form

a magnetic flux, so that

Assume that a ∈ R. With the notation (2.11), we have (2.12)

n write u using a Fourier decomposition

dσ is an eigenfunction of (2.4) with eigenvalue λ = λ ,a such that 2 a = (2.5), we conclude that the spectrum of -Δ A is given by

adratic interpolation inequalities and consequences 2.2. Let a ∈ R and p > 2. With the notation (2.11), there exists a tone increasing function λ → μ a,p (λ) on (-Λ a , +∞) such that μ a,p (λ) constant in the inequality

roof is adapted from [13, Proposition 3.1]. For an arbitrary t ∈ (0, 1), hat

nce of Lemma 2.2 and of the diamagnetic inequality (see, e.g., [27,

Inequalities with Aharonov-Bohm magnetic potentials

), the estimate is obtained by choosing t such that λ + t Λ a 1t = 2 p -2 hat μ 0,p (2/(p -2)) = 2/(p -2). The limit as λ → -Λ a is obtained ground state of -Δ A on H 1 (S 2 ) as test function.

ame method as for the proof of (2.9), we can deduce a Hardy-type . Let a ∈ R, p > 2 and q = p/(p -2). With the notation (2.11), if φ l potential in L q (S 2 ), then ic rings: Subquadratic interpolation inequalities on S 1 results of Sec. 2.2.1 to the subquadratic range 1 < p < 2 using the ].

nt of the inequality se of (2.1) corresponding to d = 1, we have the non-magnetic interality

2). Our first result is the magnetic counterpart of this inequality.

Let a ∈ R and p ∈ [1, 2). Then there exists a concave monotone tion μ → λ a,p (μ) on R + such that

2) note by λ a,p (μ) the optimal constant in (3.1).

xistence of λ a,p (μ) is a consequence of (3.1) and of the diamagnetic ρ = |ψ| and φ be such that ψ = ρ θ) exp(i φ(θ) . Since

For all a

gnetic inequality we know that the sequence (ψ n ) n∈N is bounded in e compact Sobolev embeddings, this sequence is relatively compact in L 2 (S 1 ). The map ψ → ψi a ψ 2 L 2 (S 1 ) is lower semi-continuous ma, which proves the claim.

Asssume that a ∈ (0, 1/2), p ∈ [1, 2) and μ ≥ -a 2 . If ψ ∈ H 1 (S 1 ) is ction for (3.2) with ψ L p (S 1 ) = 1, then ψ(s) = 0 for any s ∈ S 1 . roof goes as in [14]. Let us decompose v(s) = ψ(s) e ias as a real and an t, respectively v 1 and v 2 , which both solve the same Euler-Lagrange

∈ C 0,1/2 (S 1 ) and the nonlinear term is continuous, hence v is smooth.

If both v 1 and v 2 vanish at the same vanishes identically, which means that v 1 and v 2 are proportional. /2), ψ is not 2π-periodic, a contradiction.

tion to a scalar minimization problem c. 2.1.1 if a = 0 and assume in the proofs that a > 0. The main steps n are similar to the case p > 2 of [14]. We repeat the key points for Let us define

.

u ∈ H 1 (S 1 ) is such that u(s 0 ) = 0 for some s 0 ∈ (-π, π], then

integrable. In this case, as mentioned earlier, we adopt the convention

For any a

nsider functions on S 1 as 2π-periodic functions on R. If ψ ∈ H 1 (S 1 ), s) e ias satisfies the condition

imization is taken on the set of the functions v ∈ C 0,1/2 (R) such that and (3.4) holds. e iφ written in polar form, the boundary condition becomes

, and

imization is taken on the set of the functions (u, φ) ∈ C(R) 2 such L 2 (S 1 ) and (3.5) holds. ultiplication of u by a constant so that u L p (S 1 ) = 1, the Eulertions are

by integrating the second equation and using Lemma 3.3, we find a h that φ = L/u 2 . Taking (3.5) into account, we deduce from

.

s that

imization is taken on all k ∈ Z and on all functions u ∈ H 1 (S 1 ). restriction a ∈ (0, 1/2), the minimum is achieved by k = 0. = 1/2 is a limit case that can be handled as in [14,Theorem III.7]. e result holds also true, with the minimizer being in H 1 0 (S 1 )\{0}, and

ty result as in [14], the study of (3.2) is reduced to the study of the inequality

w a real valued function. Necessary adaptations to the trivial case he limit case a = 1/2 are straightforward and left to the reader. The the analogue of Proposition 2.1 in the subcritical range.

. Let p ∈ (1, 2), a ∈ (0, 1/2), and μ > 0.

) + 4 a 2 ≤ 1, then λ a,p (μ) = a 2 + μ and equality in (3.6) is achieved e constants. +4 a 2 > 1, then λ a,p (μ) < a 2 +μ and equality (3.6) is not achieved stants.

e (i) we can write

≤ 1/(2p) and conclude using (2.7). ), let us consider the test function u ε := 1 + ε w 1 , where w 1 is the corresponding to the first non-zero eigenvalue ofd 2 /ds 2 on H 1 (S 1 ), boundary conditions, namely, w 1 (s) = cos s and λ 1 = 1. A Taylor ws that

the result. Notice that the Taylor expansion is also valid if a = 0, so he optimal constant in (3.1), and also that a similar Taylor expansion f (2.7), which formally corresponds to p = -2.

toy model for Aharonov-Bohm magnetic fields on the flat torus sider the flat torus y) with ary conditions in x and y. We denote by dσ the uniform probability dx dy/(4π 2 ) and consider the magnetic gradient

etic ground state estimate Assume that a ∈ [0, 1/2]. With the notation (3.7), we have

ake a Fourier decomposition on the basis (e i x e i k y ) k, ∈Z . We find t modes are given by k = 0, = 0 : λ 00 = a 2 , k = 1, = 0 :

is the lowest mode.

ry-Emery method applied to the 2-dimensional torus e flow given by

and. Integrations by parts show that

.

the Poincaré inequality that

As a e have the following result.

rization result without magnetic potential r than Proposition 3.1 follows from a tensorization argument that can 1, 17].

king on T 2 a function depending only on x ∈ S 1 , it is clear that the .8) cannot be improved. The proof of (3.8) can be done with the method applied to S 1 and goes as follows. sider the flow given by

= 0 on the one hand, and

and. Hence

Inequalities with Aharonov-Bohm magnetic potentials 1, because of the Poincaré inequality u 2 L 2 (S 1 ) ≥ u 2 L 2 (S 1 ) . Up to of λ, this computation also holds if p > 2 or if p = -2, as noticed is straightforward to extend it to the limit case p = 2 corresponding mic Sobolev inequality. to [11, Proposition 3.1] or [17, Theorem 2.1] and up to a straightforon to the periodic setting, the optimal constant for the inequality on s the same as for the inequality on S 1 , provided 1 ≤ p < 2. quence of Proposition 3.2, we have the inequality

p (μ) is a concave monotone increasing function on (0, +∞) such that r any μ ∈ 0, 1/(2p) .

sider the generalization of (3.9) to the case a = 0.

Assume that p ∈ [1, 2) and a ∈ [0, 1/2]. With the notation (3.7), oncave monotone increasing function μ → Λ a,p (μ) on (0, +∞) such a,p (μ) = a 2 where Λ a,p (μ) is the optimal constant in the inequality

arbitrary t ∈ (0, 1), we can write that

etry result in the subquadratic regime tion of the results on magnetic rings of Theorem 3.1, we can prove a lt for the optimal functions in (3.10) in the case p < 2. Let Λ a,p (μ) l constant in (3.10).

. Assume that a ∈ [0, 1/2] and p ∈ [1, 2). Then

2) is then constant with respect to x. Moreover, μ if and only if μ (2p) + 4 a 2 ≤ 1 and equality in (3.10) is then by the constants. use the notation f dx := 1 2π π -π f dx in order to denote a normaln with respect to the single variable x, where y is considered as a r almost every x ∈ S 1 we can apply (3.2) to the function ψ(x, •) and

e equality is achieved by functions v which are constant with respect rem 3.1 applies.

ic Hardy inequalities in dimensions 2 and 3

, we draw some consequences of our results on magnetic rings of Inequalities with Aharonov-Bohm magnetic potentials ieb-Thirring inequalities on the circle duality we obtain a spectral estimate.

Assume that a ∈ [0, 1/2] and p ∈ [1, 2). If φ is a non-negative that φ -1 ∈ L q (S 1 ), then the lowest eigenvalue λ 1 of -(∂ yi a) 2 + φ below according to

achieved by a constant potential φ if φ -1 -1 L q (S 1 ) (2p) + 4 a 2 ≤ 1.

Hölder's inequality with exponents 2/(2p) and 2/p, we get that

φ |ψ| 2 dσ p), and with μ = φ -1 -1 L q (S 1 ) , i a ψ| 2 dσ + . Let a ∈ [0, 1/2], p ∈ (1, 2) and q = p/(2p). If φ is a non-negative that φ -1 ∈ L q (S 1 ), then . Let A as in (1.1), a ∈ [0, 1/2], p ∈ (1, 2) and q = p/(2p). If φ ive potential such that φ -1 ∈ L q (S 1 ) with φ -1 L q (S 1 ) = 1, then for alued function ψ ∈ H 1 (R 2 ) we have al.

sider cylindrical coordinates (ρ, θ, z) ∈ R + × [0, 2π) × R such that . In this system of coordinates the magnetic kinetic energy is For any ψ ∈ H 1 (R 3 ), we have

ve an elementary proof. Assume that ψ is smooth and has compact inequality follows from the expansion of the square

gration by parts of the cross terms. . Let A as in (1.2), a ∈ [0, 1/2], p ∈ (1, 2) and q = p/(2p). If φ such that φ -1 ∈ L q (S 1 ) with φ -1 L q (S 1 ) = 1, then for any complex n ψ ∈ H 1 (R 3 ) we have

ase is φ ≡ 1, for which we obtain that

-Bohm Magnetic Interpolation Inequalities in R 2 polation inequalities on R 2 are considered without weights in Sec. 4.1. en introduced as in [7] in order to prove the new magnetic Caffarelli-

Inequalities with Aharonov-Bohm magnetic potentials ic interpolation inequalities without weights r on R 2 the Aharonov-Bohm magnetic potential A given by (1.1). agnetic inequality

∈ (2, ∞) and λ > 0, the Gagliardo-Nirenberg inequality

] for details. Here μ a,p (λ) is the optimal constant in (4.2) for any λ and, as a function of λ, μ a,p (λ) is monotone increasing and conat right-hand sides in (4.1) and (4.2) involve norms with respect to asure. It turns out that μ a,p (λ) is equal to the best constant of the

nstruction we know that μ a,p (λ) ≥ C p λ p/2 . By taking an optimal (4.1) and considering ψ n (x) = ψ(x + n e) with n ∈ N and e ∈ S 1 , we is equality. ve by contradiction that equality is not achieved.

) is achieved by functions with a constant phase only, this φ = 0 a.e., a contradiction with the periodicity of ψ with respect to ∈ Z.

4.1 means that the Aharonov-Bohm magnetic potential plays no ghted interpolation inequalities. This is why it is natural to introduce s with adapted scaling properties.

the optimal function in (4.5) is

and a multiplication by a constant, if

etry breaking, i.e. the optimal functions are not radially symmetric.

computation shows that λ < λ • for any a ∈ (0, 1/2), and so there is e do not know whether the optimal functions in (4.5) are symmetric heless, as shown in [7], the values of λ and λ • are numerically very ther. If λ ≤ λ , the expression of μ(λ) is explicit and given by

Inequalities with Aharonov-Bohm magnetic potentials the equivalence of (4.3) and (4.4), we prove that the magnetic inequality (4.5) is equivalent to an interpolation inequality of n-Nirenberg type in the presence of the Aharonov-Bohm magnetic (Magnetic Caffarelli-Kohn-Nirenberg inequality). Let p ∈ as in (1.1) for some a ∈ [0, 1/2] and a ≤ 0. With μ as in Theoy γ < a 2 + a 2 , we have that equality, with q = p/(p -2), for an arbitrary parameter τ > 0. For e choice of τ , we obtain the following result.

Let q ∈ (1, 2) and A as me a ∈ [0, 1/2]. Then for any function φ ∈ L q R 2 |x| -2 dx , we have

the best constant in (4.5). Finally, when a 2 ≤ 4/(12 + p 2 ), we know (0) explicitly: .B], it is proved that for all a > 0, there is a constant C(a) such that ∈ [0, 1/2] and

(5.1) an angular dependence, we have the following result.

. Let A as in (1.2), a ∈ [0, 1/2] and q ∈ (1, +∞). Then, for all

| and μ a,p is defined as in Proposition 2.2.

a ∈ [0, 1/2], according to Proposition 2.2, we find in the limit case hat μ a,2 (0) ≥ Λ a = a (a + 1) and improve the estimate (5.1) to 1) if φ ≡ 1. . Let A as in (1.2), a ∈ [0, 1/2], p > 2, q = p/(p-2) and φ ∈ L q (S 1 ).