Hölder estimates for magnetic Schrödinger semigroups in Rd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^{d}$$\end{document} from mirror coupling

We use the mirror coupling of Brownian motion to show that under a β∈(0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta \in (0,1)$$\end{document}-dependent Kato-type assumption on the possibly nonsmooth electromagnetic potential, the corresponding magnetic Schrödinger semigroup in Rd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^d$$\end{document} has a global Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{p}$$\end{document}-to-C0,β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^{0,\beta }$$\end{document} Hölder smoothing property for all p∈[1,∞]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\in [1,\infty ]$$\end{document}; in particular, his all eigenfunctions are uniformly β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document}-Hölder continuous. This result shows that the eigenfunctions of the Hamilton operator of a molecule in a magnetic field are uniformly β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document}-Hölder continuous under weak Lq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^q$$\end{document}-assumptions on the magnetic potential.


Introduction
has shown that each eigenfunction of a multi-particle Schrödinger operator H = − + W in L 2 (R 3m ) with a potential W : R 3m → R of the form with w j , w jk ∈ L p (R 3 ) + L ∞ (R 3 ) for some p ≥ 2 is uniformly β-Hölder continuous for all 0 < β < 2 − 3/ p, that is, where we have written points in R 3m in the form x = (x 1 , . . . , x m ), where x j ∈ R 3 for j = 1, . . . , m. In particular, an application of this result to multi-particle Coulombtype potentials shows that all molecular Hamilton operators (in the infinite mass limit) are uniformly α-Hölder continuous for all 0 < α < 1. Kato's proof relies on the Fourier transform and so does not apply directly to magnetic Schrödinger operators (even if one assumes a Coulomb gauge). The aim of this paper is to use probabilistic techniques to find a variant of Kato's regularity result that applies to a magnetic Schrödinger operator H (A, V ) with magnetic potential A : R d → R d and electric potential V : R d → R. To this end, we prove the following smoothing result (cf. Theorem 2.5): Let β ∈ (0, 1) and let C 0,β (R d ) denote the space of uniformly β-Hölder continuous functions on R d , with its seminorm given by and consider for q ∈ [1, ∞] the Banach space C 0,β (R d ) ∩ L q (R d ) with its norm Then for all Borel functions A : and all t > 0, 1 ≤ p ≤ q ≤ ∞ one has and the norm of this operator can be estimated explicitly. Above, K β (R d ), β ∈ [0, 1], denotes the β-Kato class (cf. Definition 2.1) of Borel functions R d → R which has been introduced in [5], so that K(R d ) := K 0 (R d ) is the classical Kato class [1] and one has K β (R d ) ⊂ K α (R d ) if β ≥ α. Note also that H (A, V ) = θ implies e −t H(A,V ) = e −tθ , so that one also obtains global β-Hölder regularity for eigenfunctions.
The mapping property is well known [2] and only requires a local Kato assumption on |A| 2 , |div(A)| and the positive part of V , and a global Kato assumption on the negative part of V .
The proof of (2) uses Brownian mirror coupling techniques (cf. Sect. 2 for the basic definitions) to deal with the magnetic potential A. Let us mention here that the use of Brownian coupling techniques in the context of Hölder estimates for semigroups that generate diffusion (which in our case would correspond to taking V = 0, A = 0) has a long history, also on Riemannian manifolds (cf. [3,9,10] for some classical results).
Our main tool (cf. Theorem 2.3) for the proof of (2) is provided by the following estimate: There exists a universal constant c 0 < ∞, such that for every q ∈ (1, ∞), every Borel function A : every t > 0, x = y in R d , and every mirror coupling (X, Y) of Brownian motions from (x, y) one has where for any Brownian motion Z, the process S t (A|Z) denotes the magnetic Euclidean action functional (cf. (7) ) which appears in the Feynman-Kac-Itô formula and where the constant C(A, t, q) < ∞ can be computed explicitly. This estimate is then combined with the Feynman-Kac-Itô formula (and perturbation theory to deal with V ) to finally obtain (2).
Let us mention that locally uniform β-Hölder continuity results for nonmagnetic Schrödinger eigenfunctions under L q -assumptions on V have also been obtained in [8] (cf. Theorem 11.7 therein) using straightforward Sobolev embedding techniques. In addition, in [11] (cf. Theorem B.3.5 therein) it is shown that β-dependent (Katotype) assumptions in V leading to locally uniform β-Hölder smoothing results for nonmagnetic Schrödinger semigroups. In the latter case, the ultimate argument relies on potential theory, while Brownian motion only enters through the Feynman-Kac formula in order to show that the Schrödinger semigroup is L ∞ -smoothing.
Using L p -criteria for the β-Kato class (cf. Remark 2.2), we show that o result directly implies the following generalization of Kato's result for multi-particle Schrödinger operators in R 3n to magnetic multi-particle Schrödinger operators: Assume there exists β ∈ (0, 1), l ∈ N and Borel functions a : where div is defined in the distributional sense, and define a vector potential, resp. a magnetic potential on R 3n through Then for all t > 0 and p ∈ [1, ∞] one has To the best of our knowledge, this is the first global Hölder-regularity result for multiparticle magnetic Schrödinger operators. Let us finally explain how this result applies to molecules in a magnetic field: Given R ∈ R 3n , l ∈ N, Z ∈ [0, ∞) l , consider the potential Given a : R 3 → R 3 (sufficiently well behaved), set as above A(x) := n i=1 a(x i ). Then the operator is the Hamilton operator corresponding to a molecule (in the infinite mass limit) with l protons and with n electrons, where the jth nucleus is located in R j , and has Z j protons, and the electrons interact with the magnetic field induced by A. Then given an arbitrary β ∈ (0, 1), one has (5) for so that the previous result gives that for all t > 0 and p ∈ [1, ∞] one has as long as

Main results
We start by recalling the definition of the mirror coupling of Brownian motions as presented in [6] and follow their exposition (pages 1-3 therein) closely before presenting our main results.
and Y are Brownian motions starting in x and y, respectively. Then, with the coupling time with the transition density of Brownian motion starting in a. The reason for this notion of maximality is that for an arbitrary coupling of Brownian motions one has ≤ in (6). Let x and y be two distinct points of R d . Then is the hyperplane orthogonal on and bisecting the segment x y. Furthermore, define the affine map This is the reflection at the hyperplane N x,y . Let L x,y be the linear part of R x,y . Note that L x,y is symmetric and idempotent.
is the hitting time of X with respect to N x,y . In other words, Y is equal to the reflection of X at N x,y before X hits N x,y , and is then equal to X. It follows that τ (X, Y) = τ x,y (X), which by an explicit calculation of P(t ≤ τ x,y (X)) implies that every mirror coupling is maximal. Whenever well defined, we consider the following action functional on the paths of any Brownian motion Z, which depends on a sufficiently regular function A : Above, i is the imaginary unit, div(A) denotes the divergence of A understood in the distributional sense and the stochastic integral is understood in Itô's sense. Let P a denote the law of Brownian motion starting in a, which is considered as a probability measure on the space of continuous paths ω : [0, ∞) → R d . Generalizing the Kato class, the following hierarchy of Kato classes has been introduced in [5]: Note that which follows straightforwardly (cf. Lemma 3.9 in [5]) from and if W ∈ K β (R d ), then also (cf. the proof of Theorem 3.10 in [5]) The following probabilistic estimate is our main technical result: where 1/q * + 1/q = 1 and Given a Brownian motion Z, we split Clearly we a.s. have Likewise, heuristically, for s < τ one has dY s = LdX s , while for s ≥ τ one has dY s = dX s , and we therefore expect that holds a.s., whereÃ To show that Eq. (13) holds, by replacing with the sequence A n := (max (A 1 , n), . . . , max(A d , n)), n ∈ N, and using the Itô isometry and dominated convergence, we can assume that A is bounded. By Theorem 6.5 in [12] we have the L 2 -convergence of the dyadic approximations where t i := it 2 n for i = 0, . . . , 2 n . We immediately note that in case t < τ, we have Y s = RX s on [0, t]; hence, in that case by the above limits we conclude that: If we now assume that τ ∈ (t k , t k+1 for some k = 0, . . . , 2 n − 1, we get the following expressions for the summands in the above limits: In the last step, we have used that L is self-adjoint and Rv − Rw For i ≥ k + 2, the summands vanish. Compiling these equations allows us to make the following estimates, because RX τ = X τ . In particular, since L is self-adjoint and idempotent, Since A is bounded by some κ > 0, and so |γ | ≤ 2κ, using we can thus estimate as follows, Since τ is an X-stopping time, we conclude by the Markov property of X, using and once more t j+1 − t j = t 2 n , that Altogether, we have found that under the assumptions of the theorem one has We are now going to estimate the L 1 -norms of M t and I t . Let us start with E (|I t |): setting we have In view of where from here on C < ∞ denotes a universal constant whose value may change from line to line. Now let us turn to the estimate for E (|M t |): Define We note that h (r ) ≥ |r | and that h is in In particular, we have |h | ≤ 1 [−2,2] and h ≤ 1. We conclude by Itô's formula is an (L 2 )-martingale, as follows from the assumption on A and the boundedness of h . Thus, Hence, similarly to the Lebesgue integral I t , we conclude Thus, we have shown Finally, noting that for all purely imaginary z, z one has the elementary estimate and (S t (Z)) = 0, the proof is complete.
then the symmetric sesquilinear form is semibounded from below and closable [2]. Thus, the closure of this form induces a self-adjoint semibounded from below operator H (A, V ) in L 2 (R d ). The corresponding magnetic Schrödinger semigroup is given by the Feynman-Kac-Itô formula [2] where Z(x) is an arbitrary Brownian motion in R d starting in x ∈ R d . Using the Feynman-Kac-Itô formula for V = 0 with Theorem 2.3 to deal with the magnetic potential A, and perturbation theory to deal with the electric potential V , we can now establish: and let t > 0, 1 ≤ p ≤ q ≤ ∞. Then one has continuously, and there exists a universal constant c 0 < ∞ and a constant C V < ∞ which only depends on V , such that is the locally bounded function from Theorem 2.3, and Remark 2.6 1) Using monotone convergence, one finds which is finite for all t > 0 by Remark 2.2.5), so that a posteriori one also has D V < ∞ a.e.
2) As our proof shows, the constant C V can be chosen to be any constant which satisfies that for all 1 ≤ p ≤ q ≤ ∞, r > 0 one has The existence of such a uniform constant has been shown in [2]. Proof of Theorem 2. 5 We start by noting that the assumptions on A together with Jensen's inequality, and that K β (R d ) ⊂ K(R d ) shows that the pair (A, V ) satisfies (15).
Set q := 1/(1 − β) ∈ (1, ∞) so that q * = 1/β and pick a mirror coupling (X, Y) from (x, y) ∈ (R d × R d )\diag(R d ) and set τ := τ (X, Y). Then, given r > 0, ∈ L 2 (R d ) ∩ L ∞ (R d ) we can estimate as follows, where c 0 < ∞ is a universal constant. Thus, we have shown Duhamel's formula 2 states that There exists [2] a constant C V such that for all 1 ≤ p ≤ q ≤ ∞, r > 0 one has so that Moreover, by what we have shown above, Given f ∈ L ∞ (R d ), x ∈ R d , and a Brownian motion Z(x) in R d starting in x we have, using |e −S s/2 (A|Z(x)) | = 1, the estimate Combining (17), (19), (20), (21), we have shown that for all ∈ L ∞ (R d ), and so The above estimate together with = e − t 2 H (A,V ) and (18) shows Finally, using (18) we end up with which completes the proof.
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