Endpoint estimates for Riesz transforms of magnetic Schrödinger operators

Let $A=-(\nabla-i\vec{a})^2+VLet be a magnetic Schr?dinger operator acting on L ²(R ⁿ ), n≥1, where and 0≤V∈L ¹ _loc. Following [1], we define, by means of the area integral function, a Hardy space H ¹ _A associated with A. We show that Riesz transforms (∂/∂x _k -i a _k )A ^-1/2 associated with A, , are bounded from the Hardy space H ¹ _A into L ¹. By interpolation, the Riesz transforms are bounded on L ^p for all 1<p≤2.     

Some estimates for commutators of Riesz transform associated with Schrödinger type operators

Let L₁ = ?Δ + V be a Schr:dinger operator and let L² = (?Δ)² + V² be a Schrödinger type operator on ?ⁿ (n ? 5), where V≠ 0 is a nonnegative potential belonging to certain reverse Hölder class B_s for s ? n/2. The Hardy type space \(H_{{L_2}}^1\) is defined in terms of the maximal function with respect to the semigroup \(\left\{ {{e^{ - t{L_2}}}} \right\}\) and it is identical to the Hardy space \(H_{{L_1}}^1\) established by Dziubański and Zienkiewicz. In this article, we prove the L^p-boundedness of the commutator R_b = bRf - R(bf) generated by the Riesz transform \(R = {\nabla ^2}L_2^{ - 1/2}\), where \(b \in BM{O_\theta }(\varrho )\), which is larger than the space BMO(?ⁿ). Moreover, we prove that R_b is bounded from the Hardy space \(H_{\mathcal{L}_1 }^1 \) into weak \(L_{weak}^1 (\mathbb{R}^n )\).     

Semigroup and Riesz transform for the Dunkl–Schrödinger operators

Semigroup Forum - Let $$L_k=-\Delta _k+V$$ be the Dunkl–Schrödinger operators, where $$\Delta _k=\sum _{j=1}^dT_j^2$$ is the Dunkl Laplace operator associated to the Dunkl operators...     

Spectral estimates for Schrödinger and Dirac-type operators on Riemannian manifolds

Supported by funds of M.U.R.S.T. (Italy). The author is grateful to S. Gallot for his encouragement and for helpful discussions and to G. Besson for some interesting remarks     

Commutators of Riesz transforms of magnetic Schrödinger operators

Let \({A=-(\nabla-i{\vec a})\cdot (\nabla-i{\vec a}) +V}\) be a magnetic Schrödinger operator acting on \({L^2({\mathbb R}^n)}\), n ≥  1, where \({{\vec a}=(a_1, \ldots, a_n)\in L^2_{\rm loc}({\mathbb R}^n, {\mathbb R}^n)}\) and \({0\leq V\in L^1_{\rm loc}({\mathbb R}^n)}\). In this paper, we show that when a function \({b\in {\rm BMO}({\mathbb R}^n)}\), the commutators [b, T _k ]f = T _k (b f) ? b T _k f, k = 1, . . . , n, are bounded on \({L^p({\mathbb R}^n)}\) for all 1 < p < 2, where the operators T _k are Riesz transforms (?/?x _k  ? i a _k )A ^?1/2 associated with A.     

Daugavet type inequalities for operators on L p–spaces

Let T be a regular operator from L ^p L ^p. Then , where T_r denotes the regular norm of T, i.e., T_r=|T| where |T| denotes the modulus operator of a regular operator T. For p=1 every bounded linear operator is regular and T=T_r, so that the above inequality generalizes the Daugavet equation for operators on L ¹–spaces. The main result of this paper (Theorem 9) is a converse of the above result. Let T be a regular linear operator on L _p and denote by T _A the operator T_A. Then for all A with (A)>0 if and only if .     

Riesz transform characterization of Hardy spaces associated with Schrödinger operators with compactly supported potentials

Let L=?Δ+V be a Schrödinger operator on ? ^d , d≥3. We assume that V is a nonnegative, compactly supported potential that belongs to L ^p (? ^d ), for some p>d /2. Let K _t be the semigroup generated by ?L. We say that an L ¹(? ^d )-function f belongs to the Hardy space \(H^{1}_{L}\) associated with L if sup?_t>0|K _t f| belongs to L ¹(? ^d ). We prove that \(f\in H^{1}_{L}\) if and only if R _j f∈L ¹(? ^d ) for j=1,…,d, where R _j =(?/? x _j )L ^?1/2 are the Riesz transforms associated with L.     

Lp resolvent estimates for magnetic Schrödinger operators with unbounded background fields

We prove L^p and smoothing estimates for the resolvent of magnetic Schrödinger operators. We allow electromagnetic potentials that are small perturbations of a smooth, but possibly unbounded background potential. As an application, we prove an estimate on the location of eigenvalues of magnetic Schrödinger and Pauli operators with complex electromagnetic potentials.     

Resolvent at low energy and Riesz transform for Schrödinger operators on asymptotically conic manifolds. I.

Let $M^\circLet be a complete noncompact manifold and g an asymptotically conic manifold on , in the sense that compactifies to a manifold with boundary M in such a way that g becomes a scattering metric on M. A special case that we focus on is that of asymptotically Euclidean manifolds, where the induced metric at infinity is equal to the standard metric on S ⁿ⁻¹; such manifolds have an end that can be identified with in such a way that the metric is asymptotic in a precise sense to the flat Euclidean metric. We analyze the asymptotics of the resolvent kernel (P + k ²)⁻¹ where is the sum of the positive Laplacian associated to g and a real potential function which vanishes to second order at the boundary (i.e. decays to second order at infinity on ) and such that if . Then we show that on a blown up version of the resolvent kernel is conormal to the lifted diagonal and polyhomogeneous at the boundary, and we are able to identify explicitly the leading behaviour of the kernel at each boundary hypersurface. Using this we show that the Riesz transform of P is bounded on for 1 < p < n if , and that this range is optimal if or if M has more than one end. The result with is new even when , g is the Euclidean metric and V is compactly supported. When V ≡ 0 with one end, the range of p becomes 1 <  p <  p _max where p _max > n depends explicitly on the first non-zero eigenvalue of the Laplacian on the boundary . Our results hold for all dimensions ≥ 3 under the assumption that P has neither zero modes nor a zero-resonance. In the follow-up paper Guillarmou and Hassell (Resolvent at low energy and Riesz transform for Schr?dinger operators on asymptotically conic manifolds, preprint) [7] we analyze the same situation in the presence of zero modes and zero-resonances.     

Maximal function characterizations of Musielak-Orlicz-Hardy spaces associated with magnetic Schr?dinger operators

Let φ be a growth function, and let A:=-(?-ia)?(?-ia)+V be a magnetic Schr?dinger operator on L2(?n),n≥2, where α:=(α1,α2,?,αn)∈Lloc2(?n,?n) and 0≤V∈Lloc1(?n). We establish the equivalent characterizations of the Musielak-Orlicz-Hardy space HA,φ(?n), defined by the Lusin area function associated with {e-t2A}t>0, in terms of the Lusin area function associated with {e-tA}t>0, the radial maximal functions and the nontangential maximal functions associated with {e-t2A}t>0 and {e-tA}t>0, respectively. The boundedness of the Riesz transforms LkA-1/2,k∈{1,2,?,n}, from HA,φ(?n) to Lφ(?n) is also presented, where L_k is the closure of ??xk-iαk in L2(?n). These results are new even when φ(x,t):=ω(x)tp for all x∈?nand t ∈(0,+∞) with p ∈(0, 1] and ω∈A∞(?n) (the class of Muckenhoupt weights on ?n).     

Spectral estimates for Schrödinger operators with sparse potentials on graphs

A construction of “sparse potentials,” suggested by the authors for the lattice \mathbbZ^d {\mathbb{Z}^d} , d > 2, is extended to a large class of combinatorial and metric graphs whose global dimension is a number D > 2. For the Schr?dinger operator − Δ − αV on such graphs, with a sparse potential V, we study the behavior (as α → ∞) of the number N_(−Δ − αV) of negative eigenvalues of − Δ − αV. We show that by means of sparse potentials one can realize any prescribed asymptotic behavior of N_(−Δ − αV) under very mild regularity assumptions. A similar construction works also for the lattice \mathbbZ² {\mathbb{Z}^2} , where D = 2. Bibliography: 13 titles.     

The BMO L space and Riesz transforms associated with Schrödinger operators

Let L =△ ＋ V be a SchrSdinger operator in Rd, d ≥ 3, where the nonnegative potential V belongs to the reverse HSlder class Sd. We establish the BMOL-boundedness of Riesz transforms З/ЗxiL-1/2, and give the Fefferman-Stein type decomposition of BMOL functions.     

Dimension free estimates for Riesz transforms of some Schr?dinger operators

We prove dimension free L ^∞ → L ^∞-estimates for the Riesz transform T = V L ⁻¹, L = −Δ + V, where Δ is the Laplacian in ℝ ^d , and the polynomial V ≥ 0 satisfies C. L. Fefferman conditions; see [7]. As a corollary we get dimension free L ^p → L p(ℓ ²)-estimates, 1 < p < ∞, for the vector of Riesz transforms.     

L~p estimates for Riesz transform and their commutators associated with Schr?dinger type operator

Let H_2 =(-△)~2 +V~2 be the Schrodinger type operator,where V satisfies reverse Holder inequality.In this paper,we establish the L~P boundedness for V~2H_2~(-1),H_2~(-1)V~2,VH_2~(-1/2)and H_2~(-1/2)V,and that of their commutators.We also prove that H_2~(-1)V~2,H_2~(-1/2)V are bounded from BMO_L to BMO_L.     

Regularity estimates in Hölder spaces for Schrödinger operators via a T1 theorem

We derive Hölder regularity estimates for operators associated with a time-independent Schrödinger operator of the form $-\Delta +V$ . The results are obtained by checking a certain condition on the function $T1$ . Our general method applies to get regularity estimates for maximal operators and square functions of the heat and Poisson semigroups, for Laplace transform type multipliers and also for Riesz transforms and negative powers $(-\Delta +V)^{-\gamma /2}$ , all of them in a unified way.     

Some estimates of Schrödinger type operators on the Heisenberg group

In this paper, we consider the Schrödinger type operator ${H = (-\Delta _{\mathbb {H}}^n)^2 +V ^{2}}In this paper, we consider the Schr?dinger type operator H = (-D_{\mathbb H}ⁿ)² +V ²{H = (-\Delta _{\mathbb {H}}^n)^2 +V ^{2}}, where the nonnegative potential V belongs to the reverse H?lder class B_q1 for q₁ 3 \frac Q 2,Q 3 6{B_{{q}_{1}}\, {\rm for}\, q_{1}\geq {\frac {Q}{ 2}},Q \geq 6}, and D_{\mathbb Hⁿ}{\Delta_{\mathbb {H}^n}} is the sublaplacian on the Heisenberg group \mathbb Hⁿ{\mathbb {H}^n}. An L ^p estimate and a weak type L ¹ estimate for the operator ?⁴_{\mathbb Hⁿ} H^-1{\nabla^4_{\mathbb {H}^n} H^{-1}} when V ? B_q1{V \in B_{{q}_{1}}} for 1 < p ￡ \fracq₁2{1 < p \leq \frac{q_{1}}{2}} are obtained.     

Some estimates for the Schrödinger type operators with nonnegative potentials

In this paper we consider the Schrödinger operator ?Δ + V on \({\mathbb R^d}\), where the nonnegative potential V belongs to the reverse Hölder class \({B_{q_{_1}}}\) for some \({q_{_1}\geq \frac{d}{2}}\) with d ≥ 3. Let \({H^1_L(\mathbb R^d)}\) denote the Hardy space related to the Schrödinger operator L = ?Δ + V and \({BMO_L(\mathbb R^d)}\) be the dual space of \({H^1_L(\mathbb R^d)}\). We show that the Schrödinger type operator \({\nabla(-\Delta +V)^{-\beta}}\) is bounded from \({H^1_L(\mathbb R^d)}\) into \({L^p(\mathbb R^d)}\) for \({p=\frac{d}{d-(2\beta-1)}}\) with \({ \frac{1}{2}<\beta<\frac{3}{2} }\) and that it is also bounded from \({L^p(\mathbb R^d)}\) into \({BMO_L(\mathbb R^d)}\) for \({p=\frac{d}{2\beta-1}}\) with \({ \frac{1}{2}<\beta< 2}\).