On Riesz transforms characterization of H^1 spaces associated with some Schr\ (original) (raw)
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Riesz transform characterization of Hardy spaces associated with certain Laguerre expansions
Tohoku Mathematical Journal, 2010
Let L = −∆ + V be a Schrödinger operator on R d , d ≥ 3. We assume that V is a nonnegative, compactly supported potential that belongs to L p (R d), for some p > d/2. Let Kt be the semigroup generated by −L. We say that an L 1 (R d)-function f belongs to the Hardy space H 1 L associated with L if sup t>0 |Ktf | belongs to L 1 (R d). We prove that f ∈ H 1 L if and only if Rj f ∈ L 1 (R d) for j = 1, ..., d, where Rj = ∂ ∂x j L −1/2 are the Riesz transforms associated with L.
A Characterization of Hardy Spaces Associated with Certain Schrödinger Operators
Potential Analysis, 2014
Let {K t } t>0 be the semigroup of linear operators generated by a Schrödinger operator −L = Δ − V (x) on R d , d ≥ 3, where V (x) ≥ 0 satisfies Δ −1 V ∈ L ∞. We say that an L 1-function f belongs to the Hardy space H 1 L if the maximal function M L f (x) = sup t>0 |K t f (x)| belongs to L 1 (R d). We prove that the operator (−Δ) 1/2 L −1/2 is an isomorphism of the space H 1 L with the classical Hardy space H 1 (R d) whose inverse is L 1/2 (−Δ) −1/2. As a corollary we obtain that the space H 1 L is characterized by the Riesz transforms R j = ∂ ∂x j L −1/2 .
2011
We investigate the Hardy space H^1_L associated to the Schr\"odinger operator L=-\Delta+V on R^n, where V=\sum_{j=1}^d V_j. We assume that each V_j depends on variables from a linear subspace VV_j of \Rn, dim VV_j \geq 3, and V_j belongs to L^q(VV_j) for certain q. We prove that there exist two distinct isomorphisms of H^1_L with the classical Hardy space. As a corollary we deduce a specific atomic characterization of H_L^1. We also prove that the space H_L^1 is described by means of the Riesz transforms R_{L,i} = \partial_i L^{-1/2}.
Hardy spaces H1 for Schrödinger operators with compactly supported potentials
Annali di Matematica Pura ed Applicata (1923 -), 2005
Let L = −∆ + V be a Schrödinger operator on R d , d ≥ 3, where V is a nonnegative compactly supported potential that belongs to L p for some p > d/2. Let {K t } t>0 denote the semigroup of linear operators generated by −L. For a function f we define its H 1 L-norm by f H 1 L = sup t>0 |K t f(x)| L 1 (dx). It is proved that for a properly defined weight w a function f belongs to H 1 L if and only if w f ∈ H 1 (R d), where H 1 (R d) is the classical real Hardy space.
Hardy spaces related to Schrödinger operators with potentials which are sums of Lp-functions
Journal of Mathematical Analysis and Applications, 2012
We investigate the Hardy space H 1 L associated to the Schrödinger operator L = −∆ + V on R n , where V = d j=1 Vj. We assume that each Vj depends on variables from a linear subspace Vj of R n , dim Vj ≥ 3, and Vj belongs to L q (Vj) for certain q. We prove that there exist two distinct isomorphisms of H 1 L with the classical Hardy space. As a corollary we deduce a specific atomic characterization of H 1 L. We also prove that the space H 1 L is described by means of the Riesz transforms RL,i = ∂iL −1/2 .
Second order Riesz transforms associated to the Schrödinger operator for
Journal of Mathematical Analysis and Applications, 2014
Let L = − + V be the Schrödinger operator on R n , where V belongs to the class of reverse Hölder weights R H q for some q > max{2, n/2}. We show that the second order Riesz transforms ∇ 2 L −1 and V L −1 are bounded from the Hardy spaces H p L (R n) associated to L into L p (R n) for 0 < p 1. We show also that the operators ∇ 2 L −1 map the classical Hardy spaces H p (R n) into H p (R n) for a restricted range of p.
Weighted Hardy spaces associated to operators and boundedness of singular integrals
Arxiv preprint arXiv:1202.2063, 2012
Let (X, d, µ) be a space of homogeneous type, i.e. the measure µ satisfies doubling (volume) property with respect to the balls defined by the metric d. Let L be a non-negative self-adjoint operator on L 2 (X). Assume that the semigroup of L satisfies the Davies-Gaffney estimates. In this paper, we study the weighted Hardy spaces H p L,w (X), 0 < p ≤ 1, associated to the operator L on the space X. We establish the atomic and the molecular characterizations of elements in H p L,w (X). As applications, we obtain the boundedness on H p L,w (X) for the generalized Riesz transforms associated to L and for the spectral multipliers of L. Contents 1. Introduction 2. Preliminaries 2.1. Doubling metric spaces 2.2. Muckenhoupt weights 3. Weighted Hardy spaces associated to operators 3.1. Definition of weighted Hardy spaces 3.2. Finite propagation speed for the wave equation 3.3. Weighted tent spaces 3.4. Atomic characterization of weighted Hardy spaces H p L,w (X) 3.5. Molecular characterization of weighted Hardy spaces H p L,w (X) 4. Boundedness of singular integrals with non-smooth kernels 4.1. Generalized Riesz transforms 4.2. Boundedness of Riesz transforms associated with magnetic Schrödinger operators 4.3. Spectral multiplier theorem on H p L,w (X) References
Behaviour of Schrödinger Riesz transforms over smoothness spaces
Journal of Mathematical Analysis and Applications
As it was shown by Shen, the Riesz transforms associated to the Schrödinger operator L = −∆ + V are not bounded on L p (R d)-spaces for all p, 1 < p < ∞, under the only assumption that the potential satisfies a reverse Hölder condition of order d/2, d ≥ 3. Furthermore, they are bounded only for p in some finite interval of the type (1, p 0), so it can not be expected to preserve regularity spaces. In this work we search for some kind of minimal additional conditions on the potential in order to obtain boundedness on appropriate weighted BM O type regularity spaces for all first and second order Riesz transforms, namely for the operators ∇L −1/