Hardy spaces related to Schrödinger operators with potentials which are sums of Lp-functions (original) (raw)
Related papers
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}.
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 .
On Hardy spaces associated with certain Schrödinger operators in dimension 2
Revista Matemática Iberoamericana, 2012
We study the Hardy space H 1 associated with the Schrödinger operator L = −Δ + V on R 2 , where V ≥ 0 is a compactly supported nonzero C 2-potential. We prove that this space, which is originally defined by means of the maximal function associated with the semigroup generated by −L, admits a special atomic decomposition with atoms satisfying a weighted cancellation condition with a weight of logarithmic growth.
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.
On Riesz transforms characterization of H^1 spaces associated with some Schr\
2010
Let Lf(x)=-\Delta f(x) + V(x)f(x), V\geq 0, V\in L^1_{loc}(R^d), be a non-negative self-adjoint Schr\"odinger operator on R^d. 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} |e^{-tL} f(x)| belongs to L^1(R^d). We prove that under certain assumptions on V the space H^1_L is also characterized by the Riesz transforms R_j=\frac{\partial}{\partial x_j}
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.
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.
On the atomic decomposition for Hardy spaces on Lipschitz domains of
Journal of Functional Analysis, 2004
Let O be a special Lipschitz domain on R n ; and L be a second-order elliptic self-adjoint operator in divergence form L ¼ ÀdivðArÞ on Lipschitz domain O subject to Neumann boundary condition. In this paper, we give a simple proof of the atomic decomposition for Hardy spaces H p N ðOÞ of O for a range of p; by means of nontangential maximal function associated with the Poisson semigroup of L: