The Second-Order Riesz Transforms Related to Schrödinger Operators Acting on BMO-Type Spaces (original) (raw)

Riesz transforms related to Schrödinger operators acting on BMO type spaces

Journal of Mathematical Analysis and Applications, 2009

In this work we obtain boundedness on suitable weighted BM O type spaces of Riesz transforms, and their adjoints, associated to the Schrödinger operator −∆ + V , where V satisfies a reverse Hölder inequality. Our results are new even in the unweighted case.

BMO spaces related to Schrödinger operators with potentials satisfying a reverse Hölder inequality

Mathematische Zeitschrift, 2004

We identify the dual space of the Hardy-type space H 1 L related to the time independent Schrödinger operator L = −∆ + V , with V a potential satisfying a reverse Hölder inequality, as a BM O-type space BM O L. We prove the boundedness in this space of the versions of some classical operators associated to L (Hardy-Littlewood, semigroup and Poisson maximal functions, square function, fractional integral operator). We also get a characterization of BM O L in terms of Carlesson measures.

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.

The Stein-Weiss type inequalities for the B-Riesz potentials

Journal of Mathematical Inequalities, 2011

We establish two inequalities of Stein-Weiss type for the Riesz potential operator I α,γ (B− Riesz potential operator) generated by the Laplace-Bessel differential operator Δ B in the weighted Lebesgue spaces L p,|x| β ,γ. We obtain necessary and sufficient conditions on the parameters for the boundedness of I α,γ from the spaces L p,|x| β ,γ to L q,|x| −λ ,γ , and from the spaces L 1,|x| β ,γ to the weak spaces W L q,|x| −λ ,γ. In the limiting case p = Q/α we prove that the modified B− Riesz potential operator I α,γ is bounded from the spaces L p,|x| β ,γ to the weighted B − BMO spaces BMO |x| −λ ,γ. As applications, we get the boundedness of I α,γ from the weighted B-Besov spaces B s pθ ,|x| β ,γ to the spaces B s qθ ,|x| −λ ,γ. Furthermore, we prove two Sobolev embedding theorems on weighted Lebesgue L p,|x| β ,γ and weighted B-Besov spaces B s pθ ,|x| β ,γ by using the fundamental solution of the B-elliptic equation Δ α/2 B .

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}

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/

Weighted inequalities for commutators of Schrödinger–Riesz transforms

Journal of Mathematical Analysis and Applications, 2012

In this work we obtain weighted L p , 1 < p < ∞, and weak L log L estimates for the commutator of the Riesz transforms associated to a Schrödinger operator −∆ + V , where V satisfies some reverse Hölder inequality. The classes of weights as well as the classes of symbols are larger than Ap and BM O corresponding to the classical Riesz transforms.