Lp boundedness of Riesz transform related to Schroedinger operators on a manifold

Badr, Nadine; Ali, Besma Ben

doi:10.2422/2036-2145.2009.4.05

Cited by 4 publications

(2 citation statements)

References 50 publications

(54 reference statements)

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“…In the following two lemmas, we prove some estimates related to Laguerre functions defined in (2). These comprise a main ingredient in the proof of the Calderón reproducing formula.…”

Section: This Implies Thatmentioning

confidence: 95%

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Laguerre operator and its associated weighted Besov and Triebel-Lizorkin spaces

Duong

2016

Trans. Amer. Math. Soc.

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Consider the space X = ( 0 , ∞ ) X=(0,\infty ) equipped with the Euclidean distance and the measure d μ α ( x ) = x α d x d\mu _\alpha (x)=x^{\alpha }dx where α ∈ ( − 1 , ∞ ) \alpha \in (-1,\infty ) is a fixed constant and d x dx is the Lebesgue measure. Consider the Laguerre operator L = − d 2 d x 2 − α x d d x + x 2 \displaystyle L=-\frac {d^2}{dx^2} -\frac {\alpha }{x}\frac {d}{dx}+x^2 on X X . The aim of this article is threefold. Firstly, we establish a Calderón reproducing formula using a suitable distribution of the Laguerre operator. Secondly, we study certain properties of the Laguerre operator such as a Harnack type inequality on the solutions and subsolutions of Laplace equations associated to Laguerre operators. Thirdly, we establish the theory of the weighted homogeneous Besov and Triebel-Lizorkin spaces associated to the Laguerre operator. We define the weighted homogeneous Besov and Triebel-Lizorkin spaces by the square functions of the Laguerre operator, then show that these spaces have an atomic decomposition. We then study the fractional powers L − γ , γ > 0 L^{-\gamma }, \gamma >0 , and show that these operators map boundedly from one weighted Besov space (or one weighted Triebel-Lizorkin space) to another suitable weighted Besov space (or weighted Triebel-Lizorkin space). We also show that in particular cases of the indices, our new weighted Besov and Triebel-Lizorkin spaces coincide with the (expected) weighted Hardy spaces, the weighted L p L^p spaces or the weighted Sobolev spaces in Laguerre settings.

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“…In the following two lemmas, we prove some estimates related to Laguerre functions defined in (2). These comprise a main ingredient in the proof of the Calderón reproducing formula.…”

Section: This Implies Thatmentioning

confidence: 95%

“…These comprise a main ingredient in the proof of the Calderón reproducing formula. be Laguerre functions defined in (2). For each m, ∈ N there exist c(m, ) and δ > 0 so that…”

Section: This Implies Thatmentioning

confidence: 99%

Laguerre operator and its associated weighted Besov and Triebel-Lizorkin spaces

Duong

2016

Trans. Amer. Math. Soc.

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Riesz transforms, fractional power and functional calculus of Schrödinger operators on weightedLp-spaces

Assaad

2013

Journal of Mathematical Analysis and Applications

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Some Estimates for Schrödinger Type Operators on Musielak-Orlicz-Hardy Spaces

Yang¹

2014

Taiwanese J. Math.

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Let L := −div(A∇) + V be a Schrödinger type operator with the nonnegative potential V belonging to the reverse Hölder class RH q0 (R n ) for some q 0 ∈ [n, ∞) with n ≥ 3, where A satisfies the uniformly elliptic condition. Assume that ϕ :class of uniformly Muckenhoupt weights) and its uniformly critical lower type index i(ϕ) ∈ ( n n+α0 , 1], where α 0 ∈ (0, 1] measures the regularity of kernels of the semigroup generalized by L 0 := −div(A∇). In this article, we first prove that operators V L −1 , V 1/2 ∇L −1 and ∇ 2 L −1 are bounded from the Musielak-Orlicz-Hardy space associated with L, H ϕ, L (R n ), to the Musielak-Orlicz space L ϕ (R n ). Moreover, we also obtain the boundedness of V L −1 and ∇ 2 L −1 on H ϕ, L (R n ). All these results are new even when ϕ(x, t) := t p , with p ∈ ( n n+α0 , 1], for all x ∈ R n and t ∈ [0, ∞).

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Lp boundedness of Riesz transform related to Schroedinger operators on a manifold

Cited by 4 publications

References 50 publications

Laguerre operator and its associated weighted Besov and Triebel-Lizorkin spaces

Laguerre operator and its associated weighted Besov and Triebel-Lizorkin spaces

Riesz transforms, fractional power and functional calculus of Schrödinger operators on weightedLp-spaces

Some Estimates for Schrödinger Type Operators on Musielak-Orlicz-Hardy Spaces

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