Sharp weak type estimates for Riesz transforms

Oşekowski, Adam

doi:10.1007/s00605-014-0613-7

Cited by 4 publications

(1 citation statement)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Proof of (1.12) and (1.13). For the restricted weak type bound (1.12) we argue similarly using the restricted weak type (p, q) bound for the Laplacian resolvent ([23, Theorem 1.4]) and the L p,1 −L p,1 estimate of the Riesz transforms (see [30,Theorem 1.1]). Indeed,…”

Section: )mentioning

confidence: 89%

Resolvent estimates for the Lamé operator and failure of Carleman estimates

Kwon¹,

Lee²,

Seo³

2019

Preprint

View full text Add to dashboard Cite

In this paper, we consider the Lamé operator −∆ * and study resolvent estimate, uniform Sobolev estimate, and Carlman estimate for −∆ * . First, we obtain sharp L p -L q resolvent estimates for −∆ * for admissible p, q. This extends the particular case q = p p−1 due to Cossetti [6]. Secondly, we show failure of uniform Sobolev estimate and Carlman estimate for −∆ * . Our approach is different from that in [6] which relies heavily on the Helmholtz decomposition. Instead, we directly analyze the Fourier multiplier of the resolvent. This direct approach allows us to prove not only the upper bound but also the lower bound on the resolvent, so we get the sharp L p -L q bounds for the resolvent of −∆ * . Strikingly, the relevant uniform Sobolev and Carleman estimates turn out to be false for the Lamé operator −∆ * even though the uniform resolvent estimates for −∆ * are valid for certain range of p, q. This contrasts with the classical result regarding the Laplacian ∆ due to Kenig, Ruiz, and Sogge [21] in which the uniform resolvent estimate plays crucial role in proving the uniform Sobolev and Carlman estimates for ∆. We also describe locations of the L q -eigenvalues of −∆ * + V with complex potential V by making use of the sharp L p -L q resolvent estimates for −∆ * .

show abstract

Section: )mentioning

confidence: 89%