Weighted Littlewood-Paley Theory and Exponential-Square Integrability

2008 ◽  
Keyword(s):  
Square Integrability

Square Integrable Representations and the Standard Module Conjecture for General Spin Groups

2009 ◽  
Vol 61 (3) ◽  
pp. 617-640 ◽  
Author(s):  
Wook Kim
Keyword(s):  
Half Plane ◽  
Spin Groups ◽  
Standard Module ◽  
Square Integrability ◽  
Integrability Criterion

Abstract.In this paper we study square integrable representations and L -functions for quasisplit general spin groups over a p-adic field. In the first part, the holomorphy of L -functions in a half plane is proved by using a variant formof Casselman's square integrability criterion and the Langlands–Shahidi method. The remaining part focuses on the proof of the standard module conjecture. We generalize Muić's idea via the Langlands–Shahidimethod towards a proof of the conjecture. It is used in the work of M. Asgari and F. Shahidi on generic transfer for general spin groups.

scholarly journals Exponential-Square Integrability, Weighted Inequalities for the Square Functions Associated to Operators, and Applications

2020 ◽  
Author(s):  
Peng Chen ◽  
Xuan Thinh Duong ◽  
Liangchuan Wu ◽  
Lixin Yan
Keyword(s):  
Analytic Semigroup ◽  
Adjoint Operator ◽  
Square Function ◽  
Weighted Norm ◽  
Square Functions ◽  
Spectral Multipliers ◽  
Eigenvalue Estimates ◽  
Preservation Condition ◽  
Square Integrability ◽  
The Laplace Operator

Abstract Let $X$ be a metric space with a doubling measure. Let $L$ be a nonnegative self-adjoint operator acting on $L^2(X)$, hence $L$ generates an analytic semigroup $e^{-tL}$. Assume that the kernels $p_t(x,y)$ of $e^{-tL}$ satisfy Gaussian upper bounds and Hölder continuity in $x$, but we do not require the semigroup to satisfy the preservation condition $e^{-tL}1 = 1$. In this article we aim to establish the exponential-square integrability of a function whose square function associated to an operator $L$ is bounded, and the proof is new even for the Laplace operator on the Euclidean spaces ${\mathbb R^n}$. We then apply this result to obtain: (1) estimates of the norm on $L^p$ as $p$ becomes large for operators such as the square functions or spectral multipliers; (2) weighted norm inequalities for the square functions; and (3) eigenvalue estimates for Schrödinger operators on ${\mathbb R}^n$ or Lipschitz domains of ${\mathbb R}^n$.

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