A T(1) theorem for singular Radon transforms

Greenblatt, Michael

doi:10.1007/s00208-007-0126-y

Math. Ann.

2007

DOI: 10.1007/s00208-007-0126-y

|View full text |Cite

A T(1) theorem for singular Radon transforms

Michael Greenblatt

Abstract: We prove an analogue for singular Radon transforms to the T (1) theorem of David and Journe.

Help me understand this report

Search citation statements

Order By: Relevance

Paper Sections

Select...

Citation Types

Supporting

Mentioning

Contrasting

Year Published

2022

Publication Types

Select...

Article1

Relationship

Self Cite0

Independent1

Authors

Journals

Cited by 1 publication

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

Real analytic multi-parameter singular Radon transforms: Necessity of the Stein-Street condition

Zhang

2022

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

We study operators of the form T f ( x ) = ψ ( x ) ∫ f ( γ t ( x ) ) K ( t ) d t , \begin{equation*} Tf(x)= \psi (x) \int f(\gamma _t(x))K(t)\,dt, \end{equation*} where γ t ( x ) \gamma _t(x) is a real analytic function of ( t , x ) (t,x) mapping from a neighborhood of ( 0 , 0 ) (0,0) in R N × R n \mathbb {R}^N \times \mathbb {R}^n into R n \mathbb {R}^n satisfying γ 0 ( x ) ≡ x \gamma _0(x)\equiv x , ψ ( x ) ∈ C c ∞ ( R n ) \psi (x) \in C_c^\infty (\mathbb {R}^n) , and K ( t ) K(t) is a “multi-parameter singular kernel” with compact support in R N \mathbb {R}^N ; for example when K ( t ) K(t) is a product singular kernel. The celebrated work of Christ, Nagel, Stein, and Wainger studied such operators with smooth γ t ( x ) \gamma _t(x) , in the single-parameter case when K ( t ) K(t) is a Calderón-Zygmund kernel. Street and Stein generalized their work to the multi-parameter case, and gave sufficient conditions for the L p L^p -boundedness of such operators. This paper shows that when γ t ( x ) \gamma _t(x) is real analytic, the sufficient conditions of Street and Stein are also necessary for the L p L^p -boundedness of T T , for all such kernels K K .

show abstract

Real analytic multi-parameter singular Radon transforms: Necessity of the Stein-Street condition

Zhang

2022

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.

Contact Info

customersupport@researchsolutions.com

10624 S. Eastern Ave., Ste. A-614

Henderson, NV 89052, USA

Blog Terms and Conditions API Terms Privacy Policy Contact Cookie Preferences Do Not Sell or Share My Personal Information

Made with 💙 for researchers

Part of the Research Solutions Family.

A T(1) theorem for singular Radon transforms

Abstract: We prove an analogue for singular Radon transforms to the T (1) theorem of David and Journe.

Cited by 1 publication

References 18 publications

Real analytic multi-parameter singular Radon transforms: Necessity of the Stein-Street condition

Real analytic multi-parameter singular Radon transforms: Necessity of the Stein-Street condition

Contact Info

Product

Resources

About