# A polynomial Carleson operator along the paraboloid

@article{Pierce2019APC, title={A polynomial Carleson operator along the paraboloid}, author={L. B. Pierce and Po-Lam Yung}, journal={Revista Matem{\'a}tica Iberoamericana}, year={2019} }

In this work we extend consideration of the polynomial Carleson operator to the setting of a Radon transform acting along the paraboloid in $\mathbb{R}^{n+1}$ for $n \geq 2$. Inspired by work of Stein and Wainger on the original polynomial Carleson operator, we develop a method to treat polynomial Carleson operators along the paraboloid via van der Corput estimates. A key new step in the approach of this paper is to approximate a related maximal oscillatory integral operator along the…

## 9 Citations

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## References

SHOWING 1-10 OF 13 REFERENCES

Pointwise Convergence of Fourier Series

- Mathematics
- 1973

In this paper, we present a new proof of a theorem of Carleson and Hunt: The Fourier series of an LP function on [0, 2J] converges almost everywhere (p > 1). (See [1], [51.) Our proof is very much in…

Convergence almost everywhere of certain singular integrals and multiple Fourier series

- Mathematics
- 1971

T s Ts = o ( l o g l o g I~]), t~]-+ 0% for almost eve ry x in T,. I n Sections 1 to 3 in this pape r we prove among o ther things the following theorem, which generalizes the L p es t imate of the…

Problems in harmonic analysis related to curvature

- Mathematics
- 1978

X \R(1 r\\ f /OO # / ( * ) --> e-*0 \B(e9 X)\ JB(t,x) for certain types of «-dimensional sets B(e9 x) shrinking to x as e -* 0. (\B(e9 x)\ is of course the Lebesgue measure of B(e9 x).) Some standard…

The (Weak-L2) Boundedness of the Quadratic Carleson Operator

- Mathematics
- 2007

We prove that the generalized Carleson operator with polynomial phase function of degree two is of weak type (2,2). For this, we introduce a new approach to the time-frequency analysis of the…

A proof of boundedness of the Carleson operator

- Mathematics
- 2000

We give a simplified proof that the Carleson operator is of weak type (2, 2). This estimate is the main ingredient in the proof of Carleson’s theorem on almost everywhere convergence of Fourier…

Harmonic analysis on nilpotent groups and singular integrals I. Oscillatory integrals

- Mathematics
- 1987

Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals

- Mathematics
- 1993

PrefaceGuide to the ReaderPrologue3IReal-Variable Theory7IIMore About Maximal Functions49IIIHardy Spaces87IVH[superscript 1] and BMO139VWeighted Inequalities193VIPseudo-Differential and Singular…

On the convergence of Fourier series

- Mathematics
- 1984

We define the space Bp={f:(−π,π]→R, f(t)=∑n=0∞cnbn(t), ∑n=0∞|cn|<∞}. Each bn is a special p-atom, that is, a real valued function, defined on (−π,π], which is either b(t)=1/2π or…

THE SURVEY OF H i IN EXTREMELY LOW-MASS DWARFS (SHIELD)

- Physics
- 2011

We present first results from the Survey of H i in Extremely Low-mass Dwarfs (SHIELD), a multi-configuration Expanded Very Large Array (EVLA) study of the neutral gas contents and dynamics of…