Zhong Gao

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Benchmarks and Challenges in Pose Estimation for Egocentric Hand Interactions with Objects

2024-03-25

Zicong Fan , Takehiko Ohkawa , Linlin Yang +7 more

We interact with the world with our hands and see it through our own (egocentric) perspective. A holistic 3D understanding of such interactions from egocentric views is important for tasks … We interact with the world with our hands and see it through our own (egocentric) perspective. A holistic 3D understanding of such interactions from egocentric views is important for tasks in robotics, AR/VR, action recognition and motion generation. Accurately reconstructing such interactions in 3D is challenging due to heavy occlusion, viewpoint bias, camera distortion, and motion blur from the head movement. To this end, we designed the HANDS23 challenge based on the AssemblyHands and ARCTIC datasets with carefully designed training and testing splits. Based on the results of the top submitted methods and more recent baselines on the leaderboards, we perform a thorough analysis on 3D hand(-object) reconstruction tasks. Our analysis demonstrates the effectiveness of addressing distortion specific to egocentric cameras, adopting high-capacity transformers to learn complex hand-object interactions, and fusing predictions from different views. Our study further reveals challenging scenarios intractable with state-of-the-art methods, such as fast hand motion, object reconstruction from narrow egocentric views, and close contact between two hands and objects. Our efforts will enrich the community's knowledge foundation and facilitate future hand studies on egocentric hand-object interactions.

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Restriction estimates for hyperbolic cone in high dimensions

2023-02-15

Zhong Gao , Jiqiang Zheng

In this paper, we obtain the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript p"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">L^p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> restriction estimates for the truncated conic … In this paper, we obtain the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript p"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">L^p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> restriction estimates for the truncated conic surface <disp-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Sigma equals StartSet left-parenthesis xi prime comma xi Subscript n Baseline comma minus xi Subscript n Superscript negative 1 Baseline mathematical left-angle xi prime comma upper N xi Superscript prime Baseline mathematical right-angle right-parenthesis colon left-parenthesis xi prime comma xi Subscript n Baseline right-parenthesis element-of upper B Superscript n minus 1 Baseline left-parenthesis 0 comma 1 right-parenthesis times left-bracket 1 comma 2 right-bracket EndSet"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal">Σ</mml:mi> <mml:mo>=</mml:mo> <mml:mstyle scriptlevel="0"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo maxsize="1.2em" minsize="1.2em">{</mml:mo> </mml:mrow> </mml:mstyle> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>ξ</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>ξ</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>−</mml:mo> <mml:msubsup> <mml:mi>ξ</mml:mi> <mml:mi>n</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msubsup> <mml:mo fence="false" stretchy="false">⟨</mml:mo> <mml:msup> <mml:mi>ξ</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo>,</mml:mo> <mml:mi>N</mml:mi> <mml:msup> <mml:mi>ξ</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo fence="false" stretchy="false">⟩</mml:mo> <mml:mo stretchy="false">)</mml:mo> <mml:mo>:</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>ξ</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>ξ</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mo>∈</mml:mo> <mml:msup> <mml:mi>B</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mo>×</mml:mo> <mml:mo stretchy="false">[</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">]</mml:mo> <mml:mstyle scriptlevel="0"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo maxsize="1.2em" minsize="1.2em">}</mml:mo> </mml:mrow> </mml:mstyle> </mml:mrow> <mml:annotation encoding="application/x-tex">\begin{equation*} \Sigma =\big \{(\xi ’,\xi _n,-\xi _n^{-1}\langle \xi ’,N\xi ’\rangle ): (\xi ’,\xi _n)\in B^{n-1}(0,1)\times [1,2]\big \} \end{equation*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N equals upper I Subscript n minus 1 minus m Baseline circled-plus left-parenthesis minus upper I Subscript m Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>I</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> <mml:mo>−</mml:mo> <mml:mi>m</mml:mi> </mml:mrow> </mml:msub> <mml:mo>⊕</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mo>−</mml:mo> <mml:msub> <mml:mi>I</mml:mi> <mml:mi>m</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">N=I_{n-1-m}\oplus (-I_m)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m less-than-or-equal-to left floor StartFraction n minus 3 Over 2 EndFraction right floor"> <mml:semantics> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>≤</mml:mo> <mml:mo fence="false" stretchy="false">⌊</mml:mo> <mml:mstyle displaystyle="false" scriptlevel="0"> <mml:mfrac> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>−</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mstyle> <mml:mo fence="false" stretchy="false">⌋</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">m\leq \lfloor \tfrac {n-3}2\rfloor</mml:annotation> </mml:semantics> </mml:math> </inline-formula> provided <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p greater-than StartFraction 2 left-parenthesis n plus 3 right-parenthesis Over n plus 1 EndFraction"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mstyle displaystyle="false" scriptlevel="0"> <mml:mfrac> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>3</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:mfrac> </mml:mstyle> </mml:mrow> <mml:annotation encoding="application/x-tex">p>\tfrac {2(n+3)}{n+1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The main ingredients of the proof are the bilinear estimates of strongly separated property and a geometric distribution about caps.

Fourier Restriction Implies Maximal and Variational Fourier Restriction in Lorentz Space

2024-04-02

Zhong Gao , Chengbin Xu , Jiqiang Zheng

In this paper, we show that Fourier restriction theorem implies the maximal and variational Fourier restriction in Lorentz space frame. The key ingredient is that we work out the induction … In this paper, we show that Fourier restriction theorem implies the maximal and variational Fourier restriction in Lorentz space frame. The key ingredient is that we work out the induction argument in Lorentz space, based on some good properties of Lorentz space. This extends Kovač's result [J. Funct. Anal., 2019, 277(10): 3355–3372] to Lorentz space frame.

Hörmander oscillatory integral operators: A revisit

2024-12-02

Chuanwei Gao , Zhong Gao , Changxing Miao

Hörmander oscillatory integral operators: A revisit

2024-12-02

Chuanwei Gao , Zhong Gao , Changxing Miao

Fourier Restriction Implies Maximal and Variational Fourier Restriction in Lorentz Space

2024-04-02

Zhong Gao , Chengbin Xu , Jiqiang Zheng

Benchmarks and Challenges in Pose Estimation for Egocentric Hand Interactions with Objects

2024-03-25

Zicong Fan , Takehiko Ohkawa , Linlin Yang +7 more

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Restriction estimates for hyperbolic cone in high dimensions

2023-02-15

Zhong Gao , Jiqiang Zheng

Coauthor	Papers Together
Jiqiang Zheng	2
Angela Yao	1
Jiajun Liang	1
Otmar Hilliges	1
Aditya Prakash	1
Takehiko Ohkawa	1
Shihao Zhou	1
Xuanyang Zhang	1
Karim Abou Zeid	1
Chuanwei Gao	1
Yoichi Sato	1
Bastian Leibe	1
Feng Lu	1
Kun He	1
Xue Zhang	1
Saurabh Gupta	1
Zheng Liu	1
Seungryul Baek	1
Lin Nie	1
Linlin Yang	1
Zicong Fan	1
Zhishan Zhou	1
Jeongwan On	1
Chengbin Xu	1
Changxing Miao	1
Hyung Jin Chang	1
Fei Li	1

Bounds on Oscillatory Integral Operators Based on Multilinear Estimates

2011-11-04

Jean Bourgain , Larry Guth

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Bilinear restriction estimates for surfaces with curvatures of different signs

2005-08-01

Sanghyuk Lee

Recently, the sharp <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L squared"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">L^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-bilinear (adjoint) restriction estimates for the cone and the paraboloid were … Recently, the sharp <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L squared"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">L^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-bilinear (adjoint) restriction estimates for the cone and the paraboloid were established by Wolff and Tao, respectively. Their results rely on the fact that for the cone and the paraboloid, the nonzero principal curvatures have the same sign. We generalize those bilinear restriction estimates to surfaces with curvatures of different signs.

A Sharp Bilinear Cone Restriction Estimate

2001-05-01

Thomas Wolff

The purpose of this paper is to prove an essentially sharp L^2 Fourier restriction estimate for light cones, of the type which is called bilinear in the recent literature. The purpose of this paper is to prove an essentially sharp L^2 Fourier restriction estimate for light cones, of the type which is called bilinear in the recent literature.

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A sharp bilinear restriction estimate for paraboloids

2003-12-01

Terence Tao

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Besicovitch type maximal operators and applications to fourier analysis

1991-06-01

Jean Bourgain

Restriction estimates for hyperbolic paraboloids in higher dimensions via bilinear estimates

2021-11-02

Alex Barron

Let \mathbb{H} be a (d-1) -dimensional hyperbolic paraboloid in \mathbb{R}^d and let Ef be the Fourier extension operator associated to \mathbb{H} , with f supported in B^{d-1}(0,2) . We prove … Let \mathbb{H} be a (d-1) -dimensional hyperbolic paraboloid in \mathbb{R}^d and let Ef be the Fourier extension operator associated to \mathbb{H} , with f supported in B^{d-1}(0,2) . We prove that \lVert Ef \rVert_{L^p (B(0,R))} \leq C_{\varepsilon}R^{\varepsilon}\lVert f \rVert_{L^p} for all p \geq 2(d+2)/d whenever d/2\geq m + 1 , where m is the minimum between the number of positive and negative principal curvatures of \mathbb{H} . Bilinear restriction estimates for \mathbb{H} proved by S. Lee and Vargas play an important role in our argument.

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A cone restriction estimate using polynomial partitioning

2021-11-17

Yumeng Ou , Hong Wang

We obtain an improved Fourier restriction estimate for a truncated cone using the method of polynomial partitioning in dimension n\geq 3 , which in particular solves the cone restriction conjecture … We obtain an improved Fourier restriction estimate for a truncated cone using the method of polynomial partitioning in dimension n\geq 3 , which in particular solves the cone restriction conjecture for n=5 , and recovers the sharp range for 3\leq n\leq 4 . The main ingredient of the proof is a k -broad estimate for the cone extension operator, which is a weak version of the k -linear cone restriction conjecture for 2\leq k\leq n .

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A restriction estimate in R3 using brooms

2022-04-14

Hong Wang

If f is a function supported on the truncated paraboloid in R3 and E is the corresponding extension operator, then we prove that for all p>3+3∕13, ‖Ef‖Lp(R3)≤C‖f‖L∞. The proof combines … If f is a function supported on the truncated paraboloid in R3 and E is the corresponding extension operator, then we prove that for all p>3+3∕13, ‖Ef‖Lp(R3)≤C‖f‖L∞. The proof combines Wolff's two ends argument with polynomial partitioning techniques. We also observe some geometric structures in the wave packets.

Restriction estimates using polynomial partitioning II

2018-01-01

Larry Guth

We improve the estimates in the restriction problem in dimension $n \geqslant 4$. To do so, we establish a weak version of a $k$-linear restriction estimate for any $k$. The … We improve the estimates in the restriction problem in dimension $n \geqslant 4$. To do so, we establish a weak version of a $k$-linear restriction estimate for any $k$. The exponents in this weak $k$-linear estimate are sharp for all $k$ and $n$.

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On the restriction of the Fourier transform to a conical surface

1985-01-01

Bartolome Barcelo Taberner

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma"> <mml:semantics> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:annotation encoding="application/x-tex">\Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the surface of a circular cone in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" … Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma"> <mml:semantics> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:annotation encoding="application/x-tex">\Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the surface of a circular cone in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper R cubed"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">R</mml:mi> </mml:mrow> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{{\mathbf {R}}^3}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We show that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 less-than-or-slanted-equals p greater-than 4 slash 3"> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>⩽</mml:mo> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mn>4</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">1 \leqslant p > 4/3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 slash q equals 3 left-parenthesis 1 minus 1 slash p right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>q</mml:mi> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>p</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">1/q = 3(1 - 1/p)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f element-of upper L Superscript p Baseline left-parenthesis bold upper R cubed right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>∈</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">R</mml:mi> </mml:mrow> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">f \in {L^p}({{\mathbf {R}}^3})</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, then the Fourier transform of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f"> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding="application/x-tex">f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> belongs to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript q Baseline left-parenthesis normal upper Gamma comma d sigma right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>q</mml:mi> </mml:msup> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>d</mml:mi> <mml:mi>σ</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{L^q}(\Gamma ,d\sigma )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for a certain natural measure <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma"> <mml:semantics> <mml:mi>σ</mml:mi> <mml:annotation encoding="application/x-tex">\sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma"> <mml:semantics> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:annotation encoding="application/x-tex">\Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Following P. Tomas we also establish bounds for restrictions of Fourier transforms to conic annuli at the endpoint <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p equals 4 slash 3"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mn>4</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p = 4/3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, with logarithmic growth of the bound as the thickness of the annulus tends to zero.

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A restriction estimate using polynomial partitioning

2015-02-25

Larry Guth

If <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a smooth compact surface in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R cubed"> <mml:semantics> <mml:msup> … If <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a smooth compact surface in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R cubed"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {R}^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with strictly positive second fundamental form, and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E Subscript upper S"> <mml:semantics> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>S</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">E_S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the corresponding extension operator, then we prove that for all <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p greater-than 3.25"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mn>3.25</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p > 3.25</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar upper E Subscript upper S Baseline f double-vertical-bar Subscript upper L Sub Superscript p Subscript left-parenthesis double-struck upper R cubed right-parenthesis Baseline less-than-or-equal-to upper C left-parenthesis p comma upper S right-parenthesis double-vertical-bar f double-vertical-bar Subscript upper L Sub Superscript normal infinity Subscript left-parenthesis upper S right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">‖</mml:mo> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>S</mml:mi> </mml:msub> <mml:mi>f</mml:mi> <mml:msub> <mml:mo fence="false" stretchy="false">‖</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msub> <mml:mo>≤</mml:mo> <mml:mi>C</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mi>S</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo fence="false" stretchy="false">‖</mml:mo> <mml:mi>f</mml:mi> <mml:msub> <mml:mo fence="false" stretchy="false">‖</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi mathvariant="normal">∞</mml:mi> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>S</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\| E_S f\|_{L^p(\mathbb {R}^3)} \le C(p,S) \| f \|_{L^\infty (S)}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The proof uses polynomial partitioning arguments from incidence geometry.

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A restriction theorem for the Fourier transform

1975-01-01

Peter A. Tomas

Let ƒ be a Schwartz function on R n , and let ƒ(0) denote the restriction of the Fourier transform of ƒ to the unit sphere S n ~x in … Let ƒ be a Schwartz function on R n , and let ƒ(0) denote the restriction of the Fourier transform of ƒ to the unit sphere S n ~x in R n .We prove THEOREM.Iff is in L p (R n ) for some p with 1 < p < 2(n +1)/(« + 3), then 5 s n-i\Ke)\ 2 de<c p \\f\\ 2 p .PROOF.Jl/(0)l 2 d0 = ƒƒ* f(x)fà(x)dx = fmdè*f(x)dx<\\f\\ p \\âd *f\\ p , for conjugate indices p and p .Thus it suffices to prove that the operator given by convolution with *3$ is bounded from LP to LP for p in the appropriate range.Let K(x) be a radial Schwartz function with K(x) = 1 for \x\ < 100, and let).It suffices to show there exists e = e(p) > 0 such that \\T k * ƒ \\ p , < C2~€ fc || ƒ || p .This follows from interpolating the estimates \\T k * ƒ IL < C2-( "~1>*/ 2 || f\\ x and ||r fc *f\\ 2 < 2 k \\f\\ 2 .Professor E. M. Stein has extended the range of this result to include p = 2(n + l)/(n + 3).His proof uses complex interpolation of the operators given by convolution with the functions B a (x) = J 0 (27t\x\)/\x\°.Then A great deal was previously known about such restriction theorems.E. M. Stein originally established the theorem for 1 </? < 4n/(3n + 1).For n = 2, this was extended by Fefferman and Stein [2] to the range 1 < p < 6/5.P. Sjolin (see [1]) proved the theorem for n = 3 and 1 <p < 4/3.Finally, A. Zygmund [3] determined for two dimensions all p and q such that the Fourier transform of an LP function restricts to L q (S x ).Since a

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Some Maximal Inequalities

1971-01-01

Charles Fefferman , E. M. Stein

On Fourier coefficients and transforms of functions of two variables

1974-01-01

A. Zygmund

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L p -Estimates for oscillatory integrals in several variables

1991-12-01

Jean Bourgain

Endpoint Strichartz estimates

1998-10-01

M. Keel , Terence Tao

We prove an abstract Strichartz estimate, which implies previously unknown endpoint Strichartz estimates for the wave equation (in dimension n ≥ 4) and the Schrödinger equation (in dimension n ≥ … We prove an abstract Strichartz estimate, which implies previously unknown endpoint Strichartz estimates for the wave equation (in dimension n ≥ 4) and the Schrödinger equation (in dimension n ≥ 3). Three other applications are discussed: local existence for a nonlinear wave equation; and Strichartz-type estimates for more general dispersive equations and for the kinetic transport equation.

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A bilinear approach to the restriction and Kakeya conjectures

1998-01-01

Terence Tao , Ana Vargas , Luis Vega

Bilinear restriction estimates have appeared in work of Bourgain, Klainerman, and Machedon. In this paper we develop the theory of these estimates (together with the analogues for Kakeya estimates). As … Bilinear restriction estimates have appeared in work of Bourgain, Klainerman, and Machedon. In this paper we develop the theory of these estimates (together with the analogues for Kakeya estimates). As a consequence we improve the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper L Superscript p Baseline comma upper L Superscript p Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:mo>,</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(L^p,L^p)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> spherical restriction theorem of Wolff from <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p greater-than 42 slash 11"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mn>42</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>11</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p > 42/11</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p greater-than 34 slash 9"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mn>34</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>9</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p > 34/9</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and also obtain a sharp <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper L Superscript p Baseline comma upper L Superscript q Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:mo>,</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>q</mml:mi> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(L^p,L^q)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> spherical restriction theorem for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q greater-than 4 minus five twenty-sevenths"> <mml:semantics> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>></mml:mo> <mml:mn>4</mml:mn> <mml:mo>−</mml:mo> <mml:mfrac> <mml:mn>5</mml:mn> <mml:mn>27</mml:mn> </mml:mfrac> </mml:mrow> <mml:annotation encoding="application/x-tex">q> 4 - \frac {5}{27}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.

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A note on maximal Fourier restriction for spheres in all dimensions

2022-12-30

Marco Vitturi

We prove a maximal Fourier restriction theorem for hypersurfaces in $\mathbb{R}^{d}$ for any dimension $d\geq 3$ in a restricted range of exponents given by the Tomas-Stein theorem (spheres being the … We prove a maximal Fourier restriction theorem for hypersurfaces in $\mathbb{R}^{d}$ for any dimension $d\geq 3$ in a restricted range of exponents given by the Tomas-Stein theorem (spheres being the most canonical example). The proof consists of a simple observation. When $d=3$ the range corresponds exactly to the full Tomas-Stein one, but is otherwise a proper subset when $d>3$. We also present an application regarding the Lebesgue points of functions in $\mathcal{F}(L^p)$ when $p$ is sufficiently close to 1.

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A maximal restriction theorem and Lebesgue points of functions in $\mathcal F(L^p)$

2019-04-01

Detlef Müller , Fulvio Ricci , James Wright

Fourier restriction theorems, whose study had been initiated by E.M. Stein, usually describe a family of a priori estimates of the L^q -norm of the restriction of the Fourier transform … Fourier restriction theorems, whose study had been initiated by E.M. Stein, usually describe a family of a priori estimates of the L^q -norm of the restriction of the Fourier transform of a function f in L^p(\mathbb R^n) to a given subvariety S , endowed with a suitable measure. Such estimates allow to define the restriction \mathcal{R} f of the Fourier transform of an L^p -function to S in an operator theoretic sense. In this article, we begin to investigate the question what is the „intrinsic" pointwise relation between \mathcal{R} f and the Fourier transform of f , by looking at curves in the plane, for instance with non-vanishing curvature. To this end, we bound suitable maximal operators, including the Hardy–Littlewood maximal function of the Fourier transform of f restricted to S .

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A variational restriction theorem

2021-05-07

Vjekoslav Kovač , Diogo Oliveira e Silva

We establish variational estimates related to the problem of restricting the Fourier transform of a three-dimensional function to the two-dimensional Euclidean sphere. At the same time, we give a short … We establish variational estimates related to the problem of restricting the Fourier transform of a three-dimensional function to the two-dimensional Euclidean sphere. At the same time, we give a short survey of the recent field of maximal Fourier restriction theory.

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Fourier restriction implies maximal and variational Fourier restriction

2019-04-01

Vjekoslav Kovač

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Improved restriction estimate for hyperbolic surfaces in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">R</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:math>

2017-05-03

Chu-Hee Cho , Jungjin Lee

Sharp $L^2$ estimates of the Schrödinger maximal function in higher dimensions

2019-05-01

Xiumin Du , Ruixiang Zhang

We show that, for $n\ge 3$, $\mathrm{lim}_{t\to 0} e^{it\Delta} f(x) f(x)$ holds almost everywhere for all $f\in H^s(\mathbb{R}^n)$ provided that $s> \frac{n}{2(n+1)}$. Due to a counterexample by Bourgain, up to … We show that, for $n\ge 3$, $\mathrm{lim}_{t\to 0} e^{it\Delta} f(x) f(x)$ holds almost everywhere for all $f\in H^s(\mathbb{R}^n)$ provided that $s> \frac{n}{2(n+1)}$. Due to a counterexample by Bourgain, up to the endpoint, this result is sharp and fully resolves a problem raised by Carleson. Our main theorem is a fractal $L^2$ restriction estimate, which also gives improved results on the size of the divergence set of the Schrödinger solutions, the Falconer distance set problem and the spherical average Fourier decay rates of fractal measures. The key ingredients of the proof include multilinear Kakeya estimates, decoupling and induction on scales.

Extensions of the Stein–Tomas Theorem

2011-01-01

Jong-Guk Bak , Andreas Seeger

We prove an endpoint version of the Stein-Tomas restriction theorem, for a general class of measures, and with a strengthened Lorentz space estimate.A similar improvement is obtained for Stein's estimate … We prove an endpoint version of the Stein-Tomas restriction theorem, for a general class of measures, and with a strengthened Lorentz space estimate.A similar improvement is obtained for Stein's estimate on oscillatory integrals of Carleson-Sjölin-Hörmander type and some spectral projection operators on compact manifolds, and for classes of oscillatory integral operators with one-sided fold singularities.

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A sharp Schrödinger maximal estimate in $\mathbb{R}^2$

2017-07-31

Xiumin Du , Larry Guth , Xiaochun Li

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Convolution operators and L(p,q) spaces

1963-03-01

Richard O’Neil

Decouplings for curves and hypersurfaces with nonzero Gaussian curvature

2017-10-01

Jean Bourgain , Ciprian Demeter

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Linear adjoint restriction estimates for paraboloid

2019-02-09

Changxing Miao , Junyong Zhang , Jiqiang Zheng

We prove a class of modified paraboloid restriction estimates with a loss of angular derivatives for the full set of paraboloid restriction conjecture indices. This result generalizes the paraboloid restriction … We prove a class of modified paraboloid restriction estimates with a loss of angular derivatives for the full set of paraboloid restriction conjecture indices. This result generalizes the paraboloid restriction estimate in radial case from [Shao, Rev. Mat. Iberoam. 25(2009), 1127–1168], as well as the result from [Miao et al. Proc. AMS 140(2012), 2091–2102]. As an application, we show a local smoothing estimate for a solution of the linear Schrödinger equation under the assumption that the initial datum has additional angular regularity.

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Fourier restriction to a hyperbolic cone

2020-04-09

Benjamin Baker Bruce

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A Fourier restriction theorem for a perturbed hyperbolic paraboloid

2019-08-05

Stefan Buschenhenke , Detlef Müller , Ana Vargas

In contrast to elliptic surfaces, the Fourier restriction problem for hypersurfaces of non-vanishing Gaussian curvature which admit principal curvatures of opposite signs is still hardly understood. In fact, even for … In contrast to elliptic surfaces, the Fourier restriction problem for hypersurfaces of non-vanishing Gaussian curvature which admit principal curvatures of opposite signs is still hardly understood. In fact, even for 2-surfaces, the only case of a hyperbolic surface for which Fourier restriction estimates could be established that are analogous to the ones known for elliptic surfaces is the hyperbolic paraboloid or 'saddle' z = x y . The bilinear method gave here sharp results for p > 10 / 3 , and this result was recently improved to p > 3.25 . This paper aims to be the first step in extending those results to more general hyperbolic surfaces. We consider a specific cubic perturbation of the saddle and obtain the sharp result, up to the endpoint, for p > 10 / 3 . In the application of the bilinear method, we show that the behavior at small scales in our surface is drastically different from the saddle. Indeed, as it turns out, in some regimes the perturbation term assumes a dominant role, which necessitates the introduction of a number of new techniques that should also be useful for the study of more general hyperbolic surfaces. This specific perturbation has turned out to be of fundamental importance also to the understanding of more general classes of perturbations.

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Improved Fourier restriction estimates in higher dimensions

2019-01-01

Jonathan Hickman , Keith M. Rogers

We consider Guth's approach to the Fourier restriction problem via polynomial partitioning.By writing out his induction argument as a recursive algorithm and introducing new geometric information, known as the polynomial … We consider Guth's approach to the Fourier restriction problem via polynomial partitioning.By writing out his induction argument as a recursive algorithm and introducing new geometric information, known as the polynomial Wolff axioms, we obtain improved bounds for the restriction conjecture, particularly in high dimensions.Consequences for the Kakeya conjecture are also considered.

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Maximal restriction estimates and the maximal function of the Fourier transform

2019-10-30

João P. G. Ramos

We prove inequalities concerning the restriction of the strong maximal function of the Fourier transform to the circle, providing an answer to a question left open by Müller, Ricci, and … We prove inequalities concerning the restriction of the strong maximal function of the Fourier transform to the circle, providing an answer to a question left open by Müller, Ricci, and Wright. We employ methods similar in spirit to the classical proofs of the two-dimensional restriction theorem, with the addition of a suitable trick to help us linearise our maximal function. In the end, we comment on how to use the same linearisation trick in combination with Vitturi’s duality argument to obtain sharper high-dimensional results for the Hardy–Littlewood maximal function.

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Sharp estimates for oscillatory integral operators via polynomial partitioning

2019-01-01

Larry Guth , Jonathan Hickman , Marina Iliopoulou

2 ) Given a (possibly empty) list of objects L, for real numbers Ap, Bp 0 depending on some Lebesgue exponent p, the notation Ap L Bp or Bp L … 2 ) Given a (possibly empty) list of objects L, for real numbers Ap, Bp 0 depending on some Lebesgue exponent p, the notation Ap L Bp or Bp L Ap signifies that Ap CBp for some constant C =C L,n,p 0 depending on the objects in the list, n and p.In addition, Ap ∼ L Bp is used to signify that Ap L Bp and Ap L Bp.( 4 ) In particular, Lee [19] proved that for positive-definite phases (1.4) holds for p 2(n+2)/n in all dimensions, extending the range in Theorem 1.1 when n is odd.( 7 ) The connection with Bochner-Riesz multipliers is made via the classical reduction of Carleson-Sjölin [10], [17], as discussed in the previous subsection.

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A Fourier restriction theorem for a perturbed hyperbolic paraboloid: polynomial partitioning

2022-02-07

Stefan Buschenhenke , Detlef Müller , Ana Vargas

Abstract We consider a surface with negative curvature in $${{\mathbb {R}}}^3,$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> which is a cubic perturbation of the … Abstract We consider a surface with negative curvature in $${{\mathbb {R}}}^3,$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> which is a cubic perturbation of the saddle. For this surface, we prove a new restriction theorem, analogous to the theorem for paraboloids proved by L. Guth in 2016. This specific perturbation has turned out to be of fundamental importance also to the understanding of more general classes of one-variate perturbations, and we hope that the present paper will further help to pave the way for the study of general perturbations of the saddle by means of the polynomial partitioning method.

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Fourier restriction for smooth hyperbolic 2-surfaces

2022-08-27

Stefan Buschenhenke , Detlef Müller , Ana Vargas

Abstract We prove Fourier restriction estimates by means of the polynomial partitioning method for compact subsets of any sufficiently smooth hyperbolic hypersurface in $$\mathbb {R}^3.$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mrow> … Abstract We prove Fourier restriction estimates by means of the polynomial partitioning method for compact subsets of any sufficiently smooth hyperbolic hypersurface in $$\mathbb {R}^3.$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> <mml:mo>.</mml:mo> </mml:mrow> </mml:math> Our approach exploits in a crucial way the underlying hyperbolic geometry, which leads to a novel notion of strong transversality and corresponding “exceptional” sets. For the division of these exceptional sets we make crucial and perhaps surprising use of a lemma on level sets for sufficiently smooth one-variate functions from a previous article of ours.

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Partitions of Flat One-Variate Functions and a Fourier Restriction Theorem for Related Perturbations of the Hyperbolic Paraboloid

2021-02-18

Stefan Buschenhenke , Detlef Müller , Ana Vargas

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Princeton Mathematical Series

1948-12-01

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Classical Fourier Analysis

2008-01-01

Loukas Grafakos

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Microlocal decoupling inequalities and the distance problem on Riemannian manifolds

2022-12-01

Alex Iosevich , Bochen Liu , Yakun Xi

We study the generalization of the Falconer distance problem to the Riemannian setting. In particular, we extend the result of Guth--Iosevich--Ou--Wang for the distance set in the plane to general … We study the generalization of the Falconer distance problem to the Riemannian setting. In particular, we extend the result of Guth--Iosevich--Ou--Wang for the distance set in the plane to general Riemannian surfaces. Key new ingredients include a family of refined microlocal decoupling inequalities, which are related to the work of Beltran--Hickman--Sogge on Wolff-type inequalities, and an analog of Orponen's radial projection lemma which has proved quite useful in recent work on distance sets.

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A Stein--Tomas restriction theorem for general measures

2002-01-01

Themis Mitsis

We prove a general Stein-Tomas type restriction theorem for measures of given dimension and Fourier exponent. We prove a general Stein-Tomas type restriction theorem for measures of given dimension and Fourier exponent.

Improved local smoothing estimate for the wave equation in higher dimensions

2023-02-14

Chuanwei Gao , Bochen Liu , Changxing Miao +1 more

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A note on the Schrödinger maximal function

2016-11-01

Jean Bourgain

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Oscillatory integrals and multipliers on FLp

1973-12-01

Lars Hörmander

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Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations

1977-09-01

Robert S. Strichartz

Linear and bilinear estimates for oscillatory integral operators related to restriction to hypersurfaces

2006-07-19

Sanghyuk Lee

Maximal Functions Associated to Filtrations

2001-02-01

Michael Christ , Alexander Kiselev

The Bochner-Riesz conjecture implies the restriction conjecture

1999-02-01

Terence Tao

L p regularity of averages over curves and bounds for associated maximal operators

2007-02-01

Malabika Pramanik , Andreas Seeger

We prove that for a finite type curve in â3 the maximal operator generated by dilations is bounded on Lp for sufficiently large p. We also show the endpoint Lp … We prove that for a finite type curve in â3 the maximal operator generated by dilations is bounded on Lp for sufficiently large p. We also show the endpoint Lp â Lp1/p regularity result for the averaging operators for large p. The proofs make use of a deep result of Thomas Wolff about decompositions of cone multipliers.

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Some analytic problems related to statistical mechanics

1980-01-01

Lennart Carleson

Lower bounds at infinity for solutions of differential equations with constant coefficients

1973-03-01

Lars Hörmander