43 results on '"PO-LAM YUNG"'
Search Results
2. MATRIX-MFO Tandem Workshop/Small Collaboration: Rough Wave Equations
- Author
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Dorothee Frey, Zihua Guo, Andrew Hassell, Pierre Portal, Jan Rozendaal, and Po-Lam Yung
- Subjects
General Medicine - Published
- 2022
3. A short proof of ℓ2 decoupling for the moment curve
- Author
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Shaoming Guo, Zane Kun Li, Po-Lam Yung, and Pavel Zorin-Kranich
- Subjects
Mathematics - Number Theory ,Mathematics - Classical Analysis and ODEs ,General Mathematics ,010102 general mathematics ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Number Theory (math.NT) ,11L07 (Primary) 11L15, 42B25, 26D05 (Secondary) ,0101 mathematics ,01 natural sciences - Abstract
We give a short and elementary proof of the $\ell^{2}$ decoupling inequality for the moment curve in $\mathbb{R}^k$, using a bilinear approach inspired by the nested efficient congruencing argument of Wooley (arXiv:1708.01220)., v2: 10 pages, minor corrections
- Published
- 2021
4. Spaces of Besov-Sobolev type and a problem on nonlinear approximation
- Author
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Óscar Domínguez, Andreas Seeger, Brian Street, Jean Van Schaftingen, Po-Lam Yung, and UCL - SST/IRMP - Institut de recherche en mathématique et physique
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Mathematics - Functional Analysis ,Mathematics - Classical Analysis and ODEs ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Mathematics - Numerical Analysis ,Numerical Analysis (math.NA) ,Analysis ,Functional Analysis (math.FA) - Abstract
We study fractional variants of the quasi-norms introduced by Brezis, Van Schaftingen, and Yung in the study of the Sobolev space $\dot W^{1,p}$. The resulting spaces are identified as a special class of real interpolation spaces of Sobolev-Slobodecki\u{\i} spaces. We establish the equivalence between Fourier analytic definitions and definitions via difference operators acting on measurable functions. We prove various new results on embeddings and non-embeddings, and give applications to harmonic and caloric extensions. For suitable wavelet bases we obtain a characterization of the approximation spaces for best $n$-term approximation from a wavelet basis via smoothness conditions on the function; this extends a classical result by DeVore, Jawerth and Popov., Comment: 45 pages
- Published
- 2023
5. Sobolev spaces revisited
- Author
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Haïm Brezis, Andreas Seeger, Jean Van Schaftingen, Po-Lam Yung, and UCL - SST/IRMP - Institut de recherche en mathématique et physique
- Subjects
Mathematics - Functional Analysis ,Mathematics - Classical Analysis and ODEs ,General Mathematics ,Mathematics::Analysis of PDEs ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Functional Analysis (math.FA) - Abstract
We describe a recent, one-parameter family of characterizations of Sobolev and BV functions on $\mathbb{R}^n$, using sizes of superlevel sets of suitable difference quotients. This provides an alternative point of view to the BBM formula by Bourgain, Brezis and Mironescu, and complements in the case of BV some results of Cohen, Dahmen, Daubechies and DeVore about the sizes of wavelet coefficients of such functions. An application towards Gagliardo-Nirenberg interpolation inequalities is then given. We also establish a related one-parameter family of formulae for the $L^p$ norm of functions in $L^p(\mathbb{R}^n)$., Comment: 21 pages
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- 2022
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6. Decoupling for two quadratic forms in three variables: a complete characterization.
- Author
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Shaoming Guo, Changkeun Oh, Roos, Joris, Po-Lam Yung, and Zorin-Kranich, Pavel
- Subjects
MATHEMATICAL decoupling ,QUADRATIC forms ,CLASSIFICATION - Abstract
We prove sharp decoupling inequalities for all degenerate surfaces of codimension two in R
5 given by two quadratic forms in three variables. Together with previous work by Demeter, Guo, and Shi in the non-degenerate case, this provides a classification of decoupling inequalities for pairs of quadratic forms in three variables. [ABSTRACT FROM AUTHOR]- Published
- 2023
- Full Text
- View/download PDF
7. A polynomial Carleson operator along the paraboloid
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Lillian B. Pierce and Po-Lam Yung
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Class (set theory) ,Pure mathematics ,Paraboloid ,Polynomial ,Radon transform ,General Mathematics ,010102 general mathematics ,Mathematics::Classical Analysis and ODEs ,Function (mathematics) ,01 natural sciences ,Operator (computer programming) ,Quadratic equation ,0101 mathematics ,Constant (mathematics) ,Mathematics - Abstract
In this work we extend consideration of the well-known polynomial Carleson operator to the setting of a Radon transform acting along the paraboloid in Rn+1 for n≥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 paraboloid by a smoother operator, which we accomplish via a Littlewood–Paley decomposition and the use of a square function. The most technical aspect then arises in the derivation of bounds for oscillatory integrals involving integration over lower-dimensional sets. The final theorem applies to polynomial Carleson operators with phase belonging to a certain restricted class of polynomials with no linear terms and whose homogeneous quadratic part is not a constant multiple of the defining function |y|2 of the paraboloid in Rn+1.
- Published
- 2019
8. Improved discrete restriction for the parabola
- Author
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Shaoming Guo, Zane Kun Li, and Po-Lam Yung
- Subjects
Mathematics - Number Theory ,Mathematics - Classical Analysis and ODEs ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Number Theory (math.NT) - Abstract
Using ideas from Guth-Maldague-Wang and working over $\mathbb{Q}_p$, we show that the discrete restriction constant for the parabola is $O_{\varepsilon}((\log M)^{2 + \varepsilon})$., 21 pages; revised version incorporating referee comments
- Published
- 2021
9. Going to Lorentz when fractional Sobolev, Gagliardo and Nirenberg estimates fail
- Author
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Jean Van Schaftingen, Po-Lam Yung, Haim Brezis, and UCL - SST/IRMP - Institut de recherche en mathématique et physique
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Mathematics::Functional Analysis ,Applied Mathematics ,Lorentz transformation ,010102 general mathematics ,Mathematics::Analysis of PDEs ,01 natural sciences ,Functional Analysis (math.FA) ,010101 applied mathematics ,Sobolev space ,Mathematics - Functional Analysis ,symbols.namesake ,Mathematics - Analysis of PDEs ,Mathematics - Classical Analysis and ODEs ,symbols ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,0101 mathematics ,26D10 (Primary), 26A33, 35A23, 46E30, 46E35 (Secondary) ,Nirenberg and Matthaei experiment ,Analysis ,Mathematical physics ,Mathematics ,Analysis of PDEs (math.AP) - Abstract
In the cases where there is no Sobolev-type or Gagliardo-Nirenberg-type fractional estimate involving $\lvert u\rvert_{W^{s,p}}$, we establish alternative estimates where the strong $L^p$ norms are replaced by Lorentz norms., Comment: Fixed up some typos. arXiv admin note: text overlap with arXiv:2003.05216
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- 2021
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10. Bourgain-Brezis-Mironescu Convergence via Triebel-Lizorkin Spaces
- Author
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Denis Brazke, Armin Schikorra, and Po-Lam Yung
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Mathematics - Functional Analysis ,Mathematics - Analysis of PDEs ,Mathematics - Classical Analysis and ODEs ,Applied Mathematics ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Analysis ,Analysis of PDEs (math.AP) ,Functional Analysis (math.FA) - Abstract
We study a convergence result of Bourgain--Brezis--Mironescu (BBM) using Triebel-Lizorkin spaces. It is well known that as spaces $W^{s,p} = F^{s}_{p,p}$, and $H^{1,p} = F^{1}_{p,2}$. When $s\to 1$, the $F^{s}_{p,p}$ norm becomes the $F^{1}_{p,p}$ norm but BBM showed that the $W^{s,p}$ norm becomes the $H^{1,p} = F^{1}_{p,2}$ norm. Naively, for $p \neq 2$ this seems like a contradiction, but we resolve this by providing embeddings of $W^{s,p}$ into $F^{s}_{p,q}$ for $q \in \{p,2\}$ with sharp constants with respect to $s \in (0,1)$. As a consequence we obtain an $\mathbb{R}^N$-version of the BBM-result, and obtain several more embedding and convergence theorems of BBM-type that to the best of our knowledge are unknown., Comment: changes in presentation, new references
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- 2021
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11. Families of functionals representing Sobolev norms
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Haim Brezis, Andreas Seeger, Jean Van Schaftingen, and Po-Lam Yung
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Mathematics - Functional Analysis ,Mathematics - Analysis of PDEs ,Mathematics - Classical Analysis and ODEs ,Astrophysics::High Energy Astrophysical Phenomena ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,26D10 (26A33, 35A23, 42B25, 42B35, 46E30, 46E3) ,Functional Analysis (math.FA) ,Analysis of PDEs (math.AP) - Abstract
We obtain new characterizations of the Sobolev spaces $\dot W^{1,p}(\mathbb{R}^N)$ and the bounded variation space $\dot{BV}(\mathbb{R}^N)$. The characterizations are in terms of the functionals $\nu_{\gamma} (E_{\lambda,\gamma/p}[u])$ where \[ E_{\lambda,\gamma/p}[u]= \Big\{(x,y )\in \mathbb{R}^N \times \mathbb{R}^N \colon x \neq y, \, \frac{|u(x)-u(y)|}{|x-y|^{1+\gamma/p}}>\lambda\Big\} \] and the measure $\nu_{\gamma}$ is given by $\mathrm{d} \nu_\gamma(x,y)=|x-y|^{\gamma-N} \mathrm{d} x \mathrm{d} y$. We provide characterizations which involve the $L^{p,\infty}$-quasi-norms $\sup_{\lambda>0} \lambda \, \nu_{\gamma} (E_{\lambda,\gamma/p}[u]) ^{1/p}$ and also exact formulas via corresponding limit functionals, with the limit for $\lambda\to\infty$ when $\gamma>0$ and the limit for $\lambda\to 0^+$ when $\gamma1$ the characterizations hold for all $\gamma \neq 0$. For $p=1$ the upper bounds for the $L^{1,\infty}$ quasi-norms fail in the range $\gamma\in [-1,0) $; moreover in this case the limit functionals represent the $L^1$ norm of the gradient for $C^\infty_c$-functions but not for generic $\dot W^{1,1}$-functions. For this situation we provide new counterexamples which are built on self-similar sets of dimension $\gamma+1$. For $\gamma=0$ the characterizations of Sobolev spaces fail; however we obtain a new formula for the Lipschitz norm via the expressions $\nu_0(E_{\lambda,0}[u])$., Comment: 40 pages
- Published
- 2021
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12. Limiting Sobolev and Hardy inequalities on stratified homogeneous groups.
- Author
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VAN SCHAFTINGEN, JEAN and PO-LAM YUNG
- Subjects
- *
SOBOLEV spaces , *MATHEMATICAL equivalence , *AUSDEHNUNGSLEHRE , *RIEMANNIAN manifolds , *GEOMETRY - Abstract
We give a sufficient condition for limiting Sobolev and Hardy inequalities to hold on stratified homogeneous groups. In the Euclidean case, this condition reduces to the known cancelling necessary and sufficient condition. We obtain in particular endpoint Korn-Sobolev and Korn-Hardy inequalities on stratified homogeneous groups. [ABSTRACT FROM AUTHOR]
- Published
- 2022
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13. Sobolev spaces revisited.
- Author
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Brezis, Haïm, Seeger, Andreas, Van Schaftingen, Jean, and Po-Lam Yung
- Subjects
SOBOLEV spaces ,FUNCTIONS of bounded variation ,INTERPOLATION - Abstract
We describe a recent, one-parameter family of characterizations of Sobolev and BV functions on Rn, using sizes of superlevel sets of suitable difference quotients. This provides an alternative point of view to the BBM formula by Bourgain, Brezis, and Mironescu, and complements in the case of BV some results of Cohen, Dahmen, Daubechies, and DeVore about the sizes of wavelet coefficients of such functions. An application towards Gagliardo--Nirenberg interpolation inequalities is then given. We also establish a related one-parameter family of formulae for the L
p norm of functions in Lp (ℝn . [ABSTRACT FROM AUTHOR]- Published
- 2022
- Full Text
- View/download PDF
14. Limiting Sobolev and Hardy inequalities on stratified homogeneous groups
- Author
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Jean Van Schaftingen, Po-Lam Yung, and UCL - SST/IRMP - Institut de recherche en mathématique et physique
- Subjects
symmetric derivative ,General Mathematics ,Mathematics::Analysis of PDEs ,homogeneous differential operator ,Mathematics - Analysis of PDEs ,maximally hypoelliptic operator ,canceling operator ,Korn-Sobolev inequality ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Hodge inequality ,Complex Variables (math.CV) ,overdetermined elliptic operator ,compatibility conditions ,35R03 (26D15, 35A23, 35H20, 43A80, 46E35) ,cocanceling operator ,Mathematics::Functional Analysis ,Mathematics - Complex Variables ,Articles ,Sobolev embedding ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,Mathematics - Classical Analysis and ODEs ,Saint-Venant compatibility conditions ,exterior derivative ,Analysis of PDEs (math.AP) - Abstract
We give a sufficient condition for limiting Sobolev and Hardy inequalities to hold on stratified homogeneous groups. In the Euclidean case, this condition reduces to the known cancelling necessary and sufficient condition. We obtain in particular endpoint Korn--Sobolev and Korn--Hardy inequalities on stratified homogeneous groups., Comment: To appear in Ann. Fenn. Math
- Published
- 2020
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15. A surprising formula for Sobolev norms
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Po-Lam Yung, Haim Brezis, and Jean Van Schaftingen
- Subjects
Pure mathematics ,Multidisciplinary ,Computer Science::Information Retrieval ,010102 general mathematics ,Astrophysics::Instrumentation and Methods for Astrophysics ,Mathematics::Analysis of PDEs ,Computer Science::Computation and Language (Computational Linguistics and Natural Language and Speech Processing) ,01 natural sciences ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,Sobolev space ,Mathematics - Classical Analysis and ODEs ,Physical Sciences ,0103 physical sciences ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Computer Science::General Literature ,46E30 (Primary) 26A33, 26D10, 35A23, 46E35 (Secondary) ,010307 mathematical physics ,0101 mathematics ,Equivalence (measure theory) ,Mathematics - Abstract
We establish the equivalence between the Sobolev semi-norm $\|\nabla u\|_{L^p}$ and a quantity obtained when replacing the strong $L^p$ by a weak $L^p$ norm in the Gagliardo semi-norm $|u|_{W^{s,p}}$ computed at $s = 1$. As corollaries we derive alternative estimates in some exceptional cases (involving $W^{1,1}$) where the "anticipated" fractional Sobolev and Gagliardo-Nirenberg inequalities fail., Comment: 14 pages
- Published
- 2020
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16. A new formula for the L norm
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Po-Lam Yung and Qingsong Gu
- Subjects
010102 general mathematics ,Bessel potential ,Function (mathematics) ,Characterization (mathematics) ,Space (mathematics) ,01 natural sciences ,Combinatorics ,Homogeneous ,Norm (mathematics) ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Lp space ,Analysis ,Mathematics - Abstract
Recently, Brezis, Van Schaftingen and the second author [4] established a new formula for the W ˙ 1 , p norm of a function in C c ∞ ( R N ) . The formula was obtained by replacing the L p ( R 2 N ) norm in the Gagliardo semi-norm for W ˙ s , p ( R N ) with a weak- L p ( R 2 N ) quasi-norm and setting s = 1 . This provides a characterization of such W ˙ 1 , p norms, which complements the celebrated Bourgain-Brezis-Mironescu (BBM) formula [1] . In this paper, we obtain an analog for the case s = 0 . In particular, we present a new formula for the L p norm of any function in L p ( R N ) , which involves only the measures of suitable level sets, but no integration. This provides a characterization of the norm on L p ( R N ) , which complements a formula by Maz′ya and Shaposhnikova [12] . As a result, by interpolation, we obtain a new embedding of the Triebel-Lizorkin space F 2 s , p ( R N ) (i.e. the Bessel potential space ( I − Δ ) − s / 2 L p ( R N ) ), as well as its homogeneous counterpart F ˙ 2 s , p ( R N ) , for s ∈ ( 0 , 1 ) , p ∈ ( 1 , ∞ ) .
- Published
- 2021
17. Bourgain–Brezis inequalities on symmetric spaces of non-compact type
- Author
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Jean Van Schaftingen, Po-Lam Yung, and Sagun Chanillo
- Subjects
Pure mathematics ,Triple system ,010102 general mathematics ,Mathematical analysis ,Duality (optimization) ,02 engineering and technology ,Type (model theory) ,021001 nanoscience & nanotechnology ,01 natural sciences ,Sobolev inequality ,Sobolev space ,Symmetric space ,Interpolation space ,Vector field ,0101 mathematics ,0210 nano-technology ,Analysis ,Mathematics - Abstract
Let M be a global Riemannian symmetric space of non-compact type. We prove a duality estimate, for pairings of divergence-free L 1 vector fields, with vector fields in a critical Sobolev space on M : | ∫ M 〈 f , ϕ 〉 d V | ≤ C ‖ f ‖ L 1 ( d V ) ‖ ∇ ϕ ‖ L m ( d V ) . This estimate provides a remedy for the failure of a critical Sobolev embedding on such symmetric spaces.
- Published
- 2017
18. Variations on a proof of a borderline Bourgain-Brezis Sobolev embedding theorem
- Author
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Po-Lam Yung, Jean Van Schaftingen, and Sagun Chanillo
- Subjects
Mathematics::Functional Analysis ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematics::Analysis of PDEs ,01 natural sciences ,Sobolev inequality ,010101 applied mathematics ,Mathematics - Analysis of PDEs ,Mathematics - Classical Analysis and ODEs ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,0101 mathematics ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
We offer a variant of a proof of a borderline Bourgain-Brezis Sobolev embedding theorem on $\mathbb{R}^n$. We use this idea to extend the result to real hyperbolic spaces $\mathbb{H}^n$., 21 pages; to appear in a Special Issue of Chinese Annals of Mathematics in honour of Haim Brezis
- Published
- 2017
19. A SHORT PROOF OF l² DECOUPLING FOR THE MOMENT CURVE.
- Author
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SHAOMING GUO, ZANE KUN LI, PO-LAM YUNG, and ZORIN-KRANICH, PAVEL
- Subjects
ARGUMENT - Abstract
We give a short and elementary proof of the l² decoupling inequality for the moment curve in R
k , using a bilinear approach inspired by the nested efficient congruencing argument of Wooley. [ABSTRACT FROM AUTHOR]- Published
- 2021
- Full Text
- View/download PDF
20. Reversing a philosophy: from counting to square functions and decoupling
- Author
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Shaoming Guo, Po-Lam Yung, Lillian B. Pierce, Joris Roos, and Philip T. Gressman
- Subjects
Mathematics - Number Theory ,Operator (physics) ,Diophantine equation ,010102 general mathematics ,Extension (predicate logic) ,Decoupling (cosmology) ,01 natural sciences ,Square (algebra) ,Combinatorics ,Permutation ,symbols.namesake ,Differential geometry ,Fourier analysis ,Mathematics - Classical Analysis and ODEs ,0103 physical sciences ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,symbols ,Number Theory (math.NT) ,010307 mathematical physics ,Geometry and Topology ,0101 mathematics ,Mathematics ,42B20, 42B25, 11D45 - Abstract
Breakthrough work of Bourgain, Demeter, and Guth recently established that decoupling inequalities can prove powerful results on counting integral solutions to systems of Diophantine equations. In this note we demonstrate that in appropriate situations this implication can also be reversed. As a first example, we observe that a count for the number of integral solutions to a system of Diophantine equations implies a discrete decoupling inequality. Second, in our main result we prove an $L^{2n}$ square function estimate (which implies a corresponding decoupling estimate) for the extension operator associated to a non-degenerate curve in $\mathbb{R}^n$. The proof is via a combinatorial argument that builds on the idea that if $\gamma$ is a non-degenerate curve in $\mathbb{R}^n$, then as long as $x_1,\ldots, x_{2n}$ are chosen from a sufficiently well-separated set, then $ \gamma(x_1)+\cdots+\gamma(x_n) = \gamma(x_{n+1}) + \cdots + \gamma(x_{2n}) $ essentially only admits solutions in which $x_1,\ldots,x_n$ is a permutation of $x_{n+1},\ldots, x_{2n}$., Comment: 16 pages
- Published
- 2019
21. Maximal functions associated with families of homogeneous curves: $L^p$ bounds for $p\le 2$
- Author
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Shaoming Guo, Po-Lam Yung, Joris Roos, and Andreas Seeger
- Subjects
Mathematics::Functional Analysis ,General Mathematics ,010102 general mathematics ,Parabola ,Mathematics::Classical Analysis and ODEs ,16. Peace & justice ,01 natural sciences ,Combinatorics ,symbols.namesake ,Mathematics::Probability ,Homogeneous ,Mathematics - Classical Analysis and ODEs ,0103 physical sciences ,symbols ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Maximal operator ,Maximal function ,010307 mathematical physics ,Hilbert transform ,0101 mathematics ,Mathematics - Abstract
Let $M^{(u)}$, $H^{(u)}$ be the maximal operator and Hilbert transform along the parabola $(t, ut^2) $. For $U\subset(0,\infty)$ we consider $L^p$ estimates for the maximal functions $\sup_{u\in U}|M^{(u)} f|$ and $\sup_{u\in U}|H^{(u)} f|$, when $1, Comment: 14 pages, final version to appear in Proceedings of the Edinburgh Mathematical Society
- Published
- 2019
22. Solution of the tangential Kohn Laplacian on a class of non-compact CR manifolds
- Author
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Po-Lam Yung and Chin-Yu Hsiao
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Mathematics - Complex Variables ,Applied Mathematics ,Operator (physics) ,010102 general mathematics ,Yamabe problem ,Boundary (topology) ,Conformal map ,01 natural sciences ,Square (algebra) ,Domain (mathematical analysis) ,010101 applied mathematics ,Mathematics - Analysis of PDEs ,Differential Geometry (math.DG) ,Mathematics - Classical Analysis and ODEs ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,CR manifold ,Complex Variables (math.CV) ,0101 mathematics ,Laplace operator ,Analysis ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
We solve $\square_b$ on a class of non-compact 3-dimensional strongly pseudoconvex CR manifolds via a certain conformal equivalence. The idea is to make use of a related $\square_b$ operator on a compact 3-dimensional strongly pseudoconvex CR manifold, which we solve using a pseudodifferential calculus. The way we solve $\square_b$ works whenever $\overline{\partial}_b$ on the compact CR manifold has closed range in $L^2$; in particular, as in Beals and Greiner, it does not require the CR manifold to be the boundary of a strongly pseudoconvex domain in $\mathbb{C}^2$. Our result provides in turn a key step in the proof of a positive mass theorem in 3-dimensional CR geometry, by Cheng, Malchiodi and Yang, which they then applied to study the CR Yamabe problem in 3 dimensions., Comment: to appear in Calc. Var. Partial Differential Equations
- Published
- 2019
23. A maximal function for families of Hilbert transforms along homogeneous curves
- Author
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Shaoming Guo, Andreas Seeger, Po-Lam Yung, and Joris Roos
- Subjects
General Mathematics ,010102 general mathematics ,Parabola ,16. Peace & justice ,01 natural sciences ,Combinatorics ,symbols.namesake ,Homogeneous ,Mathematics - Classical Analysis and ODEs ,0103 physical sciences ,symbols ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Maximal function ,010307 mathematical physics ,Hilbert transform ,0101 mathematics ,Operator norm ,Mathematics - Abstract
Let $H^{(u)}$ be the Hilbert transform along the parabola $(t, ut^2)$ where $u\in \mathbb R$. For a set $U$ of positive numbers consider the maximal function $\mathcal{H}^U \!f= \sup\{|H^{(u)}\! f|: u\in U\}$. We obtain an (essentially) optimal result for the $L^p$ operator norm of $\mathcal{H}^U$ when $2, Comment: 44 pages, fixed typos. A shortened version of this paper will appear in Mathematische Annalen
- Published
- 2019
- Full Text
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24. A bilinear proof of decoupling for the cubic moment curve
- Author
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Shaoming Guo, Po-Lam Yung, and Zane Kun Li
- Subjects
Mathematics - Number Theory ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Bilinear interpolation ,01 natural sciences ,Mathematics - Classical Analysis and ODEs ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Number Theory (math.NT) ,0101 mathematics ,Decoupling (electronics) ,Moment curve ,Mathematics - Abstract
Using a bilinear method that is inspired by the method of efficient congruencing of Wooley [Woo16], we prove a sharp decoupling inequality for the moment curve in $\mathbb{R}^3$., Comment: 28 pages; revised version incorporating referee comments
- Published
- 2019
- Full Text
- View/download PDF
25. Decoupling for two quadratic forms in three variables: a complete characterization
- Author
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Shaoming Guo, Changkeun Oh, Joris Roos, Po-Lam Yung, and Pavel Zorin-Kranich
- Subjects
Mathematics - Number Theory ,Mathematics - Classical Analysis and ODEs ,General Mathematics ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Number Theory (math.NT) ,42B25 (Primary) 11L15, 26D05 (Secondary) - Abstract
We prove sharp decoupling inequalities for all degenerate surfaces of codimension two in $\mathbb{R}^5$ given by two quadratic forms in three variables. Together with previous work by Demeter, Guo, and Shi in the non-degenerate case (arXiv:1609.04107), this provides a classification of decoupling inequalities for pairs of quadratic forms in three variables., Comment: v2: 19 pages, small corrections following referee report
- Published
- 2019
- Full Text
- View/download PDF
26. Sharp variation-norm estimates for oscillatory integrals related to Carleson's theorem
- Author
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Shaoming Guo, Joris Roos, and Po-Lam Yung
- Subjects
Numerical Analysis ,Applied Mathematics ,010102 general mathematics ,Mathematics::Classical Analysis and ODEs ,01 natural sciences ,polynomial Carleson ,Mathematics - Analysis of PDEs ,Mathematics - Classical Analysis and ODEs ,Norm (mathematics) ,Carleson's theorem ,0103 physical sciences ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Applied mathematics ,Ergodic theory ,010307 mathematical physics ,0101 mathematics ,42B20 ,variation-norm ,42B25 ,Analysis ,Smoothing ,Mathematics ,Analysis of PDEs (math.AP) - Abstract
We prove variation-norm estimates for certain oscillatory integrals related to Carleson's theorem. Bounds for the corresponding maximal operators were first proven by Stein and Wainger. Our estimates are sharp in the range of exponents, up to endpoints. Such variation-norm estimates have applications to discrete analogues and ergodic theory. The proof relies on square function estimates for Schr\"odinger-like equations due to Lee, Rogers and Seeger. In dimension one, our proof additionally relies on a local smoothing estimate. Though the known endpoint local smoothing estimate by Rogers and Seeger is more than sufficient for our purpose, we also give a proof of certain local smoothing estimates using Bourgain-Guth iteration and the Bourgain-Demeter $\ell^2$ decoupling theorem. This may be of independent interest, because it improves the previously known range of exponents for spatial dimensions $n \ge 4$., Comment: 46 pages, 1 figure
- Published
- 2017
27. Solving the Kohn Laplacian on asymptotically flat CR manifolds of dimension 3
- Author
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Chin-Yu Hsiao and Po-Lam Yung
- Subjects
Mathematics - Differential Geometry ,Mathematics - Complex Variables ,General Mathematics ,010102 general mathematics ,Structure (category theory) ,Inverse ,01 natural sciences ,Projection (linear algebra) ,Volume form ,Combinatorics ,Kernel (algebra) ,Mathematics - Analysis of PDEs ,Differential Geometry (math.DG) ,0103 physical sciences ,Several complex variables ,FOS: Mathematics ,CR manifold ,010307 mathematical physics ,Complex Variables (math.CV) ,0101 mathematics ,Laplace operator ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
Let $(\hat{X}, T^{1,0} \hat{X})$ be a compact orientable CR embeddable three dimensional strongly pseudoconvex CR manifold, where $T^{1,0} \hat{X}$ is a CR structure on $\hat{X}$. Fix a point $p \in \hat{X}$ and take a global contact form $\hat{\theta}$ so that $\hat{\theta}$ is asymptotically flat near $p$. Then $(\hat{X}, T^{1,0} \hat{X}, \hat{\theta})$ is a pseudohermitian $3$-manifold. Let $G_p \in C^{\infty} (\hat{X} \setminus \{p\})$, $G_p > 0$, with $G_p(x) \sim \vartheta(x,p)^{-2}$ near $p$, where $\vartheta(x,y)$ denotes the natural pseudohermitian distance on $\hat{X}$. Consider the new pseudohermitian $3$-manifold with a blow-up of contact form $(\hat{X} \setminus \{p\}, T^{1,0} \hat{X}, G^2_p \hat{\theta})$ and let $\Box_{b}$ denote the corresponding Kohn Laplacian on $\hat{X} \setminus \{p\}$. In this paper, we prove that the weighted Kohn Laplacian $G^2_p \Box_b$ has closed range in $L^2$ with respect to the weighted volume form $G^2_p \hat{\theta} \wedge d\hat{\theta}$, and that the associated partial inverse and the Szeg\"{o} projection enjoy some regularity properties near $p$. As an application, we prove the existence of some special functions in the kernel of $\Box_{b}$ that grow at a specific rate at $p$. The existence of such functions provides an important ingredient for the proof of a positive mass theorem in 3-dimensional CR geometry by Cheng-Malchiodi-Yang., Comment: 72 pages
- Published
- 2015
28. A surprising formula for Sobolev norms.
- Author
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Brezis, Haïm, Van Schaftingen, Jean, and Po-Lam Yung
- Subjects
SOBOLEV spaces - Abstract
We establish the equivalence between the Sobolev seminorm ||∇||L
p and a quantity obtained when replacing strong Lp by weak Lp in the Gagliardo seminorm |u|Ws,p computed at s = 1. As corollaries we derive alternative estimates in some exceptional cases (involving W1,1 ) where the "anticipated" fractional Sobolev and Gagliardo-Nirenberg inequalities fail. [ABSTRACT FROM AUTHOR]- Published
- 2021
- Full Text
- View/download PDF
29. The Incompressible Navier Stokes Flow in Two Dimensions with Prescribed Vorticity
- Author
-
Po-Lam Yung, Jean Van Schaftingen, Sagun Chanillo, and UCL - SST/IRMP - Institut de recherche en mathématique et physique
- Subjects
Physics ,Mathematics::Analysis of PDEs ,Vorticity ,Stokes flow ,Stokes stream function ,Sobolev space ,Physics::Fluid Dynamics ,symbols.namesake ,Vorticity equation ,Incompressible flow ,Stokes' law ,Stream function ,symbols ,Mathematical physics - Abstract
We study the incompressible two dimensional Navier–Stokes equation with initial vorticity in the homogeneous Sobolev space \(\dot{W}^{1,1}(\mathbb{R}^{2})\). This complements our earlier work for the case when the initial vorticity is in the inhomogeneous Sobolev space \(W^{1,1}(\mathbb{R}^{2})\).
- Published
- 2017
30. A subelliptic Bourgain–Brezis inequality
- Author
-
Yi Wang and Po-Lam Yung
- Subjects
Combinatorics ,Sobolev space ,Mathematics::Functional Analysis ,Lemma (mathematics) ,Homogeneous ,Applied Mathematics ,General Mathematics ,Mathematics::Analysis of PDEs ,Heisenberg group ,Function (mathematics) ,Mathematics - Abstract
We prove an approximation lemma on (stratified) homogeneous groups that allows one to approximate a function in the non-isotropic Sobolev space $\dot{NL}^{1,Q}$ by $L^{\infty}$ functions, generalizing a result of Bourgain-Brezis \cite{MR2293957}. We then use this to obtain a Gagliardo-Nirenberg inequality for $\bar{\partial}_b$ on the Heisenberg group $\mathbb{H}^n$.
- Published
- 2014
31. Pseudodifferential operators of mixed type adapted to distributions of $k$-planes
- Author
-
Elias M. Stein and Po-Lam Yung
- Subjects
Pure mathematics ,Rank (linear algebra) ,Mathematics - Complex Variables ,Pseudodifferential operators ,General Mathematics ,010102 general mathematics ,Mixed type ,Tangent ,01 natural sciences ,Linear subspace ,010101 applied mathematics ,Mathematics - Analysis of PDEs ,Mathematics - Classical Analysis and ODEs ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Complex Variables (math.CV) ,0101 mathematics ,Invariant (mathematics) ,Constant (mathematics) ,Distribution (differential geometry) ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
We study the phenomena that arise when we combine the standard pseudodifferential operators with those operators that appear in the study of some sub-elliptic estimates, and on strongly pseudoconvex domains. The algebra of operators we introduce is geometrically invariant, and is adapted to a smooth distribution of tangent subspaces of constant rank. We isolate certain ideals in the algebra whose analysis is of particular interest., Comment: typos fixed
- Published
- 2013
32. Polynomial Carleson operators along monomial curves in the plane
- Author
-
Joris Roos, Lillian B. Pierce, Shaoming Guo, and Po-Lam Yung
- Subjects
42B20, 42B25, 44A12 ,Polynomial ,Pure mathematics ,Monomial ,Plane (geometry) ,010102 general mathematics ,Phase (waves) ,01 natural sciences ,symbols.namesake ,Quadratic equation ,Operator (computer programming) ,Differential geometry ,Fourier analysis ,Mathematics - Classical Analysis and ODEs ,0103 physical sciences ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,symbols ,010307 mathematical physics ,Geometry and Topology ,0101 mathematics ,Mathematics - Abstract
We prove $L^p$ bounds for partial polynomial Carleson operators along monomial curves $(t,t^m)$ in the plane $\mathbb{R}^2$ with a phase polynomial consisting of a single monomial. These operators are "partial" in the sense that we consider linearizing stopping-time functions that depend on only one of the two ambient variables. A motivation for studying these partial operators is the curious feature that, despite their apparent limitations, for certain combinations of curve and phase, $L^2$ bounds for partial operators along curves imply the full strength of the $L^2$ bound for a one-dimensional Carleson operator, and for a quadratic Carleson operator. Our methods, which can at present only treat certain combinations of curves and phases, in some cases adapt a $TT^*$ method to treat phases involving fractional monomials, and in other cases use a known vector-valued variant of the Carleson-Hunt theorem., 27 pages
- Published
- 2016
33. On a trilinear singular integral form with determinantal kernel
- Author
-
Danqing He, Christoph Thiele, Philip T. Gressman, Po-Lam Yung, Brian Street, and Vjekoslav Kovač
- Subjects
Pure mathematics ,General Mathematics ,Bilinear interpolation ,Symmetry group ,01 natural sciences ,law.invention ,03 medical and health sciences ,Matrix (mathematics) ,symbols.namesake ,law ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,0101 mathematics ,multilinear form ,bilinear Hilbert transform ,Calderon commutator ,invariant trilinear form ,030304 developmental biology ,Mathematics ,0303 health sciences ,Applied Mathematics ,010102 general mathematics ,Commutator (electric) ,Singular integral ,Kernel (algebra) ,Fourier transform ,Multilinear form ,Mathematics - Classical Analysis and ODEs ,symbols ,42B20 - Abstract
We study a trilinear singular integral form acting on two-dimensional functions and possessing invariances under arbitrary matrix dilations and linear modulations. One part of the motivation for introducing it lies in its large symmetry groups acting on the Fourier side. Another part of the motivation is that this form stands between the bilinear Hilbert transforms and the first Calder\'on commutator, in the sense that it can be reduced to a superposition of the former, while it also successfully encodes the latter. As the main result we determine the exact range of exponents in which the ${L}^p$ estimates hold for the considered form., Comment: 12 pages
- Published
- 2015
34. Sobolev Inequalities for $(0,q)$ forms on CR manifolds of finite type
- Author
-
Po-Lam Yung
- Subjects
Khovanov homology ,Pure mathematics ,Mathematics - Complex Variables ,General Mathematics ,Invariant (physics) ,Type (model theory) ,Mathematics::Algebraic Topology ,Mathematics::Geometric Topology ,Sobolev inequality ,Mathematics - Analysis of PDEs ,Floer homology ,Mathematics::K-Theory and Homology ,Spectral sequence ,Mutation (genetic algorithm) ,FOS: Mathematics ,Quantitative Biology::Populations and Evolution ,Complex Variables (math.CV) ,Mathematics::Symplectic Geometry ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
Let $M^{2n+1}$ ($n \geq 2$) be a compact pseudoconvex CR manifold of finite commutator type whose $\dbarb$ has closed range in $L^2$ and whose Levi form has comparable eigenvalues. We prove a sharp $L^1$ Sobolev inequality for the $\dbarb$ complex for $(0,q)$ forms when $q \ne 1$ nor $n-1$. We also prove an analogous $L^1$ inequality when $M$ satisfies condition $Y(q)$. The main technical ingredient is a new kind of $L^1$ duality inequality for vector fields that satisfy Hormander's condition., Comment: to appear in Math. Res. Lett.
- Published
- 2010
35. Doubling Properties of Self-similar Measures
- Author
-
Po-Lam Yung
- Subjects
Discrete mathematics ,Combinatorics ,Bernoulli's principle ,General Mathematics ,Open set ,Golden ratio ,Measure (mathematics) ,Mathematics - Abstract
Let {F i } N i=1 be a system of similitudes in R n . We study necessary and sufficient conditions for their associated self-similar measures to be doubling on its support. An equivalent condition is obtained when {F i } satisfies the open set condition. The condition allows us to construct many examples of interest. In the case where the open set condition is not satisfied, we study an infinitely convoluted Bernoulli measure (associated with the golden ratio p = (√5 - 1) /2) and give a necessary and sufficient condition for it to be doubling on its support [0,1].
- Published
- 2007
36. A sharp subelliptic Sobolev embedding theorem with weights
- Author
-
Po-Lam Yung
- Subjects
Pure mathematics ,Mathematics - Classical Analysis and ODEs ,General Mathematics ,Mathematical analysis ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Mathematics::Analysis of PDEs ,Smoothing ,Sobolev inequality ,Mathematics - Abstract
The purpose of this short article is to prove some potential estimates that naturally arise in the study of subelliptic Sobolev inequalities for functions. This will allow us to prove a local subelliptic Sobolev inequality with the optimal amount of smoothing, as well as a variant of that which describes quantitatively an improvement of the inequality as one gets away from certain characteristic varieties.
- Published
- 2015
37. Applications of Bourgain-Brezis inequalities to Fluid Mechanics and Magnetism
- Author
-
Sagun Chanillo, Jean Van Schaftingen, Po-Lam Yung, and UCL - SST/IRMP - Institut de recherche en mathématique et physique
- Subjects
Spacetime ,Magnetism ,010102 general mathematics ,Mathematical analysis ,Mathematics::Analysis of PDEs ,Fluid mechanics ,General Medicine ,Vorticity ,01 natural sciences ,Magnetic field ,Physics::Fluid Dynamics ,General Relativity and Quantum Cosmology ,Mathematics - Analysis of PDEs ,Bounded function ,0103 physical sciences ,Bounded variation ,FOS: Mathematics ,010307 mathematical physics ,0101 mathematics ,Mathematics ,Analysis of PDEs (math.AP) - Abstract
We apply the borderline Sobolev inequalities of Bourgain-Brezis to the vorticity equation and Navier-Stokes equation in 2D. We take the initial vorticity to be in the space of functions of Bounded variation(BV). We obtain the subsequent vorticity to be in the space of functions of bounded variation, uniformly for small time, and the velocity vector to be uniformly bounded for small time. Such a conclusion cannot follow for initial vorticity taken to be just a measure or in L^1 from the Lamb-Oseen vortex example. Secondly we apply an improved Strichartz inequality obtained earlier by the first and third authors to the Maxwell equations of Electromagnetism. In particular we estimate the size of the magnetic field vector in terms of the gradient of the current density vector. The main point is that in this inequality only the L^1 norm in space appears for the gradient of the current density vector. Such a result is only possible because of a vanishing divergence inhomogeneity in the wave equation for the Magnetic field vector stemming from the Maxwell equations. A key ingredient in the proof of the improved Strichartz inequality is the Bourgain-Brezis borderline Sobolev inequalities., Comment: References added to the work of M. Ben-Artzi and also Haim Brezis in ARMA. Introduction revised
- Published
- 2015
- Full Text
- View/download PDF
38. Absence of Self-Similar Blow-up and Local Well-posedness for the Constant Mean-Curvature Wave Equation
- Author
-
Sagun Chanillo and Po-Lam Yung
- Subjects
Mean curvature ,010102 general mathematics ,Mathematical analysis ,Wave equation ,Condensed Matter::Mesoscopic Systems and Quantum Hall Effect ,01 natural sciences ,Mathematics - Analysis of PDEs ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,0101 mathematics ,Constant (mathematics) ,Analysis ,Well posedness ,Mathematics ,Analysis of PDEs (math.AP) - Abstract
In this note, we consider the constant-mean-curvature wave equation in ( 1 + 2 ) -dimensions. We show that it does not admit any self-similar blow-up. We also remark that the equation is locally well-posed for initial data in H ˙ 3 / 2 .
- Published
- 2015
- Full Text
- View/download PDF
39. A polynomial Carleson operator along the paraboloid.
- Author
-
Pierce, Lillian B. and Po-Lam Yung
- Subjects
- *
POLYNOMIAL operators , *PARABOLOID , *RADON transforms , *INTEGRAL operators , *MAXIMAL functions , *INTEGRALS - Abstract
In this work we extend consideration of the well-known polynomial Carleson operator to the setting of a Radon transform acting along the paraboloid in Rn+1 for n = 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 paraboloid by a smoother operator, which we accomplish via a Littlewood-Paley decomposition and the use of a square function. The most technical aspect then arises in the derivation of bounds for oscillatory integrals involving integration over lower-dimensional sets. The final theorem applies to polynomial Carleson operators with phase belonging to a certain restricted class of polynomials with no linear terms and whose homogeneous quadratic part is not a constant multiple of the defining function |y|² of the paraboloid in Rn+1. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
40. An Improved Strichartz Estimate for Systems with Divergence Free Data
- Author
-
Sagun Chanillo and Po-Lam Yung
- Subjects
Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,Novelty ,Mathematics::Analysis of PDEs ,01 natural sciences ,Schrödinger equation ,010101 applied mathematics ,symbols.namesake ,Mathematics - Analysis of PDEs ,Mathematics - Classical Analysis and ODEs ,Strichartz estimate ,symbols ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Vector field ,0101 mathematics ,Divergence (statistics) ,Analysis ,Mathematics ,Analysis of PDEs (math.AP) - Abstract
Using the div-curl inequalities of Bourgain-Brezis [?MR2057026] and van Schaftingen [?MR2078071], we prove an improved Strichartz estimate for systems of inhomogeneous wave and Schrodinger equations, for which the inhomogeneity is a divergence-free vector field at each given time. The novelty of the result is that one can allow $L^1_x$ norms of the inhomogeneity in the right hand side of the estimate., Comment: 9 pages
- Published
- 2012
41. Wave equations associated to Liouville systems and constant mean curvature equations
- Author
-
Po-Lam Yung and Sagun Chanillo
- Subjects
Mathematics - Differential Geometry ,Mathematics(all) ,General Mathematics ,Bubble ,Antipodal point ,01 natural sciences ,symbols.namesake ,Mathematics - Analysis of PDEs ,0103 physical sciences ,Gaussian curvature ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,0101 mathematics ,Mathematics ,Mean curvature ,Liouville equation ,010102 general mathematics ,Mathematical analysis ,Threshold energy ,Wave equation ,Differential Geometry (math.DG) ,Mathematics - Classical Analysis and ODEs ,symbols ,010307 mathematical physics ,Single point ,Analysis of PDEs (math.AP) - Abstract
We study the wave analog of the Liouville equation for constant Gauss curvature on S 2 . We show the global existence in time for the sub-critical equation with data in H 1 ( S 2 ) × L 2 ( S 2 ) . For the super-critical and the critical equation we show the global existence in time assuming the data is symmetric with respect to antipodal points on S 2 . Use is made of the Moser–Trudinger inequalities for proving this result. For general data without symmetry assumptions and for the critical equation, we establish that if there is a blow-up, then it will occur at a single point of S 2 . We also consider the wave analog of the constant mean curvature equation in the Plateau formalism. This is a system. We establish finite time blow up if the energy of the initial data is larger than 8 π and with an additional subsidiary assumption. The threshold energy of 8 π arises because it is the energy of the basic bubble discovered by Brezis and Coron.
- Published
- 2011
42. Calculus on the Sierpinski Gasket I: Polynomials, Exponentials and Power Series
- Author
-
Jonathan Needleman, Po-Lam Yung, Alexander Teplyaev, and Robert S. Strichartz
- Subjects
Power series ,Pure mathematics ,28A80 ,Uniform convergence ,010102 general mathematics ,Mathematical analysis ,010103 numerical & computational mathematics ,01 natural sciences ,Exponential polynomial ,Exponential function ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,symbols.namesake ,Mathematics - Classical Analysis and ODEs ,Taylor series ,symbols ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Non-analytic smooth function ,0101 mathematics ,Analysis ,Heat kernel ,Mathematics ,Analytic function - Abstract
We study the analog of power series expansions on the Sierpinski gasket, for analysis based on the Kigami Laplacian. The analog of polynomials are multiharmonic functions, which have previously been studied in connection with Taylor approximations and splines. Here the main technical result is an estimate of the size of the monomials analogous to x n / n !. We propose a definition of entire analytic functions as functions represented by power series whose coefficients satisfy exponential growth conditions that are stronger than what is required to guarantee uniform convergence. We present a characterization of these functions in terms of exponential growth conditions on powers of the Laplacian of the function. These entire analytic functions enjoy properties, such as rearrangement and unique determination by infinite jets, that one would expect. However, not all exponential functions (eigenfunctions of the Laplacian) are entire analytic, and also many other natural candidates, such as the heat kernel, do not belong to this class. Nevertheless, we are able to use spectral decimation to study exponentials, and in particular to create exponentially decaying functions for negative eigenvalues.
- Published
- 2003
- Full Text
- View/download PDF
43. ON A TRILINEAR SINGULAR INTEGRAL FORM WITH DETERMINANTAL KERNEL.
- Author
-
GRESSMAN, PHILIP, DANQING HE, JEKOSLAV KOVAČ, V., STREET, BRIAN, THIELE, CHRISTOPH, and PO-LAM YUNG
- Subjects
INTEGRAL operators ,INTEGRAL transforms ,MATHEMATICAL symmetry ,SYMMETRY ,HILBERT functions ,MATHEMATICAL models - Abstract
We study a trilinear singular integral form acting on twodimensional functions and possessing invariances under arbitrary matrix dilations and linear modulations. One part of the motivation for introducing it lies in its large symmetry groups acting on the Fourier side. Another part of the motivation is that this form stands between the bilinear Hilbert transforms and the first Calderón commutator, in the sense that it can be reduced to a superposition of the former, while it also successfully encodes the latter. As the main result we determine the exact range of exponents in which the L
p estimates hold for the considered form. [ABSTRACT FROM AUTHOR]- Published
- 2016
- Full Text
- View/download PDF
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