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Shaoming Guo

Publications and source records attributed to Shaoming Guo.

At least 55 records · Page 3Linked to original sources
arXiv2019

Sharp variation-norm estimates for oscillatory integrals related to Carleson's theorem

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 Schrödinger-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 $n \ge 4$.

math.CA↗
arXiv2019

Polynomial Roth theorems on sets of fractional dimensions

Let $E\subset \mathbb{R}$ be a closed set of Hausdorff dimension $α\in (0, 1)$. Let $P: \mathbb{R}\to \mathbb{R}$ be a polynomial without a constant term whose degree is bigger than one. We prove that if $E$ supports a probability measure satisfying certain dimension condition and Fourier decay condition, then $E$ contains three points $x, x+t, x+P(t)$ for some $t>0$. Our result extends the one of Laba and the third author to the polynomial setting, under the same assumption. It also gives an affirmative answer to a question in Henriot, Laba and the third author.

math.CA↗
arXiv2019

A polynomial Roth theorem on the real line

For a polynomial $P$ of degree greater than one, we show the existence of patterns of the form $(x,x+t,x+P(t))$ with a gap estimate on $t$ in positive density subsets of the reals. This is an extension of an earlier result of Bourgain. Our proof is a combination of Bourgain's approach and more recent methods that were originally developed for the study of the bilinear Hilbert transform along curves.

math.CO↗
arXiv2018

On a binary system of Prendiville: The cubic case

We prove sharp decoupling inequalities for a class of two dimensional non-degenerate surfaces in R^5, introduced by Prendiville. As a consequence, we obtain sharp bounds on the number of integer solutions of the Diophantine systems associated with these surfaces.

math.NT↗
arXiv2017

Square functions for bi-Lipschitz maps and directional operators

First we prove a Littlewood-Paley diagonalization result for bi-Lipschitz perturbations of the identity map on the real line. This result entails a number of corollaries for the Hilbert transform along lines and monomial curves in the plane. Second, we prove a square function bound for a single scale directional operator. As a corollary we give a new proof of part of a theorem of Katz on direction fields with finitely many directions.

math.CA↗
arXiv2016

Remarks on Wolff's inequality for hypersurfaces

We run an iteration argument due to Pramanik and Seeger, to provide a proof of sharp decoupling inequalities for conical surfaces and for $k$-cones. These are extensions of results Łaba and Pramanik to sharp exponents.

math.CA↗
arXiv2016

On the 1D Cubic Nonlinear Schrodinger Equation in an Almost Critical Space

We obtain the local well-posedness of the one dimensional cubic nonlinear Schrödinger Equation for initial data in the modulation space $M_{2, p}$ for all $2\le p<\infty$, which covers all the subcritical cases from the viewpoint of scaling. Moreover, in order to approach the endpoint space $M_{2,\infty}$, we will prove the almost global well-posedness in some Orlicz-type space, which is a natural generalisation of $M_{2,p}$ for large $p$. The new ingredient is an endpoint version of the two dimensional restriction estimate.

math.AP↗
arXiv2016

Hilbert transforms along Lipschitz direction fields: A lacunary model

We prove bounds for the truncated directional Hilbert transform in $L^p(\mathbb{R}^2)$ for any $1<p<\infty$ under a combination of a Lipschitz assumption and a lacunarity assumption. It is known that a lacunarity assumption alone is not sufficient to yield boundedness for $p=2$, and it is a major question in the field whether a Lipschitz assumption alone suffices, at least for some $p$.

math.CA↗
arXiv2016

Single annulus estimates for the variation-norm Hilbert transforms along Lipschitz vector fields

Let v be a planar Lipschitz vector field. We prove that the r-th variation-norm Hilbert transform along v, composed with a standard Littlewood-Paley projection operator P_k, is bounded from L^2 to L^{2, \infty}, and from L^p to itself for all p>2. Here r>2 and the operator norm is independent of k\in \Z. This generalises Lacey and Li's result for the case of the Hilbert transform. However, their result only assumes measurability for vector fields. In contrast to that, we need to assume vector fields to be Lipschitz.

math.CA↗
arXiv2016

Maximal operators and Hilbert transforms along variable non-flat homogeneous curves

We prove that the maximal operator associated with variable homogeneous planar curves $(t, u t^α)_{t\in \mathbb{R}}$, $α\not=1$ positive, is bounded on $L^p(\mathbb{R}^2)$ for each $p>1$, under the assumption that $u:\mathbb{R}^2 \to \mathbb{R}$ is a Lipschitz function. Furthermore, we prove that the Hilbert transform associated with $(t, ut^α)_{t\in \mathbb{R}}$, $α\not=1$ positive, is bounded on $L^p(\mathbb{R}^2)$ for each $p>1$, under the assumption that $u:\mathbb{R}^2\to \mathbb{R}$ is a measurable function and is constant in the second variable. Our proofs rely on stationary phase methods, $TT^*$ arguments, local smoothing estimates and a pointwise estimate for taking averages along curves.

math.CA↗
arXiv2016

Local Well-Posedness for the Derivative Nonlinear Schrödinger Equations with $L^2$ Subcritical Data

We will show its local well-posedness in modulation spaces $M^{1/2}_{2,q}({\Real})$ $(2\leq q<\infty) $. It is well-known that $H^{1/2}$ is a critical Sobolev space of DNLS so that it is locally well-posedness in $H^s$ for $s\geq 1/2$ and ill-posed in $H^{s'}$ with $s'<1/2.$ Noticing that that $M^{1/2}_{2,q} \subset B^{1/q}_{2,q}$ is a sharp embedding and $L^2 \subset B^0_{2,\infty}$, our result contains all of the subcritical data in $M^{1/2}_{2,q}$, which contains a class of functions in $L^2\setminus H^{1/2}$.

math.AP↗
arXiv2016

Polynomial Carleson operators along monomial curves in the plane

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.

math.CA↗