arXiv subjects

Naijia Liu

Publications and source records attributed to Naijia Liu.

14 recordsLinked to original sources

arXiv2026

Spherical maximal functions and Hardy spaces for Fourier integral operators

We use the Hardy spaces for Fourier integral operators to obtain bounds for spherical maximal functions in $L^{p}(\mathbb{R}^{n})$, $n\geq2$, where the radii of the spheres are restricted to a compact interval in $(0,\infty)$. These bounds extend to general hypersurfaces with non-vanishing Gaussian curvature, to the complex spherical means, and to geodesic spheres on compact manifolds. We also obtain improved maximal function bounds and pointwise convergence statements for wave equations, both on $\mathbb{R}^{n}$ and on compact manifolds. The maximal function bounds are essentially sharp for all $p\in[1,2]\cup [\frac{2(n+1)}{n-1},\infty)$, for each such hypersurface, every complex spherical mean, and on every manifold.

math.CA↗

arXiv2025

Global solutions to 3D quadratic nonlinear Schrödinger-type equation

We consider the Cauchy problem to the 3D fractional Schrödinger equation with quadratic interaction of $u\bar u$ type. We prove the global existence of solutions and scattering properties for small initial data. For the proof, one novelty is that we combine the normal form methods and the space-time resonance methods. Using the normal form transform enables us more flexibilities in designing the resolution spaces so that we can control various interactions. It is also convenient for the final data problem.

math.AP↗

arXiv2025

The Hardy spaces $\mathcal{H}^{p}_{FIO}(\mathbb{R}^{n})$ for Fourier integral operators for $p<1$

We introduce the Hardy spaces $\mathcal{H}^{p}_{FIO}(\mathbb{R}^{n})$ for Fourier integral operators for $0<p<1$, thereby extending earlier constructions for $1\leq p\leq \infty$. We then establish various properties of these spaces, including their behavior under complex interpolation and duality, and their invariance under Fourier integral operators. We also obtain Sobolev embeddings, equivalent characterizations, and a molecular decomposition. These spaces are used in the companion article arXiv:2502.02511 to determine the sharp $\mathcal{H}^{1}(\mathbb{R}^{n})$ and $\mathrm{bmo}(\mathbb{R}^{n})$ regularity of wave equations with rough coefficients.

math.AP↗

arXiv2025

$\mathcal{H}^{1}$ and $\mathrm{bmo}$ regularity for wave equations with rough coefficients

We consider second-order hyperbolic equations with rough time-independent coefficients. Our main result is that such equations are well posed on the Hardy spaces $\mathcal{H}^{s,1}_{FIO}(\mathbb{R}^{n})$ and $\mathcal{H}^{s,\infty}_{FIO}(\mathbb{R}^{n})$ for Fourier integral operators if the coefficients have $C^{1,1}\cap C^{r}$ regularity in space, for $r>\frac{n+1}{2}$, where $s$ ranges over an $r$-dependent interval. As a corollary, we obtain the sharp fixed-time $\mathcal{H}^{1}(\mathbb{R}^{n})$ and $\mathrm{bmo}(\mathbb{R}^{n})$ regularity for such equations, extending work by Seeger, Sogge and Stein in the case of smooth coefficients.

math.AP↗

arXiv2025

$L^p\to L^q$ estimates for Stein's spherical maximal operators

In this article we consider a modification of the Stein's spherical maximal operator of complex order $α$ on ${\mathbb R^n}$: $$ {\mathfrak M}^α_{[1,2]} f(x) =\sup\limits_{t\in [1,2]} \big| {1\over Γ(α) } \int_{|y|\leq 1} \left(1-|y|^2 \right)^{α-1} f(x-ty) dy\big|. $$ We show that when $n\geq 2$, suppose $\|{\mathfrak M}^α_{[1,2]} f \|_{L^q({\mathbb R^n})} \leq C\|f \|_{L^p({\mathbb R^n})}$ holds for some $α\in \mathbb{C}$, $p,q\geq1$, then we must have that $q\geq p$ and $${\rm Re}\,α\geq σ_n(p,q):=\max\left\{\frac{1}{p}-\frac{n}{q},\ \frac{n+1}{2p}-\frac{n-1}{2}\left(\frac{1}{q}+1\right),\frac{n}{p}-n+1\right\}.$$ Conversely, we show that ${\mathfrak M}^α_{[1,2]}$ is bounded from $L^p({\mathbb R^n})$ to $L^q({\mathbb R^n})$ provided that $q\geq p$ and ${\rm Re}\,α>σ_2(p,q)$ for $n=2$; and ${\rm Re}\,α>\max\left\{σ_n(p,q), 1/(2p)- (n-2)/(2q) -(n-1)/4\right\}$ for $n>2$. The range of $α,p$ and $q$ is almost optimal in the case either $n=2$, or $α=0$, or $(p,q)$ lies in some regions for $n>2$.

math.CA↗

arXiv2024

$L_x^p\rightarrow L^q_{x,u}$ estimates for dilated averages over planar curves

In this paper, we consider the $L_x^p(\mathbb{R}^2)\rightarrow L_{x,u}^q(\mathbb{R}^2\times [1,2])$ estimate for the operator $T$ along a dilated plane curve $(ut,uγ(t))$, where $$Tf(x,u):=\int_{0}^{1}f(x_1-ut,x_2-u γ(t))\,\textrm{d}t,$$ $x:=(x_1,x_2)$ and $γ$ is a general plane curve satisfying some suitable smoothness and curvature conditions. We show that $T$ is $L_x^p(\mathbb{R}^2)$ to $L_{x,u}^q(\mathbb{R}^2\times [1,2])$ bounded whenever $(\frac{1}{p},\frac{1}{q})\in \square \cup \{(0,0)\}\cup \{(\frac{2}{3},\frac{1}{3})\}$ and $1+(1 +ω)(\frac{1}{q}-\frac{1}{p})>0$, where the trapezium $\square:=\{(\frac{1}{p},\frac{1}{q}):\ \frac{2}{p}-1\leq\frac{1}{q}\leq \frac{1}{p}, \frac{1}{q}>\frac{1}{3p}, \frac{1}{q}>\frac{1}{p}-\frac{1}{3}\}$ and $ω:=\limsup_{t\rightarrow 0^{+}}\frac{\ln|γ(t)|}{\ln t}$. This result is sharp except for some borderline cases. On the other hand, in a smaller $(\frac{1}{p},\frac{1}{q})$ region, we also obtain the almost sharp estimate $T : L_x^p(\mathbb{R}^2)\rightarrow L_{x}^q(\mathbb{R}^2)$ uniformly for $u\in [1,2]$. These results imply that the operator $T$ has the so called local smoothing phenomenon, i.e., the $L^q$ integral about $u$ on $[1,2]$ extends the region of $(\frac{1}{p},\frac{1}{q})$ in uniform estimate $T : L_x^p(\mathbb{R}^2)\rightarrow L_{x}^q(\mathbb{R}^2)$.

math.CA↗

arXiv2023

Local smoothing and Hardy spaces for Fourier integral operators on manifolds

We introduce the Hardy spaces for Fourier integral operators on Riemannian manifolds with bounded geometry. We then use these spaces to obtain improved local smoothing estimates for Fourier integral operators satisfying the cinematic curvature condition, and for wave equations on compact manifolds. The estimates are essentially sharp, for all $2<p<\infty$ and on each compact manifold. We also apply our local smoothing estimates to nonlinear wave equations with initial data outside of $L^{2}$-based Sobolev spaces.

math.AP↗

arXiv2023

$L^p$-improving bounds of maximal functions along planar curves

In this paper, we study the $L^p(\mathbb{R}^2)$-improving bounds, i.e., $L^p(\mathbb{R}^2)\rightarrow L^q(\mathbb{R}^2)$ estimates, of the maximal function $M_γ$ along a plane curve $(t,γ(t))$, where $$M_γf(x_1,x_2):=\sup_{u\in [1,2]}\left|\int_{0}^{1}f(x_1-ut,x_2-u γ(t))\,\textrm{d}t\right|,$$ and $γ$ is a general plane curve satisfying some suitable smoothness and curvature conditions. We obtain $M_γ : L^p(\mathbb{R}^2)\rightarrow L^q(\mathbb{R}^2)$ if $(\frac{1}{p},\frac{1}{q})\in Δ\cup \{(0,0)\}$ and $(\frac{1}{p},\frac{1}{q})$ satisfying $1+(1 +ω)(\frac{1}{q}-\frac{1}{p})>0$, where $Δ:=\{(\frac{1}{p},\frac{1}{q}):\ \frac{1}{2p}<\frac{1}{q}\leq \frac{1}{p}, \frac{1}{q}>\frac{3}{p}-1 \}$ and $ω:=\limsup_{t\rightarrow 0^{+}}\frac{\ln|γ(t)|}{\ln t}$. This result is sharp except for some borderline cases. As Hickman stated in [J. Funct. Anal. 270 (2016), pp. 560--608], this is a very different situation.

math.CA↗

arXiv2023

Lp bounds for Stein's spherical maximal operators

Let ${\frak M}^α$ be the spherical maximal operators of complex order $α$ on ${\mathbb R^n}$. In this article we show that when $n\geq 2$, suppose \begin{eqnarray*} \|{\frak M}^α f \|_{L^p({\mathbb R^n})} \leq C\|f \|_{L^p({\mathbb R^n})} \end{eqnarray*} holds for some $α$ and $p\geq 2$, then we must have ${\rm Re}\,α\geq \max \{1/p-(n-1)/2,\ -(n-1)/p \}.$ When $n=2$, we prove that $\|{\frak M}^α f \|_{L^p({\mathbb R^2})} \leq C\|f \|_{L^p({\mathbb R^2})}$ if ${\rm Re}\ \ α>\max\{1/p-1/2,\ -1/p\}$, and hence the range of $α$ is sharp in the sense the estimate fails for ${\rm Re}\ α<\max\{1/p-1/2, -1/ p\}.$

math.AP↗

arXiv2023

A Framework for Ductility in Metallic Glasses

The understanding and quantification of ductility in crystalline metals, which has led to their widespread and effective usage as a structural material, is lacking in metallic glasses (MGs). Here, we introduce such a framework for ductility. This very practical framework is based on a MGs ability to support stable shear band growth, quantified in a stress gradient, gradSDB, which we measure and calculate for a range of MGs. Whether a MG behaves ductile or brittle in an application is determined by the comparison between gradsDB the applied stress field gradient, gradsapp. If gradsDB > gradsapp, the MG will behave brittle, if gradsDB < gradsapp, the MG will behave ductile, and gradsapp - gradsDB indicates how ductile. This framework can explain observed plastic properties of MGs and their apparent contradicting brittle and ductile characteristics. Looking forward, proposed framework provides the constitutive relation to quantitatively model their plastic behavior in any application, a requirement to use MGs as structural materials.

cond-mat.mtrl-sci↗

arXiv2022

Characterizations of the Hardy space $\mathcal{H}_{FIO}^{1}(\mathbb{R}^{n})$ for Fourier integral operators

The Hardy spaces for Fourier integral operators $\mathcal{H}_{FIO}^{p}(\mathbb{R}^{n})$, for $1\leq p\leq \infty$, were introduced by Smith in [Smith,1998] and Hassell et al. in [Hassell-Portal-Rozendaal,2020]. In this article, we give several equivalent characterizations of $\mathcal{H}_{FIO}^{1}(\mathbb{R}^{n})$, for example in terms of Littlewood--Paley $g$ functions and maximal functions. This answers a question from [Rozendaal,2021]. We also give several applications of the characterizations.

math.AP↗

arXiv2022

Singular spherical maximal operators on a class of degenerate two-step nilpotent Lie groups

Let $G\cong\mathbb{R}^{d} \ltimes \mathbb{R}$ be a finite-dimensional two-step nilpotent group with the group multiplication $(x,u)\cdot(y,v)\rightarrow(x+y,u+v+x^{T}Jy)$ where $J$ is a skew-symmetric matrix satisfying a degeneracy condition with $2\leq {\rm rank}\, J 0}\big|\int_Σ f(x-ty, u- t x^{T}Jy) dμ(y)\big|, $$ where $Σ$ is a smooth convex hypersurface and $dμ$ is a compactly supported smooth density on $Σ$ such that the Gaussian curvature of $Σ$ is nonvanishing on supp $dμ$. In this paper we prove that when $d\geq 4$, the maximal operator ${\frak M}$ is bounded on $L^{p}(G)$ for the range $(d-1)/(d-2)<p\leq\infty$.

math.CA↗

arXiv2021

Hilbert transforms along variable planar curves: Lipschitz regularity

In this paper, for $1<p<\infty$, we obtain the $L^p$-boundedness of the Hilbert transform $H^γ$ along a variable plane curve $(t,u(x_1, x_2)γ(t))$, where $u$ is a Lipschitz function with small Lipschitz norm, and $γ$ is a general curve satisfying some suitable smoothness and curvature conditions.

math.CA↗

arXiv2020

$L^p(\mathbb{R}^2)$-boundedness of Hilbert Transforms and Maximal Functions along Plane Curves with Two-variable Coefficients

In this paper, for general plane curves $γ$ satisfying some suitable smoothness and curvature conditions, we obtain the single annulus $L^p(\mathbb{R}^2)$-boundedness of the Hilbert transforms $H^\infty_{U,γ}$ along the variable plane curves $(t,U(x_1, x_2)γ(t))$ and the $L^p(\mathbb{R}^2)$-boundedness of the corresponding maximal functions $M^\infty_{U,γ}$, where $p>2$ and $U$ is a measurable function. The range on $p$ is sharp. Furthermore, for $1<p\leq 2$, under the additional conditions that $U$ is Lipschitz and making a $\varepsilon_0$-truncation with $γ(2 \varepsilon_0)\leq 1/4\|U\|_{\textrm{Lip}}$, we also obtain similar boundedness for these two operators $H^{\varepsilon_0}_{U,γ}$ and $M^{\varepsilon_0}_{U,γ}$.

math.CA↗