arXiv subjects
Publications and source records attributed to Danqing He.
We prove a quantitative Roth theorem in the plane for the two-dimensional polynomial pattern $(x_1,x_2), (x_1,x_2)+(t_1,t_2), (x_1,x_2)+(t_1^2+t_2^2,t_1^3+t_2^3)$. A pointwise convergence result for the associated polynomial ergodic average is also obtained. A new bilinear Sobolev improving estimate serves as the primary analytic tool, derived from a new sublevel set estimate.
We prove sharp local smoothing estimates for wave equations on compact Riemannian manifolds in $n+1$ dimensions for odd $n$ and obtain improved estimates in even dimensions. This is achieved by deriving local smoothing estimates for certain Fourier integral operators. We also obtain improved local smoothing estimates for wave equations in Euclidean spaces.
We improve the range of indices when the multilinear Bochner-Riesz means converges pointwisely. We obtain this result by establishing the $L^p$ estimates and weighted estimates of $k$-linear maximal Bochner-Riesz operators inductively, which is new when $p<2/k$ in higher dimensions. To prove these estimates, we make use of a variant of Stein's square function and its multilinear generalization.
For $p\ge 2$, and $\lambda>\max\{n|\tfrac 1p-\tfrac 12|-\tfrac12, 0\}$, we prove the pointwise convergence of cone multipliers, i.e. $$ \lim_{t\to\infty}T_t^\lambda(f)\to f \text{ a.e.},$$ where $f\in L^p(\mathbb R^n)$ satisfies $supp\ \widehat f\subset\{\xi\in\mathbb R^n:\ 1<|\xi_n|<2\}$. Our main tools are weighted estimates for maximal cone operators, which are consequences of trace inequalities for cones.
In this paper, we investigate the H\"ormander type theorems for the multi-linear and multi-parameter Fourier multipliers. When the multipliers are characterized by $L^u$-based Sobolev norms for $1<u\le 2$ , our results on the smoothness assumptions are sharp in the multi-parameter and bilinear case. In the multi-parameter and multi-linear case, our results are almost sharp. Moreover, even in the one-parameter and multi-linear case, our results improve earlier ones in the literature.
We study bilinear rough singular integral operators $\mathcal{L}_Ω$ associated with a function $Ω$ on the sphere $\mathbb{S}^{2n-1}$. In the recent work of Grafakos, He, and Slavíková (Math. Ann. 376: 431-455, 2020), they showed that $\mathcal{L}_Ω$ is bounded from $L^2\times L^2$ to $L^1$, provided that $Ω\in L^q(\mathbb{S}^{2n-1})$ for $4/3<q\le \infty$ with mean value zero. In this paper, we provide a generalization of their result. We actually prove $L^{p_1}\times L^{p_2}\to L^p$ estimates for $\mathcal{L}_Ω$ under the assumption $$Ω\in L^q(\mathbb{S}^{2n-1}) \quad \text{ for }~\max{\Big(\;\frac{4}{3}\;,\; \frac{p}{2p-1} \;\Big)<q\le \infty}$$ where $1<p_1,p_2\le\infty$ and $1/2<p<\infty$ with $1/p=1/p_1+1/p_2$ . Our result improves that of Grafakos, He, and Honzík (Adv. Math. 326: 54-78, 2018), in which the more restrictive condition $Ω\in L^{\infty}(\mathbb{S}^{2n-1})$ is required for the $L^{p_1}\times L^{p_2}\to L^p$ boundedness.
We study $m$-linear homogeneous rough singular integral operators $\mathcal{L}_Ω$ associated with integrable functions $Ω$ on $\mathbb{S}^{mn-1}$ with mean value zero. We prove boundedness for $\mathcal{L}_Ω$ from $L^{p_1}\times \cdots \times L^{p_m}$ to $L^p$ when $1<p_1,\dots, p_m<\infty$ and $1/p=1/p_1+\cdots +1/p_m$ in the largest possible open set of exponents when $Ω\in L^q(\mathbb S^{mn-1})$ and $q\ge 2$. This set can be described by a convex polyhedron in $\mathbb R^m$.
In this work we obtain the pointwise almost everywhere convergence for two families of multilinear operators: (a) truncated homogeneous singular integral operators associated with $L^q$ functions on the sphere and (b) lacunary multiplier operators of limited decay. The a.e. convergence is deduced from the $L^2\times\cdots\times L^2\to L^{2/m}$ boundedness of the associated maximal multilinear operators.
The $L^p$ boundedness theory of convolution operators is \linebreak based on an initial $L^2\to L^2$ estimate derived from the Fourier transform. The corresponding theory of multilinear operators lacks such a simple initial estimate in view of the unavailability of Plancherel's identity in this setting, and up to now it has not been clear what a natural initial estimate might be. In this work we achieve exactly this goal, i.e., obtain an initial $L^2\times\cdots\times L^2\to L^{2/m}$ estimate for general building blocks of $m$-linear multiplier operators. We apply this result to deduce analogous bounds for multilinear rough singular integrals, multipliers of Hörmander type, and multipliers whose derivatives satisfy qualitative estimates.
Let $H = -Δ+ |x|^2$ be the Hermite operator in ${\mathbb R}^n$. In this paper we study almost everywhere convergence of the Bochner-Riesz means associated with $H$ which is defined by $S_R^λ(H)f(x) = \sum\limits_{k=0}^{\infty} \big(1-{2k+n\over R^2}\big)_+^λ P_k f(x).$ Here $P_k f$ is the $k$-th Hermite spectral projection operator. For $2\le p<\infty$, we prove that $$ \lim\limits_{R\to \infty} S_R^λ(H) f=f \ \ \ \text{a.e.} $$ for all $f\in L^p(\mathbb R^n)$ provided that $λ> λ(p)/2$ and $λ(p)=\max\big\{ n\big({1/2}-{1/p}\big)-{1/ 2}, \, 0\big\}.$ Conversely, we also show the convergence generally fails if $λ< λ(p)/2$ in the sense that there is an $f\in L^p(\mathbb R^n)$ for $2n/(n-1)\le p$ such that the convergence fails. This is in surprising contrast with a.e. convergence of the classical Bochner-Riesz means for the Laplacian. For $n\geq 2$ and $p\ge 2$ our result tells that the critical summability index for a.e. convergence for $S_R^λ(H)$ is as small as only the \emph{half} of the critical index for a.e. convergence of the classical Bochner-Riesz means. When $n = 1$, we show a.e. convergence holds for $f\in L^p({\mathbb R})$ with $ p\geq 2$ whenever $λ>0$. Compared with the classical result due to Askey and Wainger who showed the optimal $L^p$ convergence for $S_R^λ(H)$ on ${\mathbb R}$ we only need smaller summability index for a.e. convergence.
We obtain sharp $L^p$ bounds for oscillatory integral operators with generic homogeneous polynomial phases in several variables. The phases considered in this paper satisfy the rank one condition which is an important notion introduced by Greenleaf, Pramanik and Tang. Under certain additional assumptions, we can establish sharp damping estimates with critical exponents to prove endpoint $L^p$ estimates.
Results analogous to those proved by Rubio de Francia are obtained for a class of maximal functions formed by dilations of bilinear multiplier operators of limited decay. We focus our attention to $L^2\times L^2\to L^1$ estimates. We discuss two applications: the boundedness of the bilinear maximal Bochner-Riesz operator and of the bilinear spherical maximal operator. For the latter we improve the known results by reducing the dimension restriction from $n\ge 8$ to $n\ge 4$.
We obtain a sharp $L^2\times L^2 \to L^1$ boundedness criterion for a class of bilinear operators associated with a multiplier given by a signed sum of dyadic dilations of a given function, in terms of the $L^q$ integrability of this function; precisely we show that boundedness holds if and only if $q<4$. We discuss applications of this result concerning bilinear rough singular integrals and bilinear dyadic spherical maximal functions. Our second result is an optimal $L^2\times L^2\to L^1$ boundedness criterion for bilinear operators associated with multipliers with $L^\infty$ derivatives. This result provides the main tool in the proof of the first theorem and is also manifested in terms of the $L^q$ integrability of the multiplier. The optimal range is $q<4$ which, in the absence of Plancherel's identity on $L^1$, should be compared to $q=\infty$ in the classical $L^2\to L^2$ boundedness for linear multiplier operators.
We establish a quantitative weighted inequality for the bilinear rough singular integral, where the bound is controlled by the cube of the weight constant.
We obtain boundedness for the bilinear spherical maximal function in a range of exponents that includes the Banach triangle and a range of $L^p$ with $p<1$. We also obtain counterexamples that are asymptotically optimal with our positive results on certain indices as the dimension tends to infinity.
We obtain new multilinear multiplier theorems for symbols of restricted smoothness which lie locally in certain Sobolev spaces. We provide applications concerning the boundedness of the commutators of Calderón and Calderón-Coifman-Journé.
We prove that the bilinear maximal Bochner-Riesz operator $T_*^λ$ is bounded from $L^{p_1}(\mathbb R^n)\times L^{p_2}(\mathbb R^n)$ to $L^p(\mathbb R^n)$ for appropriate $(p_1,p_2,p)$ when $λ>(4n+3)/5$.
We discuss $L^p(\mathbb R^n)$ boundedness for Fourier multiplier operators that satisfy the hypotheses of the Hörmander multiplier theorem in terms of an optimal condition that relates the distance $|\frac 1p-\frac12|$ to the smoothness $s$ of the associated multiplier measured in some Sobolev norm. We provide new counterexamples to justify the optimality of the condition $|\frac 1p-\frac12|<\frac sn$ and we discuss the endpoint case $|\frac 1p-\frac12|=\frac sn$.