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We study the Bochner-Riesz problem for the twisted Laplacian $\mathcal L$ on $\mathbb R^2$. For $p\in [1, \infty]\setminus\{2\}$, it has been conjectured that the Bochner-Riesz means $S_\lambda^\delta(\mathcal L) f$ of order $\delta$ converges in $L^p$ for every $f\in L^p$ if and only if $\delta> \max(0,|(p-2)/p|-1/2)$. We prove the conjecture by obtaining uniform $L^p$ bounds on $S_\lambda^\delta(\mathcal L)$ up to the sharp summability indices., Comment: 15 pages

In this note we show that the strong spherical maximal function in $\mathbb R^d$ is bounded on $L^p$ if $p>2(d+1)/(d-1)$ for $d\ge 3$., Comment: 13 pages

We study the elliptic maximal functions defined by averages over ellipses and rotated ellipses which are multi-parametric variants of the circular maximal function. We prove that those maximal functions are bounded on $L^p$ for some $p\neq \infty$. For this purpose, we obtain some sharp multi-parameter local smoothing estimates., Comment: to appear in Journal of Functional Analysis

Virtual Reality (VR), Augmented Reality (AR), Mixed Reality (MR), digital twin, Metaverse and other related digital technologies have attracted much attention in recent years. These new emerging technologies are changing the world significantly. This research introduces a fusion model, i.e. Fusion Universe (FU), where the virtual, physical, and cognitive worlds are merged together. Therefore, it is crucial to establish a set of principles for the fusion model that is compatible with our physical universe laws and principles. This paper investigates several aspects that could affect immersive and interactive experience; and proposes the fundamental principles for Fusion Universe that can integrate physical and virtual world seamlessly., Comment: 5 pages, 8 figures

Meta-learning performs adaptation through a limited amount of support set, which may cause a sample bias problem. To solve this problem, transductive meta-learning is getting more and more attention, going beyond the conventional inductive learning perspective. This paper proposes so-called task-adaptive pseudo labeling for transductive meta-learning. Specifically, pseudo labels for unlabeled query sets are generated from labeled support sets through label propagation. Pseudo labels enable to adopt the supervised setting as it is and also use the unlabeled query set in the adaptation process. As a result, the proposed method is able to deal with more examples in the adaptation process than inductive ones, which can result in better classification performance of the model. Note that the proposed method is the first approach of applying task adaptation to pseudo labeling. Experiments show that the proposed method outperforms the state-of-the-art (SOTA) technique in 5-way 1-shot few-shot classification.

Recently, the idea of using FP8 as a number format for neural network training has been floating around the deep learning world. Given that most training is currently conducted with entire networks in FP32, or sometimes FP16 with mixed-precision, the step to having some parts of a network run in FP8 with 8-bit weights is an appealing potential speed-up for the generally costly and time-intensive training procedures in deep learning. A natural question arises regarding what this development means for efficient inference on edge devices. In the efficient inference device world, workloads are frequently executed in INT8. Sometimes going even as low as INT4 when efficiency calls for it. In this whitepaper, we compare the performance for both the FP8 and INT formats for efficient on-device inference. We theoretically show the difference between the INT and FP formats for neural networks and present a plethora of post-training quantization and quantization-aware-training results to show how this theory translates to practice. We also provide a hardware analysis showing that the FP formats are somewhere between 50-180% less efficient in terms of compute in dedicated hardware than the INT format. Based on our research and a read of the research field, we conclude that although the proposed FP8 format could be good for training, the results for inference do not warrant a dedicated implementation of FP8 in favor of INT8 for efficient inference. We show that our results are mostly consistent with previous findings but that important comparisons between the formats have thus far been lacking. Finally, we discuss what happens when FP8-trained networks are converted to INT8 and conclude with a brief discussion on the most efficient way for on-device deployment and an extensive suite of INT8 results for many models.

Let $\mathcal L$ denote the twisted Laplacian in $\mathbb C^d$. We study almost everywhere convergence of the Bochner--Riesz mean $S^\delta_{t}(\mathcal L) f$ of $f\in L^p(\mathbb C^d)$ as $t\to \infty$, which is an expansion of $f$ in the special Hermite functions. For $2\le p\le \infty$, we obtain the sharp range of the summability indices $\delta$ for which the convergence of $S^\delta_{t}(\mathcal L) f$ holds for all $f\in L^p(\mathbb C^d)$., Comment: 23 pages

We study $L^{p}\times L^{q}\rightarrow L^{r}$-boundedness of (sub)bilinear maximal functions associated with degenerate hypersurfaces. First, we obtain the maximal bound on the sharp range of exponents $p,q,r$ (except some border line cases) for the bilinear maximal functions given by the model surface $\big\{(y,z)\in\mathbb{R}^{n}\times \mathbb{R}^{n}:|y|^{l_{1}}+|z|^{l_{2}}=1\big\}$, $(l_{1},l_{2})\in [1,\infty)^2$, $n\ge 2$. Our result manifests that nonvanishing Gaussian curvature is not good enough, in contrast with $L^p$-boundedness of the (sub)linear maximal operator associated to hypersurfaces, to characterize the best possible maximal boundedness. Secondly, we consider the bilinear maximal function associated to the finite type curve in $\mathbb R^2$ and obtain a complete characterization of the maximal bound. We also prove multilinear generalizations of the aforementioned results., Comment: 21 pages, 3 figures

We investigate $L^p$ boundedness of the maximal function defined by the averaging operator $f\to \mathcal{A}_t^s f$ over the two-parameter family of tori $\mathbb{T}_t^{s}:=\{ ( (t+s\cos\theta)\cos\phi,\,(t+s\cos\theta)\sin\phi,\, s\sin\theta ): \theta, \phi \in [0,2\pi) \}$ with $c_0t>s>0$ for some $c_0\in (0,1)$. We prove that the associated (two-parameter) maximal function is bounded on $L^p$ if and only if $p>2$. We also obtain $L^p$--$L^q$ estimates for the local maximal operator on a sharp range of $p,q$. Furthermore, the sharp smoothing estimates are proved including the sharp local smoothing estimates for the operators $f\to \mathcal A_t^s f$ and $f\to \mathcal A_t^{c_0t} f$. For the purpose, we make use of Bourgain--Demeter's decoupling inequality for the cone and Guth--Wang--Zhang's local smoothing estimates for the $2$ dimensional wave operator., Comment: 30 pages, 2 figures

We study $L^p$-$L^q$ bounds on the spectral projection operator $\Pi_\lambda$ associated to the Hermite operator $H=|x|^2-\Delta$ in $\mathbb R^d$. We are mainly concerned with a localized operator $\chi_E\Pi_\lambda\chi_E$ for a subset $E\subset\mathbb R^d$ and undertake the task of characterizing the sharp $L^p$--$L^q$ bounds. We obtain sharp bounds in extended ranges of $p,q$. First, we provide a complete characterization of the sharp $L^p$--$L^q$ bounds when $E$ is away from $\sqrt{\lambda}\mathbb S^{d-1}$. Secondly, we obtain the sharp bounds as the set $E$ gets close to $\sqrt\lambda\mathbb S^{d-1}$. Thirdly, we extend the range of $p,q$ for which the operator $\Pi_\lambda$ is uniformly bounded from $L^p(\mathbb R^d)$ to $L^q(\mathbb R^d)$., Comment: The paper is a modified version of a part of the paper Hermite spectral projection operator (arXiv:2006.11762v3). The previous paper will remain unpublished

We study an analogue of Marstrand's circle packing problem for curves in higher dimensions. We consider collections of curves which are generated by translation and dilation of a curve $\gamma$ in $\mathbb R^d$, i.e., $ x + t \gamma$, $(x,t) \in \mathbb R^d \times (0,\infty)$. For a Borel set $F \subset \mathbb R^d\times (0,\infty)$, we show the unions of curves $\bigcup_{(x,t) \in F} ( x+t\gamma )$ has Hausdorff dimension at least $\alpha+1$ whenever $F$ has Hausdorff dimension bigger than $\alpha$, $\alpha\in (0, d-1)$. We also obtain results for unions of curves generated by multi-parameter dilation of $\gamma$. One of the main ingredients is a local smoothing type estimate (for averages over curves) relative to fractal measures., Comment: 15 pages

We study pointwise convergence of the fractional Schr\"odinger means along sequences $t_n$ which converge to zero. Our main result is that bounds on the maximal function $\sup_{n} |e^{it_n(-\Delta)^{\alpha/2}} f| $ can be deduced from those on $\sup_{0

We obtain a complete characterization of $L^p-L^q$ Carleman estimates with weight $e^{v\cdot x}$ for the polyharmonic operators. Our result extends the Carleman inequalities for the Laplacian due to Kenig--Ruiz--Sogge. Consequently, we obtain new unique continuation properties of higher order Schr\"odinger equations relaxing the integrability assumption on the solution spaces.

We establish the optimal $L^p$, $p=2(d+3)/(d+1),$ eigenfunction bound for the Hermite operator $\mathcal H=-\Delta+|x|^2$ on $\mathbb R^d$. Let $\Pi_\lambda$ denote the projection operator to the vector space spanned by the eigenfunctions of $\mathcal H$ with eigenvalue $\lambda$. The optimal $L^2$--$L^p$ bounds on $\Pi_\lambda$, $2\le p\le \infty$, have been known by the works of Karadzhov and Koch-Tataru except $p=2(d+3)/(d+1)$. For $d\ge 3$, we prove the optimal bound for the missing endpoint case. Our result is built on a new phenomenon: improvement of the bound due to asymmetric localization near the sphere $\sqrt\lambda \mathbb S^{d-1}$., Comment: Final version, to appear in JEMS. The paper is an extended revision of a part of the paper Hermite spectral projection operator (arXiv:2006.11762). The earlier paper will remain unpublished permanently

We study $L^p$-Sobolev regularity estimate for the restricted X-ray transforms generated by nondegenerate curves. Making use of the inductive strategy in the recent work by the authors, we establish the sharp $L^p$-regularity estimates for the restricted X-ray transforms in $\mathbb R^{d+1}$, $d\ge 3$. This extends the result due to Pramanik and Seeger in $\mathbb R^3$ to every dimension., Comment: 18 pages

We prove new $L^p$- $L^q$ estimates for averages over dilates of the circle with respect to $\alpha$-dimensional fractal measure, which unify different types of maximal estimates for the circular average. Our results are consequences of $L^p$- $L^q$ smoothing estimates for the wave operator relative to fractal measures. We also discuss similar results concerning the spherical averages., Comment: 25pages

Model-agnostic meta-learning (MAML) is a well-known optimization-based meta-learning algorithm that works well in various computer vision tasks, e.g., few-shot classification. MAML is to learn an initialization so that a model can adapt to a new task in a few steps. However, since the gradient norm of a classifier (head) is much bigger than those of backbone layers, the model focuses on learning the decision boundary of the classifier with similar representations. Furthermore, gradient norms of high-level layers are small than those of the other layers. So, the backbone of MAML usually learns task-generic features, which results in deteriorated adaptation performance in the inner-loop. To resolve or mitigate this problem, we propose contextual gradient scaling (CxGrad), which scales gradient norms of the backbone to facilitate learning task-specific knowledge in the inner-loop. Since the scaling factors are generated from task-conditioned parameters, gradient norms of the backbone can be scaled in a task-wise fashion. Experimental results show that CxGrad effectively encourages the backbone to learn task-specific knowledge in the inner-loop and improves the performance of MAML up to a significant margin in both same- and cross-domain few-shot classification., Comment: Accepted to WACV2022

We investigate the spectral structure of the Neumann-Poincar\'e operator on thin ellipsoids. Two types of thin ellipsoids are considered: long prolate ellipsoids and flat oblate ellipsoids. We show that the totality of eigenvalues of the Neumann-Poincar\'e operators on a sequence of the prolate spheroids is densely distributed in the interval [0,1/2] as their eccentricities tend to 1, namely, as they become longer. We then prove that eigenvalues of the Neumann-Poincar\'e operators on the oblate ellipsoids are densely distributed in the interval [-1/2, 1/2] as the ellipsoids become flatter. In particular, this shows that even if there are at most finitely many negative eigenvalues on the oblate ellipsoids, more and more negative eigenvalues appear as the ellipsoids become flatter. We also show a similar spectral property for flat three dimensional domains., Comment: 24 pages, 1 figure

We prove strong unique continuation property for the differential inequality $|(\partial_t +\Delta)u(x,t)|\le V(x,t)|u(x,t)|$ with $V$ contained in weak spaces. In particular, we establish the strong unique continuation property for $V\in L^\infty_t L^{d/2,\infty}_x$, which has been left open since the works of Escauriaza [6] and Escauriaza-Vega [8]. Our results are consequences of the Carleman estimates for the heat operator in the Lorentz spaces., Comment: The current paper is the last section of the paper Hermite spectral projection operator (arXiv:2006.11762v2). The previous paper will remain unpublished

$L^p$ boundedness of the circular maximal function $\mathcal M_{\mathbb{H}^1}$ on the Heisenberg group $\mathbb{H}^1$ has received considerable attentions. While the problem still remains open, $L^p$ boundedness of $\mathcal M_{\mathbb{H}^1}$ on Heisenberg radial functions was recently shown for $p>2$ by Beltran, Guo, Hickman, and Seeger [2]. In this paper we extend their result considering the local maximal operator $M_{\mathbb{H}^1}$ which is defined by taking supremum over $1

We prove sharp smoothing properties of the averaging operator defined by convolution with a measure on a smooth nondegenerate curve $\gamma$ in $\mathbb R^d$, $d\ge 3$. Despite the simple geometric structure of such curves, the sharp smoothing estimates have remained largely unknown except for those in low dimensions. Devising a novel inductive strategy, we obtain the optimal $L^p$ Sobolev regularity estimates, which settle the conjecture raised by Beltran-Guo-Hickman-Seeger. Besides, we show the sharp local smoothing estimates for every $d$. As a result, we establish, for the first time, nontrivial $L^p$ boundedness of the maximal average over dilations of $\gamma$ for $d\ge 4$., Comment: 31 pages, the endpoint $L^p$ Sobolev regularity results are included, and the paper is revised substantially

We consider the Bochner-Riesz means for the Hermite and special Hermite expansions and study their $L^p$ boundedness with the sharp summability index in a local setting. In two dimensions we establish the boundedness on the optimal range of $p$ and extend the previously known range in higher dimensions. Furthermore, we prove a new lower bound on the $L^p$ summability index for the Hermite Bochner-Riesz means in $\mathbb R^d$, $d\ge 2.$ This invalidates the conventional conjecture which was expected to be true.

We consider the Hausdorff dimension of the divergence set on which the pointwise convergence $\lim_{t\rightarrow 0} e^{it\sqrt{-\Delta}} f(x) = f(x)$ fails when $f \in H^s(\mathbb R^d)$. We especially prove the conjecture raised by Barcel\'o, Bennett, Carbery and Rogers \cite{BBCR} for $d=3$, and improve the previous results in higher dimensions $d\ge4$. We also show that a Strichartz type estimate for $f\to e^{it\sqrt{-\Delta}} f$ with the measure $ dt\,d\mu(x)$ is essentially equivalent to the estimate for the spherical average of $\widehat \mu$ which has been extensively studied for the Falconer distance set problem. The equivalence provides shortcuts to the recent results due to B. Liu and K. Rogers., Comment: 11 pages, 1 figure

Let $M$ be the maximal operator associated to a smooth curve in $\mathbb R^3$ which has nonvanishing curvature and torsion. We prove that $M$ is bounded on $L^p$ if and only if $p>3$., Comment: Minor changes, to appear in Invent. Math

We prove dispersive estimate for the elastic wave equation by which we extend the known Strichartz estimates for the classical wave equation to those for the elastic wave equation. In particular, the endpoint Strichartz estimates are deduced. For the purpose we diagonalize the symbols of the Lam\'e operator and its semigroup, which also gives an alternative and simpler proofs of the previous results on perturbed elastic wave equations. Furthermore, we obtain uniform Sobolev inequalities for the elastic wave operator., Comment: to appear in Proc. Amer. Math. Soc

In this paper, we consider the uniform estimate for the oscillatory integral with stationary phase, which was previously studied by Alazard-Burq-Zuily. We significantly reduce the order of required regularity condition on the phase and amplitude functions for the uniform estimate. We also study estimates for the oscillatory integrals of which phase and amplitude functions depend on the oscillation parameter. The novelty of this article lies in the use of the wave packet decomposition, which transforms the decay estimate for the oscillatory integral to the disjointness property of the supports of wave packets. The latter is geometric in its nature and less sensitive to the smoothness of the phase and amplitude functions., Comment: Section 3 in the previous version removed

Let $\Delta_{\mathbb S^n}$ denote the Laplace-Beltrami operator on the $n$-dimensional unit sphere $\mathbb S^n$. In this paper we show that $$ \| e^{it \Delta_{\mathbb S^n}}f \|_{L^4([0, 2\pi) \times \mathbb S^n)} \leq C \| f\|_{W^{\alpha, 4} (\mathbb S^n)} $$ holds provided that $n\geq 2$, $\alpha> {(n-2)/4}.$ The range of $\alpha$ is sharp up to the endpoint. As a consequence, we obtain space-time estimates for the Schr\"odinger propagator $e^{it \Delta_{\mathbb S^n}}$ on the $L^p$ spaces for $2\leq p\leq \infty.$ We also prove that for zonal functions on ${\mathbb S}^n$, the Schr\"odinger maximal operator $\sup_{0\leq t<2\pi} |e^{it\Delta_{\mathbb S^n}} f|$ is bounded from $W^{\alpha, 2}(\mathbb S^n) $ to $L^{\frac{6n}{3n-2}}(\mathbb S^n)$ whenever $\alpha>{1/3}$.

In this note we are concerned with estimates for the spectral projection operator $\mathcal{P}_\mu$ associated with the twisted Laplacian $L$. We completely characterize the optimal bounds on the operator norm of $\mathcal{P}_\mu$ from $L^p$ to $L^q$ when $1\le p\le 2\le q\le \infty$. As an application, we obtain uniform resolvent estimate for $L$., Comment: 20 pages, 1 figure

We study $L^p$-$L^q$ estimate for the spectral projection operator $\Pi_\lambda$ associated to the Hermite operator $H=|x|^2-\Delta$ in $\mathbb R^d$. Here $\Pi_\lambda$ denotes the projection to the subspace spanned by the Hermite functions which are the eigenfunctions of $H$ with eigenvalue $\lambda$. Such estimates were previously available only for $q=p'$, equivalently with $p=2$ or $q=2$ (by $TT^*$ argument) except for the estimates which are straightforward consequences of interpolation between those estimates. As shown in the works of Karadzhov, Thangavelu, and Koch and Tataru, the local and global estimates for $\Pi_\lambda$ are of different nature. Especially, $\Pi_\lambda$ exhibits complicated behaviors near the set $\sqrt\lambda\mathbb S^{d-1}$. Compared with the spectral projection operator associated to the Laplacian, $L^p$-$L^q$ estimate for $\Pi_\lambda$ is not so well understood up to now for general $p,q$. In this paper we consider $L^p$--$L^q$ estimate for $\Pi_\lambda$ in a general framework including the local and global estimates with $1\le p\le 2\le q\le \infty$ and undertake the work of characterizing the sharp bounds on $\Pi_\lambda$. We establish various new sharp estimates in extended ranges of $p,q$. First of all, we provide a complete characterization of the local estimate for $\Pi_\lambda$ which was first considered by Thangavelu. Secondly, for $d\ge5$, we prove the endpoint $L^2$--$L^{2(d+3)/(d+1)}$ estimate for $\Pi_\lambda$ which has been left open since the work of Koch and Tataru. Thirdly, we extend the range of $p,q$ for which the operator $\Pi_\lambda$ is uniformly bounded from $L^p$ to $L^q$., Comment: 90 pages, 7 figures. The last section of the previous version was taken out to be published separately

Let $H = -\Delta + |x|^2$ 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 $S_R^{\lambda}(H)f(x) = \sum\limits_{k=0}^{\infty} \big(1-{2k+n\over R^2}\big)_+^{\lambda} P_k f(x).$ Here $P_k f$ is the $k$-th Hermite spectral projection operator. For $2\le p<\infty$, we prove that $$ \lim\limits_{R\to \infty} S_R^{\lambda}(H) f=f \ \ \ \text{a.e.} $$ for all $f\in L^p(\mathbb R^n)$ provided that $\lambda> \lambda(p)/2$ and $\lambda(p)=\max\big\{ n\big({1/2}-{1/p}\big)-{1/ 2}, \, 0\big\}.$ Conversely, we also show the convergence generally fails if $\lambda< \lambda(p)/2$ 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 $S_R^{\lambda}(H)$ 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 $\lambda>0$. Compared with the classical result due to Askey and Wainger who showed the optimal $L^p$ convergence for $S_R^{\lambda}(H)$ on ${\mathbb R}$ we only need smaller summability index for a.e. convergence.

In this paper, we consider the Lam\'e operator $-\Delta^\ast$ and study resolvent estimate, uniform Sobolev estimate, and Carleman estimate for $-\Delta^\ast$. First, we obtain sharp $L^p$--$L^q$ resolvent estimates for $-\Delta^\ast$ for admissible $p,q$. This extends the particular case $q=\frac p{p-1}$ due to Barcel\'o et al. \cite{BFPRV} and Cossetti \cite{Co19}. Secondly, we show failure of uniform Sobolev estimate and Carleman estimate for $-\Delta^\ast$. For the purpose we directly analyze the Fourier multiplier of the resolvent. This allows us to prove not only the upper bound but also the lower bound on the resolvent, so we get the sharp $L^p$--$L^q$ bounds for the resolvent of $-\Delta^\ast$. Strikingly, the relevant uniform Sobolev and Carleman estimates turn out to be false for the Lam\'e operator $-\Delta^\ast$ even though the uniform resolvent estimates for $-\Delta^\ast$ are valid for certain range of $p, q$. This contrasts with the classical result regarding the Laplacian $\Delta$ due to Kenig, Ruiz, and Sogge \cite{KRS87} in which the uniform resolvent estimate plays crucial role in proving the uniform Sobolev and Carleman estimates for $\Delta$. We also describe locations of the $L^q$-eigenvalues of $-\Delta^\ast+V$ with complex potential $V$ by making use of the sharp $L^p$--$L^q$ resolvent estimates for $-\Delta^\ast$., Comment: To appear in Journal of Fourier Analysis and Applications

We establish new Strichartz estimates for orthonormal families of initial data in the case of the wave, Klein-Gordon and fractional Schr\"odinger equations. Our estimates extend those of Frank-Sabin in the case of the wave and Klein-Gordon equations, and generalize work of Frank-Lewin-Lieb-Seiringer and Frank-Sabin for the Schr\"{o}dinger equation. Due to a certain technical barrier, except for the classical Schr\"odinger equation, the Strichartz estimates for orthonormal families of initial data have not previously been established up to the sharp summability exponents in the full range of admissible pairs. We obtain the optimal estimates in various notable cases and improve the previous results. The main novelty of this paper is the use of estimates for weighted oscillatory integrals which we combine with an approach due to Frank and Sabin. This strategy also leads us to proving new estimates for weighted oscillatory integrals with optimal decay exponents which we believe to be of wider independent interest. Applications to the theory of infinite systems of Hartree type, weighted velocity averaging lemmas for kinetic transport equations, and refined Strichartz estimates for data in Besov spaces are also provided., Comment: 55 pages, substantial change to the pervious version, especially more applications included

For the one-dimensional Schr\"odinger equation, we obtain sharp maximal-in-time and maximal-in-space estimates for systems of orthonormal initial data. The maximal-in-time estimates generalize a classical result of Kenig--Ponce--Vega and allow us obtain pointwise convergence results associated with systems of infinitely many fermions. The maximal-in-space estimates simultaneously address an endpoint problem raised by Frank--Sabin in their work on Strichartz estimates for orthonormal systems of data, and provide a path toward proving our maximal-in-time estimates., Comment: 21 pages