Related papers: Gain/Loss of derivatives for complex vector fields

Let $\mathcal{G}(\frak{g}_1,\frak{g}_2)$ be the class of all probability distribution functions of positive random variables having the given first two moments $\frak{g}_1$ and $\frak{g}_2$. Let $G_1(x)$ and $G_2(x)$ be two probability…

We provide a unifying framework for $\mathcal{L}_2$-optimal reduced-order modeling for linear time-invariant dynamical systems and stationary parametric problems. Using parameter-separable forms of the reduced-model quantities, we derive…

In this note we consider the Schr\"odinger equation on compact manifolds equipped with possibly degenerate metrics. We prove Strichartz estimates with a loss of derivatives. The rate of loss of derivatives depends on the degeneracy of…

Natural gradients have been widely used in optimization of loss functionals over probability space, with important examples such as Fisher-Rao gradient descent for Kullback-Leibler divergence, Wasserstein gradient descent for…

For a principal type pseudodifferential operator, we prove that condition (psi) implies local solvability with a loss of 3/2 derivatives. We use many elements of Dencker's paper on the proof of the Nirenberg-Treves conjecture and we provide…

The representation for the sharp constant ${\rm K}_{n, p}$ in an estimate of the modulus of the $n$-th derivative of an analytic function in the upper half-plane ${\mathbb C}_+$ is considered. It is assumed that the boundary value of the…

In the present paper we prove estimates on {subsolutions of the equation $-Av+c(x)v=0$}, $x\in D$, where $D\subset \bbR^d$ is a domain (i.e. an open and connected set) and $A$ is an integro-differential operator of the Waldenfels type,…

Let $f,g_1,\dots,g_m$ be polynomials with real coefficients in a vector of variables $x=(x_1,\dots,x_n)$. Denote by $\text{diag}(g)$ the diagonal matrix with coefficients $g=(g_1,\dots,g_m)$ and denote by $\nabla g$ the Jacobian of $g$. Let…

We prove well-posedness for some abstract differential equations of the first order. Our result covers the usual case of Lipschitz composition operators. It also contains the case of some integro-differential operators acting on spaces with…

We prove higher summability for the gradient of minimizers of strongly convex integral functionals of the Calculus of Variations with (p,q)-Growth conditions in low dimension. Our procedure is set in the framework of Fractional Sobolev…

Logistic models are commonly used for binary classification tasks. The success of such models has often been attributed to their connection to maximum-likelihood estimators. It has been shown that gradient descent algorithm, when applied on…

We first prove De Giorgi type level estimates for functions in $W^{1,t}(\Omega)$, $\Omega\subset\mathbb{R}^N$, with $t>N\geq 2$. This augmented integrability enables us to establish a new Harnack type inequality for functions which do not…

In this paper we prove an energy estimate with no loss of derivatives for a strictly hyperbolic operator with Zygmund continuous second order coefficients both in time and in space. In particular, this estimate implies the well-posedness…

Let $X$ be a random vector with distribution $P_{\theta}$ where $\theta$ is an unknown parameter. When estimating $\theta$ by some estimator $\varphi(X)$ under a loss function $L(\theta,\varphi)$, classical decision theory advocates that…

In this note we prove a well-posedness result, without loss of derivatives, for strictly hyperbolic wave operators having coefficients which are Zygmund-continuous in the time variable and Lipschitz-continuous in the space variables. The…

We will show the central limit theorem for the general one-dimensional lattice where the space of symbols is a compact metric space. We consider the CLT for Lipschitz-Gibbs probabilities and in the proof we use several properties of the…

This paper studies M-estimators with gradient-Lipschitz loss function regularized with convex penalty in linear models with Gaussian design matrix and arbitrary noise distribution. A practical example is the robust M-estimator constructed…

This paper aims to give a general (possibly compact or noncompact) analog of Strichartz inequalities with loss of derivatives, obtained by Burq, G\'erard, and Tzvetkov [19] and Staffilani and Tataru [51]. Moreover we present a new approach,…

We prove optimal sampling bounds achieving $(1\pm\varepsilon)$-relative error for a broad class of Lipschitz continuous classification loss functions under various regularization terms. This includes important functions such as logistic and…

We study estimation of a multivariate function $f:{\bf R}^d \to {\bf R}$ when the observations are available from function $Af$, where $A$ is a known linear operator. Both the Gaussian white noise model and density estimation are studied.…