Related papers: Sparse Bounds for Maximal Monomial Oscillatory Hil…

We proof pointwise bounds for rough Fourier integral operators by the $L^p$ Hardy-Littlewood maximal function. We assume the Fourier integral operators have amplitudes in $L^\infty S^m_\rho$ and phases $\varphi$ such that $\varphi(x,\xi) -…

We obtain necessary and sufficient conditions on weights for a wide class of integral transforms to be bounded between weighted $L^p-L^q$ spaces, with $1\leq p\leq q\leq \infty$. The kernels $K(x,y)$ of such transforms are only assumed to…

We investigate the Hilbert transform and the maximal operator along a class of variable non-flat polynomial curves $(P(t),u(x)t)$ with measurable $u(x)$, and prove uniform $L^p$ estimates for $1<p<\infty$. In particular, via the change of…

We examine dyadic paraproducts and commutators in the non-homogeneous setting, where the underlying Borel measure $\mu$ is not assumed to be doubling. We first establish a pointwise sparse domination for dyadic paraproducts and related…

We study the bilinear Hilbert transform and bilinear maximal functions associated to polynomial curves and obtain uniform $L^r$ estimates for $r>\frac{d-1}{d}$ and this index is sharp up to the end point.

Let $r>\frac{4}{3}$ and let $\Omega \in L^{r}(\mathbb{S}^{2n-1})$ have vanishing integral. We show that the bilinear rough singular integral $$T_{\Omega}(f,g)(x)= \textrm{p.v.}…

The paper deals with a complex polynomial $H$ in two variables having - a generic highest homogeneous part (without multiple zero lines), - nonconstant lower terms. In particular, under these conditions the polynomial $H$ has at least two…

We give lower bounds for the problem of stable sparse recovery from /adaptive/ linear measurements. In this problem, one would like to estimate a vector $x \in \R^n$ from $m$ linear measurements $A_1x,..., A_mx$. One may choose each vector…

We prove that the weak-$L^{p}$ norms, and in fact the sparse $(p,1)$-norms, of the Carleson maximal partial Fourier sum operator are $\lesssim (p-1)^{-1}$ as $p\to 1^+$. This is an improvement on the Carleson-Hunt theorem, where the same…

Consider the standard Gaussian linear regression model $Y=X\theta+\epsilon$, where $Y\in R^n$ is a response vector and $ X\in R^{n*p}$ is a design matrix. Numerous work have been devoted to building efficient estimators of $\theta$ when $p$…

We prove upper and lower bounds for all the coefficients in the Hilbert Polynomial of a graded Gorenstein algebra $S=R/I$ with a quasi-pure resolution over $R$. The bounds are in terms of the minimal and the maximal shifts in the resolution…

We provide an example of a pair of weights $(u,v)$ for which the Hardy-Littlewood maximal function is bounded from $L^p(v)$ to $L^p(u)$ and from $L^{p'}(u^{1-p'})$ to $L^{p'}(v^{1-p'})$ while a dyadic sparse operator is not bounded on the…

For a Markov semigroup $P_t$ with invariant probability measure $\mu$, a constant $\ll>0$ is called a lower bound of the ultra-exponential convergence rate of $P_t$ to $\mu$, if there exists a constant $C\in (0,\infty)$ such that $$…

Sharp bounds are obtained, under a variety of assumptions on the eigenvalues of the Einstein tensor, for the ratio of the Hawking mass to the areal radius in static, spherically symmetric space-times.

Let n denote the number of variables and m the number of equations in a sparse polynomial system over the binary field. We study the inconsistency probability of randomly generated sparse polynomial systems over the binary field, where each…

For $ 0< \lambda < \frac{1}2$, let $ B_{\lambda }$ be the Bochner-Riesz multiplier of index $ \lambda $ on the plane. Associated to this multiplier is the critical index $1 < p_\lambda = \frac{4} {3+2 \lambda } < \frac{4}3$. We prove a…

By applying Fourier transformations to the natural orthogonal oscillator representations of special linear Lie algebras, Luo and the second author (2013) obtained a large family of infinite-dimensional irreducible representations of the…

In this sequence, we first prove an abstract Morse index theorem in a Hilbert space modeling a variational problem with constraints. Then, our abstract formulation is applied to study several optimization setups including closed CMC…

We consider homomorphisms of hermitian holomorphic Hilbert bundles. Assuming the homomorphism decreases curvature, we prove that its pointwise norm is plurisubharmonic.

We present a comprehensive analysis of an algorithm for evaluating high-dimensional polynomials that are invariant under permutations and rotations. The key bottleneck is the contraction of a high-dimensional symmetric and sparse tensor…