Related papers: Discretization Theorems for Entire Functions of Ex…

We prove square function estimates in $L_2$ for general operators of the form $B_1D_1+D_2B_2$, where $D_i$ are partially elliptic constant coefficient homogeneous first order self-adjoint differential operators with orthogonal ranges, and…

We prove a functional extension of an exponential inequality originally proposed by Bin Zhao and proved by Xiaosheng Mou. The main result asserts that if $\alpha_1\leq \cdots\leq \alpha_n$ and $\sum_{k=1}^n \alpha_k=0$, then \[ \sum_{k=1}^n…

We study the polynomial approximation problem in $L^2(\mu_1)$ where $\mu_1(dx) = e^{-|x|}/2 dx$. We show that for any absolutely continuous function $f$, $$ \sum_{k=1}^{\infty} \log^2(e+k) \langle f, P_k \rangle^2 \ \leq C \left(…

We prove Lp Poincare inequalities for functions on the discrete cube and their discrete gradient. We thus recover an exponential inequality and the concentration phenomenon for the uniform probability on the cube first obtained by Bobkov…

In this work we prove that certain entire $q$-functions have infinitely many nonzero roots $\left\{ \rho_{n}\right\} _{n=1}^{\infty}$, as $n\to+\infty$ the moduli $\left|\rho_{n}\right|$ grow at least exponentially. Applications to entire…

We prove concentration inequalities for functions of independent random variables {under} sub-gaussian and sub-exponential conditions. The utility of the inequalities is demonstrated by an extension of the now classical method of Rademacher…

Let $\Omega$ be an unbounded open subset of ${\mathbb R}^n$, $n \ge 2$, and $A : \Omega \times {\mathbb R}^n \to {\mathbb R}^n$ be a function such that $$ C_1 |\zeta|^p \le \zeta A (x, \zeta), \quad |A (x, \zeta)| \le C_2 |\zeta|^{p-1} $$…

We describe a `discretize-then-relax' strategy to globally minimize integral functionals over functions $u$ in a Sobolev space subject to Dirichlet boundary conditions. The strategy applies whenever the integral functional depends…

This paper is devoted to study the sharp Moser-Trudinger type inequalities in whole space $\mathbb R^N$, $N \geq 2$ in more general case. We first compute explicitly the \emph{normalized vanishing limit} and the \emph{normalized…

Let $\mathcal{B}$ be a homogeneous differential operator of order $l=1$ or $l=2$. We show that a sequence of functions of the form $(\mathcal{B}u_j)_j$ converging in the $L^1$-sense to a compact, convex set $K$ can be modified into a…

We consider incomplete exponential sums in several variables of the form S(f,n,m) = \frac{1}{2^n} \sum_{x_1 \in \{-1,1\}} ... \sum_{x_n \in \{-1,1\}} x_1 ... x_n e^{2\pi i f(x)/p}, where m>1 is odd and f is a polynomial of degree d with…

We use localized topologies to prove existence and optimal regularity results for the divergence equation $\mathrm{div} (v) = F$ in critical cases $v \in L_1(\Omega;\mathbb{R}^m)$ or $v \in C_0(\Omega;\mathbb{R}^m)$, i.e. we characterize…

For a real-valued measurable function $f$ and a nonnegative, nondecreasing function $\phi$, we first obtain a Chebyshev type inequality which provides an upper bound for $\displaystyle \phi(\lambda_{1}) \mu(\{x \in \Omega : f(x) \geq…

We study optimal dimensionless inequalities $$ \|f\|_{U^k} \leq \|f\|_{\ell^{p_{k,n}}} $$ that hold for all functions $f\colon\mathbb{Z}^d\to\mathbb{C}$ supported in $\{0,1,\ldots,n-1\}^d$ and estimates $$ \|1_A\|_{U^k}^{2^k}\leq…

A recent refinement of Ker\'ekj\'art\'o's Theorem has shown that in $\mathbb R$ and $\mathbb R^2$ all $\mathcal C^l$-solutions of the functional equation $f^n =\textrm{Id}$ are $\mathcal C^l$-linearizable, where $l\in \{0,1,\dots \infty\}$.…

In this paper we prove necessary conditions for the boundedness of fractional operators on the variable Lebesgue spaces. More precisely, we find necessary conditions on an exponent function $\pp$ for a fractional maximal operator $M_\alpha$…

For the functions $f$, which can be represented in the form of the convolution $f(x)=\frac{a_{0}}{2}+\frac{1}{\pi}\int\limits_{-\pi}^{\pi}\sum\limits_{k=1}^{\infty}e^{-\alpha k^{r}}\cos(kt-\frac{\beta\pi}{2})\varphi(x-t)dt$,…

By using Liu's $q$-partial differential equations theory, we prove that if an analytic function in several variables satisfies a system of $q$-partial differential equations, if and only if it can be expanded in terms of homogeneous…

In this paper we study the right differentiability of a parametric infimum function over a parametric set defined by equality constraints. We present a new theorem with sufficient conditions for the right differentiability with respect to…

We prove the existence of maximizers and the precompactness of $L^p$-normalized maximizing sequences modulo symmetries for all valid scale-invariant Fourier extension inequalities on the cone in $\mathbb R^{1+d}$. In the range for which…