Related papers: Integral inequalities for infimal convolution and …

Assume that $M$ is a CR compact manifold without boundary and CR Yamabe invariant $\mathcal{Y}(M)$ is positive. Here, we devote to study a class of sharp Hardy-Littlewood-Sobolev inequality as follows \begin{equation*} \Bigl| \int_M\int_M…

Let $\mathfrak{g}$ be a real finite-dimensional Lie algebra equipped with a symmetric bilinear form $\langle\cdot,\cdot\rangle$. We assume that $\langle\cdot,\cdot\rangle $ is nil-invariant. This means that every nilpotent operator in the…

Consider the discrete quadratic phase Hilbert Transform acting on $\ell^{2}$ finitely supported functions $$ H^{\alpha} f(n) : = \sum_{m \neq 0} \frac{e^{2 \pi i\alpha m^2} f(n - m)}{m}. $$ We prove that, uniformly in $\alpha \in…

The constrained minimisers of convex integral functionals of the form $\mathscr F(v)=\int_\Omega F(\nabla^k v(x))\mathrm d x $ defined on Sobolev mappings $v\in \mathrm W^{k,1}_g(\Omega , \mathbb R^N )\cap K$, where $K$ is a closed convex…

Given a probability measure $P$ on a $\sigma$-algebra of subsets of a set $\Omega$, an interval $I\subset\mathbb R$, $g\in L^1(I)$, and a function $\varphi\colon I\times\Omega\to I$ fulfilling some conditions we obtain results on the…

Assume that $g(t)\geq 0$, and $$\dot{g}(t)\leq -\gamma(t)g(t)+\alpha(t,g(t))+\beta(t),\ t\geq 0;\quad g(0)=g_0;\quad \dot{g}:=\frac{dg}{dt}, $$ on any interval $[0,T)$ on which $g$ exists and has bounded derivative from the right,…

We give the full solution of the following problem: obtain sharp inequalities between the moduli of smoothness $\omega_\alpha(f,t)_q$ and $\omega_\beta(f,t)_p$ for $0<p<q\le \infty$. A similar problem for the generalized $K$-functionals and…

Suppose that a real nonatomic function space on $[0,1]$ is equipped with two re\-arran\-ge\-ment-invariant norms $\|\cdot\|$ and $|||\cdot|||$. We study the question whether or not the fact that $(X,\|\cdot\|)$ is isometric to…

Let $M,N$ be coprime square-free integers. Let $f$ be a holomorphic cusp form of level $N$ and $g$ be either a holomorphic or a Maa{\ss} form with level $M$. Using a large sieve inequality, we establish a bound of the form…

Let $n \geq 2$, let $\Omega \subset \mathbf{R}^n$ be a bounded domain with smooth boundary, and let $1 \leq p \leq 2$. We prove a reverse-Holder inequality for functions $u$ realizing the best constant in the Sobolev inequality, that is…

We show that analytic analogs of Brunn-Minkowski-type inequalities fail for functional intrinsic volumes on convex functions. This is demonstrated both through counterexamples and by connecting the problem to results of Colesanti, Hug, and…

We improve the Gagliardo-Nirenberg inequality \[ \|\varphi\|_{L^q(\mathbb{R}^n)} \le C \|\nabla\varphi\|_{L^r(\mathbb{R}^n)} \mathcal{L}^{-(\frac 1q - \frac{n-r}{rn})} (\|\nabla\varphi\|_{L^r(\mathbb{R}^n)}), \] $r=2$,…

Let $\mathcal{X}$ be a metric space with doubling measure and $L$ a one-to-one operator of type $\omega$ having a bounded $H_\infty$-functional calculus in $L^2(\mathcal{X})$ satisfying the reinforced $(p_L, q_L)$ off-diagonal estimates on…

The paper deals with $L_p$-boundedness of the Hartley-Fourier convolutions operator and their applied aspects. We establish various new Young-type inequalities and obtain the structure of a normed ring in Banach space when equipping it with…

Let $Q$ be a Lipschitz domain in $\mathbb{R}^n$ and let $f \in L^{\infty}(Q)$. We investigate conditions under which the functional $$I_n(\varphi)=\int_Q |\nabla \varphi|^n+ f(x)\,\mathrm{det} \nabla \varphi\, \mathrm{d}x $$ obeys $I_n \geq…

We provide a comprehensive study of the convergence of the forward-backward algorithm under suitable geometric conditions, such as conditioning or {\L}ojasiewicz properties. These geometrical notions are usually local by nature, and may…

We establish a sharp upper-bound for the first non-zero even eigenvalue (corresponding to an even eigenfunction) of the Hilbert-Brunn-Minkowski operator associated to a strongly convex $C^2$-smooth origin-symmetric convex body $K$ in…

Let G be a noncompact connected Lie group and $\rho$ be the right Haar measure of G. Let $X_1,...,X_q$ be a family of left invariant vector fields which satisfy H\"ormander's condition, and let $\Delta=-\sum_{i=1}^qX_i^2$ be the…

For non-trivial solutions to the zero mode equation on a closed spin manifold \[D \varphi=iA\cdot \varphi,\] we first provide a simple proof for the sharp inequality \eq{ \norm{A}_{L^n}^2 \ge \frac {n}{4(n-1)} Y(M,[g]), } where $Y(M,[g])$…

Let $F(x) := (f_{ij}(x))_{i=1,\ldots,p; j=1,\ldots,q},$ be a ($p\times q$)-real polynomial matrix and let $f(x)$ be the smallest singular value function of $F(x).$ In this paper, we first give the following {\em nonsmooth} version of \L…