Related papers: Optimal Young's convolutions inequality and its re…
We prove sharp estimates for the decay in time of solutions to a rather general class of non-local in time subdiffusion equations on a bounded domain subject to a homogeneous Dirichlet boundary condition. Important special cases are the…
Following the works of Lyons and Oberlin, Seeger, Tao, Thiele and Wright, we relate the variation of certain discrete curves on the Lie group $\text{SU}(1,1)$ to the corresponding variation of their linearized versions on the Lie algebra.…
A nonlinear inequality is formulated in the paper. An estimate of the rate of decay of solutions to this inequality is obtained. This inequality is of interest in a study of dynamical systems and nonlinear evolution equations. It can be…
We present some reverse Young-type inequalities for the Hilbert-Schmidt norm as well as any unitarily invariant norm. Furthermore, we give some inequalities dealing with operator means. More precisely, we show that if $A, B\in {\mathfrak…
Sharp isoperimetric inequalities for the sine transform of even isotropic measures are established. The corresponding reverse inequalities are obtained in an asymptotically optimal form. These new inequalities have direct applications to…
We obtain new concavity results, up to a suitable transformation, for a class of quasi-linear equations in a convex domain involving the $p$-Laplace operator and a general nonlinearity satisfying concavity type assumptions. This provides an…
The uniform quadratic optimizatin problem (UQ) is a nonconvex quadratic constrained quadratic programming (QCQP) sharing the same Hessian matrix. Based on the second-order cone programming (SOCP) relaxation, we establish a new sufficient…
We prove tight H\"olderian error bounds for all $p$-cones. Surprisingly, the exponents differ in several ways from those that have been previously conjectured; moreover, they illuminate $p$-cones as a curious example of a class of objects…
The `full' edge isoperimetric inequality for the discrete cube (due to Harper, Bernstein, Lindsay and Hart) specifies the minimum size of the edge boundary $\partial A$ of a set $A \subset \{0,1\}^n$, as a function of $|A|$. A weaker (but…
In this paper, we study the further improvements of the reverse Young and Heinz inequalities for the wider range of $v$, namely $v\in \mathbb{R}$. These modified inequalities are used to establish corresponding operator inequalities on a…
In the present paper we characterize the existence and uniqueness of maximal Lp-regular solutions of high order convolution operator equations. Particularly, we get coercive uniform estimates with respect to spectral parameter and we show…
We study the inequality $ -\Delta u - \frac{\mu}{|x|^2} u \geq (|x|^{-\alpha} * u^p)u^q$ in an unbounded cone $\mathcal{C}_\Omega^\rho\subset \mathbb{R}^N$ ($N\geq 2$) generated by a subdomain $\Omega$ of the unit sphere $S^{N-1}\subset…
We consider the shape optimization problems for the quantities $\lambda(\Omega)T^q(\Omega)$, where $\Omega$ varies among open sets of $\mathbb{R}^d$ with a prescribed Lebesgue measure. While the characterization of the infimum is completely…
In a previous paper with the same title, we gave an upper bound for the exponent of uniform rational approximation to a quadruple of $\mathbb{Q}$-linearly independent real numbers in geometric progression. Here, we explain why this upper…
We investigate the local boundedness of solutions $u:\Omega_T\to\mathbb{R}$ to parabolic equations of the form \begin{equation*} \partial_tu-\mathrm{div}\,\mathcal{A}(x,t,Du)=0 \qquad\mbox{in }\Omega_T=\Omega\times(0,T) \end{equation*} that…
It is known that the overlap of two energy eigenstates in a decaying quantum system is bounded from above by a function of the energy detuning and the individual decay rates. This is usually traced back to the positive definiteness of an…
We prove that the weakly singular, non-linear convolution integral equation $\int_{\mathbb{R}^n}|x-y|^{-\lambda}f(y)dy=f(x)^{p-1}$, where $0<\lambda<n$, and $p=2n/(2n-\lambda)$ has at least two non-equivalent solutions. This answers a…
Jensen's operator inequality for convexifiable functions is obtained. This result contains classical Jensen's operator inequality as a particular case. As a consequence, a new refinement and a reverse of Young's inequality is given.
The hypercontractive inequality on the discrete cube plays a crucial role in many fundamental results in the Analysis of Boolean functions, such as the KKL theorem, Friedgut's junta theorem and the invariance principle. In these results the…
Let $q$ be a quadratic form over a field $F$ and let $L$ be a field extension of $F$ of odd degree. It is a classical result that if $q_L$ is isotropic (resp. hyperbolic) then $q$ is isotropic (resp. hyperbolic). In turn, given two…