# Vector- valued distributions and Hardy's uncertainty principle for operators

In this paper, we generalise Hardy's uncertainty principle to vector-valued functions, and hence to operators. The principle for operators can be formulated loosely by saying that the kernel of an operator cannot be localised near the diagonal if the spectrum is also localised.

In this paper we present a theorem that generalizes Sawyer's classic result about mixed weighted inequalities to the multilinear context. Let $\vec{w}=(w_1,.. Read More We introduce the definition of conformable derivative on time scales and develop its calculus. Fundamental properties of the conformable derivative and integral on time scales are proved. Linear conformable differential equations with constant coefficients are investigated, as well as hyperbolic and trigonometric functions. Read More This paper extends the results from arXiv:1702.04569 about sharp$A_2$-$A_\infty$estimates with matrix weights to the non-homogeneous situation. Read More In this paper we are concerned with the existence of periodic solutions for semilinear Duffing equations with impulsive effects. Firstly for the autonomous one, basing on Poincar\'{e}-Birkhoff twist theorem, we prove the existence of infinitely many periodic solutions. Secondly, as for the nonautonomous case, the impulse brings us great challenges for the study, and there are only finitely many periodic solutions, which is quite different from the corresponding equation without impulses. Read More In this paper, we consider sublinear second order differential equations with impulsive effects. Basing on the Poincar\'{e}-Bohl fixed point theorem, we first will prove the existence of harmonic solutions. The existence of subharmonic solutions is also obtained by a new twist fixed point theorem recently established by Qian etc in 2015 (\cite{Qian15}). Read More In this paper we are concerned with the periodic Hamiltonian system with one degree of freedom, where the origin is a trivial solution. We assume that the corresponding linearized system at the origin is elliptic, and the characteristic exponents of the linearized system are$\pm i\omega$with$\omega$be a Diophantine number, moreover if the system is formally linearizable, then it is analytically linearizable. As a result, the origin is always stable in the sense of Liapunov in this case. Read More In this paper we will construct a continuous positive periodic function$p(t)$such that the corresponding superlinear Duffing equation $$x"+a(x)\,x^{2n+1}+p(t)\,x^{2m+1}=0,\ \ \ \ n+2\leq 2m+1<2n+1$$ possesses a solution which escapes to infinity in some finite time, and also has infinitely many subharmonic and quasi-periodic solutions, where the coefficient$a(x)$is an arbitrary positive smooth periodic function defined in the whole real axis. Read More In recent work, we obtained the symmetry algebra for a class of Dirac operators, containing in particular the Dirac-Dunkl operator for arbitrary root system. We now consider the three-dimensional case of the Dirac-Dunkl operator associated to the root system$A_2$, and the associated Dirac equation. The corresponding Weyl group is$\mathrm{S}_3\$, the symmetric group on three elements. Read More