By using the symmetry of Dunkl Laplacian operator, we prove a sharp Shannon-type inequality and logarithmic Sobolev for transform. Combining these inequalities, obtain new, short proof Heisenberg-type uncertainty principles in setting. Moreover, by combining Nash’s inequality, Carlson’s Sobolev’s embedding theorems transform, new inequalities involving L∞-norm. Finally, Lp-spaces, from which de...

The well-known second order moment Heisenberg-Weyl inequality (or uncertainty relation) in Fourier Analysis states: Assume that f : R → C is a complex valued function of a random real variable x such that f ∈ L(R). Then the product of the second moment of the random real x for |f | and the second moment of the random real ξ for ∣∣∣f̂ ∣∣∣2 is at least E|f |2 / 4π, where f̂ is the Fourier transform...

An analogue of the classical Heisenberg inequality is given for an infinite-dimensional space. The proof relies on a commutation relationship and integration by parts formula for Gaussian measure. We also discuss when the equality holds.

the aim of this paper is to prove new quantitative uncertainty principle for the generalized fourier transform connected with a dunkl type operator on the real line. more precisely we prove an lp-lq-version of morgan's theorem.

We show that the Heisenberg groups H 2n+1 of dimension ve and higher, considered as Rieman-nian manifolds, satisfy a quadratic isoperimetric inequality. (This means that each loop of length L bounds a disk of area L 2). This implies several important results about isoperimetric inequalities for discrete groups that act either on H 2n+1 or on complex hyperbolic space, and provides interesting ex...

In this paper, we study two types of weighted Hardy–Littlewood–Sobolev (HLS) inequalities, also known as Stein–Weiss inequalities, on the Heisenberg group. More precisely, we prove the |u| weighted HLS inequality in Theorem 1.1 and the |z| weighted HLS inequality in Theorem 1.5 (where we have denoted u = (z, t) as points on the Heisenberg group). Then we provide regularity estimates of positive...

In quantum physics, measurement error and disturbance were first naively thought to be simply constrained by the Heisenberg uncertainty relation. Later, more rigorous analysis showed that the error and disturbance satisfy more subtle inequalities. Several versions of universally valid errordisturbance relations (EDR) have already been obtained and experimentally verified in the regimes where na...