<p style='text-indent:20px;'>We investigate Hardy-Rellich inequalities for perturbed Laplacians. In particular, we show that a non-trivial angular perturbation of the free operator typically improves inequality, and may also provide an estimate which does not hold in case. The main examples are related to introduction magnetic field: this is manifestation diamagnetic phenomenon, has been ...

In this paper it is shown that a new Hardy-Hilbert type integral inequality can be established by introducing a proper integral kernel function, and that the constant factor is proved to be the best possible. In particular, for case p = 2, a new Hilbert type integral inequality is given .And as the mathematics aesthetics, several important constants π and Euler number En appear simultaneously i...

We established some new α-conformable dynamic inequalities of Hardy–Knopp type. Some generalizations Hardy type in two variables on time scales are established. Furthermore, we investigated Hardy’s inequality for several functions calculus. Our results proved by using two-dimensional Jensen’s and Fubini’s theorem scales. When α=1, then obtain well-known time-scale due to Hardy. As special cases...

A Hardy inequality of the form ∫ Ω |∇f(x)|dx ≥ ( p− 1 p )p ∫ Ω {1 + a(δ, ∂Ω)(x)} |f(x)| p δ(x)p dx, for all f ∈ C∞ 0 (Ω \ R(Ω)), is considered for p ∈ (1,∞), where Ω is a domain in R, n ≥ 2, R(Ω) is the ridge of Ω, and δ(x) is the distance from x ∈ Ω to the boundary ∂Ω. The main emphasis is on determining the dependance of a(δ, ∂Ω) on the geometric properties of ∂Ω. A Hardy inequality is also e...

We establish a generalized Jensen’s inequality for analytic vector-valued functions on TN using a monotonicity property of vector-valued Hardy martingales. We then discuss how this result extends to functions on a compact abelian group G, which are analytic with respect to an order on the dual group. We also give a generalization of Helson and Lowdenslager’s version of Jensen’s inequality to ce...

Abstract. Sharp error estimates in terms of the fractional Laplacian and a weighted Besov norm are obtained for Pitt’s inequality by using the spectral representation with weights for the fractional Laplacian due to Frank, Lieb and Seiringer and the sharp Stein-Weiss inequality. Dilation invariance, group symmetry on a non-unimodular group and a nonlinear Stein-Weiss lemma are used to provide s...

We address the question of exact controllability of the wave and Schrödinger equations perturbed by a singular inverse-square potential. Exact boundary controllability is proved in the range of subcritical coefficients of the singular potential and under suitable geometric conditions. The proof relies on the method of multipliers. The key point in the proof of the observability inequality is a ...

We derive sharp Moser-Trudinger inequalities on the CR sphere. The first type is in the Adams form, for powers of the sublaplacian and for general spectrally defined operators on the space of CRpluriharmonic functions. We will then obtain the sharp Beckner-Onofri inequality for CR-pluriharmonic functions on the sphere, and, as a consequence, a sharp logarithmic Hardy-Littlewood-Sobolev inequali...

The constant is the best possible. There is a vast literature which deals with alternative proofs, various generalizations and extensions, and numerous variants and applications in analysis of inequality (1.1); see [1, 2, 3, 5, 7, 9, 8, 10, 13, 14, 15, 16, 17, 18, 19] and the references given therein. According to Hardy (see [6, Theorem 349]), Carleman’s inequality was generalized as follows. I...

In the present article, we prove some new generalizations of dynamic inequalities Hardy-type by utilizing diamond-? integrals on time scales. Furthermore, in two variables scales are proved. Moreover, Hardy for several functions using The results proved Jensen inequality and Fubini theorem Our main extend existing integral discrete inequalities. Symmetry plays an essential role determining corr...