We give a short proof of recently established Hardy-type inequality due to Keller, Pinchover, and Pogorzelski together with its optimality. Moreover, we identify the remainder term which makes it into an identity.

Let = be a non-negative matrix. Denote by the supremum of those , satisfying the following inequality: where , , and also is increasing, non-negative sequence of real numbers. If we used instead of The purpose of this paper is to establish a Hardy type formula for , where is Hausdorff matrix and A similar result is also established for where In particular, we apply o...

We prove a contractive Hardy-Littlewood type inequality for functions from $H^p(\mathbb{T})$, $0 < p \le 2$ which is sharp in the first two Taylor coefficients and asymptotically at infinity.

&lt;p style='text-indent:20px;'&gt;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, we study some systems of integral equations, including those related to Hardy-Littlewood-Sobolev (HLS) inequalities. We prove that, under some integrability conditions, the positive regular solutions to the systems are radially symmetric and monotone about some point. In particular, we established the radial symmetry of the solutions to the Euler-Lagrange equations associated wit...