Bartłomiej Dyda

Fractional Hardy inequality with a remainder term

We calculate the regional fractional Laplacian on some power function on an interval. As an application, we prove Hardy inequality with an extra term for the fractional Laplacian on the interval with the optimal constant. As a result, we obtain the fractional Hardy inequality with best constant and an extra lower-order term for general domains, following the method developed by M. Loss and C. Sloane [arXiv:0907.3054v1 [math.AP]]

Eigenvalues of the fractional Laplace operator in the unit ball

We describe a highly efficient numerical scheme for finding two-sided bounds for the eigenvalues of the fractional Laplace operator (−Δ)α/2 in the unit ball D⊂Rd, with a Dirichlet condition in the complement of D. The standard Rayleigh–Ritz variational method is used for the upper bounds, while the lower bounds involve the lesser known Aronszajn method of intermediate problems. Both require explicit expressions for the fractional Laplace operator applied to a linearly dense set of functions in L2(D). We use appropriate Jacobi-type orthogonal polynomials, which were studied in a companion paper (B. Dyda, A. Kuznetsov and M. Kwaśnicki, ‘Fractional Laplace operator and Meijer G-function’, Constr. Approx., to appear, doi:10.1007/s00365-016-9336-4). Our numerical scheme can be applied analytically when polynomials of degree two are involved. This is used to partially resolve the conjecture of Kulczycki, which claims that the second smallest eigenvalue corresponds to an antisymmetric function: we prove that this is the case when either d⩽2 and α∈(0,2], or d⩽9 and α=1, and we provide strong numerical evidence for d⩽9 and general α∈(0,2].

Hardy Inequalities and Non-explosion Results for Semigroups

We prove non-explosion results for Schrödinger perturbations of symmetric transition densities and Hardy inequalities for their quadratic forms by using explicit supermedian functions of their semigroups.

ON WEIGHTED POINCARÉ INEQUALITIES

The aim of this note is to show that Poincare inequalities imply corresponding weighted versions in a quite general setting. Fractional Poincare inequalities are considered, too. The proof is short and does not involve covering arguments.

Muckenhoupt A p -properties of Distance Functions and Applications to Hardy–Sobolev -type Inequalities

Let X be a metric space equipped with a doubling measure. We consider weights w(x) = dist(x,E)−α, where E is a closed set in X and α∈ℝ

A fractional order Hardy inequality

\alpha \in \mathbb {R}

The best constant in a fractional Hardy inequality

. We establish sharp conditions, based on the Assouad (co)dimension of E, for the inclusion of w in Muckenhoupt’s Ap classes of weights, 1 ≤ p < ∞. With the help of general Ap-weighted embedding results, we then prove (global) Hardy–Sobolev inequalities and also fractional versions of such inequalities in the setting of metric spaces.