Mark Leckband

A note on the spherical maximal operator for radial functions

The spherical maximal operator for radial functions of RI is shown to be a restricted weak type LP bounded operator for p = n/(n1), n > 2. The proof uses methods for restricted weak type single weight norm inequalities.

Hardy's Inequality and Fractal Measures

Abstract Hardys inequality and the subsequent improvement by McGehee, Pigno, and Smith are generalized from the positive integers to sets of dimension 0, dimension 1, and in between. The asymptotic estimate obtained for the Fourier transform of fractal measures is much in the spirit of recent work by Strichartz.

A note on maximal operators and reversible weak type inequalities

A class of maximal operators is shown to satisfy a weak type inequality and a corresponding converse inequality. The results are applicable to a fractionally iterated Hardy-Littlewood maximal operator.

Minimal potential results for Schrödinger equations with Neumann boundary conditions

Abstract We consider the boundary value problem - Δ p ⁢ u = V ⁢ | u | p - 2 ⁢ u - C {-\Delta_{p}u=V|u|^{p-2}u-C} , where u ∈ W 1 , p ⁢ ( D ) {u\in W^{1,p}(D)} is assumed to satisfy Neumann boundary conditions, and D is a bounded domain in ℝ n {{\mathbb{R}^{n}}} . We derive necessary conditions for the existence of nontrivial solutions. These conditions usually involve a lower bound for the product of a sharp Sobolev constant and an L p {L^{p}} norm of V. When p = n {p=n} , Orlicz norms are used. In many cases, these inequalities are best possible. Applications to linear and non-linear eigenvalue problems are also discussed.

On the local boundedness of singular integral operators

On etudie la classe des operateurs integraux singuliers dont les noyaux satisfont les conditions usuelles de regularite. Soit K un tel operateur. On etablit des conditions necessaires qui impliquent que K a des inegalites en norme L P