We use the Morrey norm estimate for the imaginary power of the Laplacian to prove an interpolation inequality for the fractional power of the Laplacian on Morrey spaces. We then prove a Hardy-type inequality and use it together with the interpolation inequality to obtain a Heisenberg-type inequality in Morrey spaces.

The theory of reproducing kernel Hilbert spaces is applied to a minimization problem with prescribed nodes. We re-prove and generalize some results previously obtained by Gunawan et al. [2,3], and also discuss the Hölder continuity of the solution to the problem.

We prove weighted Lp-Lq estimates for the maximal operators ℳα, given by , where μt denotes the normalised surface measure on the sphere of centre 0 and radius t in Rd. The techniques used involve interpolation and the Mellin transform. To do this, we also prove weighted Lp-Lq estimates for the operators of convolution with the kernels |·|−α−iη.

We study the boundedness of singular integral operators that are imaginary powers of the Laplace operator in Rn, especially from weighted Hardy spaces to weighted Lebesgue spaces where 0 < p ≤ 1. In particular, we prove some estimates for these operators when 0 < p ≤ 1 and w is in the Muckenhoupt's class Aq, for some q > 1.

We study the space lp, 1 ≤ p ≤ ∞, and its natural n-norm, which can viewed as a generalisation of its usual norm. Using a derived norm equivalent to its usual norm, we show that lp is complete with respect to its natural n-norm. In addition, we also prove a fixed point theorem for lp as an n-normed space.

Let φ denote the normalised surface measure on the unit sphere Sn−1. We shall be interested in the weighted Lp estimate for Stein's maximal function Mφf, namely

where w is an Ap weight, especially for 1 < p ≤ 2. Using the Mellin transformation approach, we prove that the estimate holds for every weight wδ where w ∈ Ap and 0 ≤ δ < (p(n − 1) − n)/(n(p − 1)), for n ≥ 3 and n/(n − 1) < p ≤ 2.