Consider the discrete cubic Hilbert transform defined on finitely supported functions f on Z byWe prove that there exists r < 2 and universal constant C such that for all finitely supported f, g on Z there exists an (r, r)-sparse form Λ r,r for whichThis is the first result of this type concerning discrete harmonic analytic operators. It immediately implies some weighted inequalities, which are also new in this setting.It is known [12,25] that this operator extends to a bounded linear operator on ℓ p (Z) to ℓ p (Z), for all 1 < p < ∞. We prove a sparse bound, which in turn proves certain weighted inequalities. Both results are entirely new. By an interval we mean a setWe say a collection of intervals S is sparse if there are subsets E S ⊂ S ⊂ Z with (a) |E S | > 1 4 |S|, uniformly in S ∈ S, and (b) the sets {E S : S ∈ S} are pairwise disjoint.

Let λ 2 ∈ N, and in dimensions d ≥ 5, let A λ f(x) denote the average of f : Z d → R over the lattice points on the sphere of radius λ centered at x. We prove ℓ p improving properties of A λ .It holds in dimension d = 4 for odd λ 2 . The dependence is in terms of ω(λ 2 ), the number of distinct prime factors of λ 2 . These inequalities are discrete versions of a classical inequality of Littman and Strichartz on the L p improving property of spherical averages on R d . In particular they are scale free, in a natural sense. The proof uses the decomposition of the corresponding multiplier whose properties were established by Magyar-Stein-Wainger, and Magyar. We then use a proof strategy of Bourgain, which dominates each part of the decomposition by an endpoint estimate.

We prove new ℓ p (Z d ) bounds for discrete spherical averages in dimensions d ≥ 5. We focus on the case of lacunary radii, first for general lacunary radii, and then for certain kinds of highly composite choices of radii. In particular, if A λ f is the spherical average of f over the discrete sphere of radius λ, we havefor any lacunary sets of integers {λ 2 k }. We follow a style of argument from our prior paper, addressing the full supremum. The relevant maximal operator is decomposed into several parts; each part requires only one endpoint estimate.

We prove that the degenerate trilinear operator C

Decomposition of tensor product representations of the unitary matrix quantum group SU q (2)We first review the spectrum of the Laplacian operator on a general Laakso space before considering modified Hamiltonians for the infinite square well, parabola, and Coulomb potentials. Additionally, we compute the spectrum for the Laplacian and its multiplicities when certain regions of a Laakso space are compressed or stretched and calculate the Casimir force experienced by two uncharged conducting plates by imposing physically relevant boundary conditions and then analytically regularizing the resulting zeta function. Lastly, we derive a general formula for the spectral zeta function and its derivative for Laakso spaces with strict self-similar structure before listing explicit spectral values for some special cases C 2012 American Institute of Physics. [http://dx.