By Xuan Duong, Jeff Hogan, Chris Meaney, Adam Sikora

The AMSI overseas convention in Harmonic research and functions was once held at Macquarie college, in Sydney, from 7 to eleven February 2011. the subjects provided incorporated research on Lie teams, capabilities areas, singular integrals, functions to partial differential equations and photograph processing, and wavelets.

This convention introduced jointly top foreign and Australian researchers, in addition to younger Australian researchers and PhD scholars, within the box of Harmonic research and comparable themes for the dissemination of the latest advancements within the box, and for discussions on destiny instructions. The goal was once to reveal the breadth and intensity of contemporary paintings in Harmonic research, to enhance present collaboration, and to forge new links.

As organisers of the convention, we're thankful to the convention members and audio system, a lot of whom travelled huge distances for his or her contributions. monetary help for the meetings used to be supplied by way of the AMSI and the dep. of arithmetic at Macquarie college. As editors of this quantity, we'd additionally prefer to thank the Centre for arithmetic and its functions in Canberra for assist in getting ready those court cases. the sleek working of the convention don't have been attainable with no the organisational abilities of Christine Hale of the dep. of arithmetic at Macquarie college.

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D} uA eA ∈ Rd , we define its even and odd parts ue and uo to be ue = |A| even uA eA and uo = |A| odd uA eA . 3. The Clifford Fourier transform. The Clifford-Fourier transform (CFT) on Rd was introduced by Brackx, De Schepper and Sommen in [1] as the exponential of a differential operator, much in the same manner that the classical Fourier transform can be defined as exp(i(π/2)Hd ) where Hd is the 1 ∂2 Hermite operator Hd = (−∆ + |x|2 − d). Here ∆ = dj=1 2 is the Lapla2 ∂xj d cian on R . Defining the angular momentum operators Lij (1 ≤ i, j ≤ d) by ∂ ∂ Lij = xi − xj and the angular Dirac operator Γ by ∂xj ∂xi (1) Γ=− ei ej Lij 1≤i

T tn/2 |x|n−2 tn/2 5. For x ∈ Rm \K, y ∈ Rn \K and all t > 1, pt (x, y) ≈ C 1 1 d(x, y)2 + exp − c t tn/2 |x|m−2 tm/2 |y|n−2 40 XUAN THINH DUONG, JI LI, AND ADAM SIKORA 6. For x, y ∈ Rm \K and all t > 1, Ct−n/2 d(x, y)2 |x|2 + |y|2 C exp − c exp − c + |x|m−2 |y|m−2 t t tm/2 n 7. For x, y ∈ R \K and all t > 1, pt (x, y) ≈ pt (x, y) ≈ |x|2 + |y|2 d(x, y)2 Ct−n/2 C exp − c exp − c + . |x|n−2 |y|n−2 t t tn/2 3. The boundedness of Hardy-Littlewood maximal function In this section we consider M = Rm Rn for m > n > 2.

Notice that Rd decomposes as Rd = Λ0 ⊕ Λ1 ⊕ · · · ⊕ Λd , where Λj = { |A|=j xA eA ; xA ∈ R}. In particular, Λ0 is the collection of scalars while Λ1 is the collection of vectors. e, [x]p = |A|=p xA eA . It is a simple matter to show that if x, y ∈ Rd are vectors, then x2 = −|x|2 (a scalar) and their Clifford product xy may be expressed as xy = − x, y + x ∧ y ∈ Λ0 ⊕ Λ2 . Here x, y is the usual dot product of x and y while x ∧ y is their wedge product. The linear involution u ¯ of u ∈ Rd is determined by the rules x ¯ = −x ¯ for all u, v ∈ Rd .