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

The AMSI overseas convention in Harmonic research and purposes used to be held at Macquarie collage, in Sydney, from 7 to eleven February 2011. the subjects offered incorporated research on Lie teams, capabilities areas, singular integrals, purposes to partial differential equations and photograph processing, and wavelets.

This convention introduced jointly best foreign and Australian researchers, in addition to younger Australian researchers and PhD scholars, within the box of Harmonic research and similar issues for the dissemination of the latest advancements within the box, and for discussions on destiny instructions. The goal used to be to exhibit the breadth and intensity of modern paintings in Harmonic research, to enhance current collaboration, and to forge new links.

As organisers of the convention, we're thankful to the convention members and audio system, lots of whom travelled huge distances for his or her contributions. monetary aid for the meetings used to be supplied through the AMSI and the dept of arithmetic at Macquarie collage. As editors of this quantity, we'd additionally wish to thank the Centre for arithmetic and its purposes in Canberra for assist in getting ready those court cases. the graceful working of the convention don't have been attainable with no the organisational abilities of Christine Hale of the dept of arithmetic at Macquarie college.

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Furthermore, we can construct balls B x such that the ball B with centre z, radius r, lying mostly in Rn by choosing z ∈ Rn , r large enough and d(z, x) = r − for sufficiently small. This implies that 1 M(f )(x) = sup χ2 (y)dy = 1. B x V (B) B Now consider the centered Hardy–Littlewood Maximal function Mc (f ). By the definition for any r > 0, 1 C f (z)dz = m dz. V (x, r) B(x,r) r B(x,r)∩(Rn \K) BOUNDEDNESS OF MAXIMAL FUNCTIONS This implies that r > |x| and the term to C rm 41 dz is comparable B(x,r)∩(Rn \K) (r − |x|)n .

Let H± d (14) 1 2 f ∈ H± (Rd ) ⇐⇒ [Fd+ f (y)] = [1 ∓ y/|y|][Fd+ f (y)]. 2 Proof. Observe that Cf (x + t) = − (15) 1 σd−1 φt ∗ f (x) x−1 and φt is the L1 -normalized dilate of φ given by |x − 1|d+1 φt (x) = t−d φ(x/t). Also, where φ(x) = Rd π (d+1)/2 e−i x,y dx = e−|y| (d+1)/2 2 Γ((d + 1)/2) (1 + |x| ) and consequently, xj e−i x,y π (d+1)/2 ∂ −|y| π (d+1)/2 yj −|y| = −i dx = i e e . 2 (d+1)/2 Γ((d + 1)/2) ∂yj Γ((d + 1)/2) |y| Rd (1 + |x| ) From this we see that the classical FT of φ is given by (16) Fd φ(y) = − π (d+1)/2 y 1+i e−|y| .

Xj → Rd we define a Dirac operator ∂ by ∂f ∂f = + ∂x0 d ej j=1 ∂f . ∂xj We say f is left monogenic on Ω ⊂ Rd (respectively Σ ⊂ Rd+1 ) if Df = 0 (respectively ∂f = 0). If d = 1 and f : Σ ⊂ R2 → R1 ≡ C, then f is left monogenic if and only if f (x, y) = u(x, y) + e1 v(x, y) is complex-analytic, or equivalently, if and only if u and v satisfy the Cauchy-Riemann equations. When d = 2, then f = f0 + f1 e1 + f2 e2 + f12 e12 : Ω ⊂ R2 → R2 ≡ H is monogenic if and only if f0 , f1 , f2 , f12 satisfy the generalised CauchyRiemann equations 0 0 − ∂x∂ 1 − ∂x∂ 2 f0 0 ∂ ∂ 0 0 f ∂x1 ∂x2 1 = 0 .