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

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

This convention introduced jointly best overseas and Australian researchers, in addition to younger Australian researchers and PhD scholars, within the box of Harmonic research and comparable subject matters for the dissemination of the latest advancements within the box, and for discussions on destiny instructions. The goal was once to demonstrate the breadth and intensity of modern paintings in Harmonic research, to augment latest collaboration, and to forge new links.

As organisers of the convention, we're thankful to the convention individuals and audio system, lots of whom travelled huge distances for his or her contributions. monetary help for the meetings was once supplied through the AMSI and the dept of arithmetic at Macquarie collage. As editors of this quantity, we might additionally prefer to thank the Centre for arithmetic and its functions in Canberra for assist in getting ready those court cases. the graceful operating of the convention should not have been attainable with no the organisational abilities of Christine Hale of the dep. of arithmetic at Macquarie collage.

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M. Stein, Harmonic analysis: Real variable methods, orthogonality and oscillatory integrals, Princeton Univ. Press, Princeton, NJ, (1993). 8. C. Segovia and J. L. Torrea, Vector-valued commutators and applications, Idiana Univ. Math. J. 38 (1989), 959-971. 9. X. Tolsa, BMO, H 1 , and Calder´ on-Zygmund operators for non doubling measures, Math. Ann. 219 (2001), 89-149. 10. X. Tolsa, A proof of the weak (1,1) inequality for singular integrals with non doubling measures based on a Calderon-Zygmund decomposition, Publ.

Suppose a ∈ H and f1 , f2 ∈ L2 (R2 , H). We denote by [a] the matrix as av . , T is right H-linear. , T is translation-invariant. A submodule Y ⊂ L2 (Rd , X) is said to be translation-invariant when τy f ∈ Y for all f ∈ Y and y ∈ Rd . In the classical case, it is well-known that the only closed translation-invariant subspaces of L2 (Rd ) = L2 (Rd , C) are of the form Y = YE = {f ∈ L2 (Rd ); Fd f (y) = 0 if y ∈ / E} where E is a measurable subset of Rd . e. y ∈ Rd . The corresponding class of closed translation-invariant submodules of L2 (R2 , H) is far richer than the the classical case, and is described by Theorem 9 below.

Let M∆ be the operator defined by (1). Then M∆ is weak type (1, 1) and for any function f ∈ Lp , 1 < p ≤ ∞, the following estimates hold M∆ f Lp (M ) ≤C f Lp (M ) . Proof. , we need to prove that there exists a positive constant C such that for any f ∈ L1 (M ) and for any λ > 0, (6) x ∈ M : sup | exp(−t∆)f (x)| > λ ≤ t>0 C f λ L1 (M ) . Fix f ∈ L1 (M ). Similarly as in Section 3 we set f1 (x) = f (x)χRm \K (x), f2 (x) = f (x)χRn \K (x) and f3 (x) = f (x)χK (x), where K is the center of M . To prove (6), it suffices to verify that the following three estimates hold: (7) x ∈ Rm \K : sup | exp(−t∆)f (x)| > λ ≤ C f λ L1 (M ) ; ≤ C f λ L1 (M ) ; t>0 (8) x ∈ Rn \K : sup | exp(−t∆)f (x)| > λ t>0 (9) x ∈ K : sup | exp(−t∆)f (x)| > λ t>0 ≤ C f λ L1 (M ) .