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The One-Third-Trick and Shift Operators

Lechner R.

Bulletin of the Polish Academy of Sciences. Mathematics

2013

Vol. 61, no 3-4

219--238

We obtain a representation as martingale transform operators for the rearrangement and shift operators introduced by T. Figiel. The martingale transforms and the underlying sigma algebras are obtained explicitly by combinatorial means. The known norm estimates for those operators are a direct consequence of our representation.

A stability property of the generalized shift operator on BMO space

Partyka D.

Zeszyty Naukowe Politechniki Rzeszowskiej. Matematyka

1999

z. 23 [175]

75-84

The supremum norm of the generalized shift operator S phi on the space BMO(R) is estimated, provided phi is an increasing and absolutely continuous homeomorphic self-mapping of R and phi' is an elemnt of BMO(R) andlog phi'II. is small. This result is extended to a locally rectifiable Jordan arc in C which is homeomorphic to R.

Square functions for wavelet type systems

Gevorkjan G., Wolnik B.

Bulletin of the Polish Academy of Sciences. Mathematics

1998

Vol. 46, no 2

155-171

We show the equivalence of the L[sub p] (0 < p [a is less than or equal to] 2) (quasi)-norms of square functions for the systems {2 [...], where f satisfies some decay condition. This implies the boundedness of the shift operator on the wavelet type unconditional basis on L[sub p], 1 < p < [infinity]. We prove also that such operator is unbounded on L[sub 1].