Translation-invariant operators on spaces of vector-valued functions

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dc.contributor Aalto-yliopisto fi
dc.contributor Aalto University en Hytönen, Tuomas 2012-01-24T14:42:24Z 2012-01-24T14:42:24Z 2003-04-25
dc.identifier.isbn 951-22-6457-9
dc.identifier.issn 0784-3143
dc.description.abstract The treatise deals with translation-invariant operators on various function spaces (including Besov, Lebesgue-Bôchner, and Hardy), where the range space of the functions is a possibly infinite-dimensional Banach space X. The operators are treated both in the convolution form T f = k ∗ f and in the multiplier form in the frequency representation, T^f = m f̂, where the kernel k and the multiplier m are allowed to take values in ℒ(X) (bounded linear operators on X). Several applications, most notably the theory of evolution equations, give rise to non-trivial instances of such operators. Verifying the boundedness of operators of this kind has been a long-standing problem whose intimate connection with certain randomized inequalities (the notion of "R-boundedness" which generalizes classical square-function estimates) has been discovered only recently. The related techniques, which are exploited and developed further in the present work, have proved to be very useful in generalizing various theorems, so far only known in a Hilbert space setting, to the more general framework of UMD Banach spaces. The main results here provide various sufficient conditions (with partial converse statements) for verifying the boundedness of operators T as described above. The treatment of these operators on the Hardy spaces of vector-valued functions is new as such, while on the Besov and Bôchner spaces the convolution point-of-view taken here complements the multiplier approach followed by various other authors. Although general enough to deal with the vector-valued situation, the methods also improve on some classical theorems even in the scalar-valued case: In particular, it is shown that the derivative condition |ξ||α| |Dαm(ξ)| ≤ C   ∀α ∈ {|α|∞ ≤ 1} ∩ {|α|1 ≤ ⌊n/2⌋ + 1} is sufficient for m to be a Fourier multiplier on Lp(ℝn), p ∈ ]1,∞[ – the set of required derivatives constitutes the intersection of the ones in the classical theorems of S. G. Mihlin and L. Hörmander. en
dc.format.extent 194
dc.format.mimetype application/pdf
dc.language.iso en en
dc.publisher Helsinki University of Technology en
dc.publisher Teknillinen korkeakoulu fi
dc.relation.ispartofseries Research reports / Helsinki University of Technology, Institute of Mathematics. A en
dc.relation.ispartofseries 458 en
dc.subject.other Mathematics en
dc.title Translation-invariant operators on spaces of vector-valued functions en
dc.type G4 Monografiaväitöskirja fi
dc.description.version reviewed en
dc.contributor.department Department of Engineering Physics and Mathematics en
dc.contributor.department Teknillisen fysiikan ja matematiikan osasto fi
dc.subject.keyword operator-valued Fourier multiplier en
dc.subject.keyword singular convolution integral en
dc.subject.keyword Lebesgue-Bôchner, Hardy, and Besov spaces of vector-valued functions en
dc.subject.keyword Mihlin's theorem en
dc.subject.keyword Hörmander's integral condition en
dc.subject.keyword R-boundedness en
dc.subject.keyword UMD space en
dc.subject.keyword Fourier embedding en
dc.subject.keyword evolution equation en
dc.subject.keyword maximal regularity en
dc.identifier.urn urn:nbn:fi:tkk-000426
dc.type.dcmitype text en
dc.type.ontasot Väitöskirja (monografia) fi
dc.type.ontasot Doctoral dissertation (monograph) en
dc.contributor.lab Institute of Mathematics en
dc.contributor.lab Matematiikan laitos fi

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