ON BOUNDEDNESS OF LITTLEWOOD{PALEY SQUARE OPERATOR ON MORREY SPACE VIA EXTRAPOLATION AND DUALITY

• Main Author: Zanu, Pebrudal
• Format: Theses
• Language: Indonesia
• Online Access: https://digilib.itb.ac.id/gdl/view/36151
• Institution: Institut Teknologi Bandung

Littlewood{Paley Square (LPS) operator is an operator maps a measurable function f to P1 j=????1 jj(f)j2 1 2 , where j is Littlewood{Paley operator. Littlewood{Paley operator is an operator maps a measurable function f via convolution relation j(f) = 2????j f, where 2????j (x) = 2jn (2jx) with : Rn ! C. With some sucient condition of , Littlewood and Paley proved boundedness of LPS operators on the classical Lebesgue spaces. Via Marcinkiewicz interpolation we obtain a weaker sucient condition to than Littlewood-Paley for boundedness of LPS operators on the Lebesgue spaces still hold. We further extend the boundedness of the LPS operator on Morrey spaces with adapting the extrapolation strategy and duality devised by Rosenthal and Schemeisser. The extrapolation strategy and duality for vector valued operators. To be able to devise the strategy we need to dene `2-Morrey spaces is set of sequence measurable function ~ f = ffjg1 j=????1 where k ~ f()k`2 in Morrey spaces. Furthermore, we construct the bipredual of `2-Morrey spaces, as the case of by Triebel in (regular) Morrey spaces. Using these tools, we are able to show boundedness of vector-valued LPS operators from regular Morrey spaces to `2-Morrey spaces.