ISSN:
1588-2632
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract Considering mixed-norm sequence spaces lp,q, p, q ≧ 1, C. N. Kellogg proved the following theorem: if 1 〈 p ≦ 2 then $$\widehat{L^p }$$ ⊂ lp′,2 and lp,2 ⊂ $$\widehat{L^{p'} }$$ , where 1/p + 1/p′ = 1. This result extends the Hausdorff-Young Theorem. We introduce here multiple mixed-norm sequence spaces $$\ell ^{p,q_1 ,q_2 ,...q_n } $$ , examine their properties and characterize the multipliers of spaces of the form lp,[s;n],q, with the index s repeated n times. By an interpolation-type argument we prove that $$\widehat{L^p }$$ ⊂ (l∞,[2;n],2, lp′,[1;n],1) for 1 〈 p ≦ 2. Using these results we obtain a further generalization of the Hausdorff-Young Theorem: if 1 〈 p ≦ 2 then $$\widehat{L^p }$$ ⋐ lp′,[2;n] and lp,[2;n] ⋐ $$\widehat{L^{p'} }$$ for each n = 0, 1, 2, ¨. The spaces lp′,[2;n] decrease and lp,[2;n] increase properly with n for 1 〈 p 〈 2 and 1/p + 1/p′ = 1. We also extend a theorem of J. H. Hedlund on multiplers of Hardy spaces $$(\widehat{H^p },\widehat{H^2 })$$ and deduce other results.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1023/A:1006555822461
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