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Generalized potentials in variable exponent Lebesgue spaces on homogeneous spaces

✍ Scribed by Mubariz G. Hajibayov; Stefan Samko

Publisher: John Wiley and Sons
Year: 2011
Tongue: English
Weight: 159 KB
Volume: 284
Category: Article
ISSN: 0025-584X
DOI: 10.1002/mana.200710204

No coin nor oath required. For personal study only.

✦ Synopsis

We consider generalized potential operators with the kernel a ([ (x ,y )])

[ (x ,y )] N on bounded quasimetric measure space (X, μ, d) with doubling measure μ satisfying the upper growth condition μB(x, r) ≤ Kr N , N ∈ (0, ∞). Under some natural assumptions on a(r) in terms of almost monotonicity we prove that such potential operators are bounded from the variable exponent Lebesgue space L p (•) (X, μ) into a certain Musielak-Orlicz space L Φ (X, μ) with the N -function Φ(x, r) defined by the exponent p(x) and the function a(r). A reformulation of the obtained result in terms of the Matuszewska-Orlicz indices of the function a(r) is also given.

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