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Generalized potentials in variable exponent Lebesgue spaces on homogeneous spaces
Author(s) -
Hajibayov Mubariz G.,
Samko Stefan
Publication year - 2011
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200710204
Subject(s) - mathematics , standard probability space , lp space , bounded function , exponent , measure (data warehouse) , monotonic function , measurable function , maximal function , space (punctuation) , hardy space , function (biology) , homogeneous , function space , pure mathematics , lebesgue integration , kernel (algebra) , mathematical analysis , banach space , combinatorics , linguistics , philosophy , database , evolutionary biology , computer science , biology
We consider generalized potential operators with the kernel $\frac{a([\varrho (x,y)])}{[\varrho (x,y)]^N}$ on bounded quasimetric measure space ( X , μ, d ) with doubling measure μ satisfying the upper growth condition μ B ( 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 p ( 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. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim