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We show that the fractional operatorI   α  ⁢   (  ⋅  ){I^{\alpha(\,\cdot\,)}}   , of variable order on a bounded open set in Ω, in a quasimetric measure space(  X  ,  d  ,  μ  ){(X,d,\mu)}    in the caseα  ⁢   (  x  )   ⁢  p  ⁢   (  x  )    ≡  n{\alpha(x)p(x)\equiv n}    (where  n  comes from the growth condition on the measure μ), is bounded from the variable exponent Lebesgue spaceL   p  ⁢   (  ⋅  )⁢   (  Ω  ){L^{p(\,\cdot\,)}(\Omega)}    intoBMO  ⁢   (  Ω  ){\mathrm{BMO}(\Omega)}    under certain assumptions onp  ⁢   (  x  ){p(x)}    andα  ⁢   (  x  ){\alpha(x)}   .

Variable exponent fractional integrals in the limiting case α( x ) p ( n ) ≡ n on quasimetric measure spaces