A necessary condition for the boundedness of the maximal operator on $$L^{p(\cdot)}$$ over reverse doubling spaces of homogeneous type

Abstract

AbstractLet $$(X,d,\mu)$$ ( X , d , μ ) be a space of homogeneous type and $$p(\cdot) \colon X \to[1,\infty]$$ p ( · ) : X → [ 1 , ∞ ] be a variable exponent. We show that if the measure $$\mu$$ μ is Borel-semiregular and reverse doubling, then the condition $${ess\,inf}_{x\in X}p(x)>1$$ e s s i n f x ∈ X p ( x ) > 1 is necessary for the boundedness of the Hardy–Littlewood maximal operator $$M$$ M on the variable Lebesgue space $$L^{p(\cdot)}(X,d,\mu)$$ L p ( · ) ( X , d , μ ) .