Theorem 1 ([1, Theorem 1.3]).

Let p ⁢ ( ⋅ ) : R d → [ 1 , ∞ ] be a measurable function satisfying 1 < p - ≤ p + < ∞ . Then p ⁢ ( ⋅ ) ∈ B M ⁢ ( R d ) if and only if for every q ∈ ( 1 , ∞ ) , there exists a number Θ p ⁢ ( ⋅ ) , q ∈ ( 0 , 1 ) such that for every θ ∈ ( 0 , Θ p ⁢ ( ⋅ ) , q ] the variable exponent r ⁢ ( ⋅ ) defined by

belongs to B M ⁢ ( R d ) .

Proof.

Take any q ∈ ( 1 , ∞ ) . It follows from [2, Theorem 4.1] (see also [1, Theorem 1.2]) that there exist the numbers q 1 ∈ ( 1 , ∞ ) and θ 1 ∈ ( 0 , 1 ) such that the variable exponent r 1 ⁢ ( ⋅ ) defined by

belongs to ℬ M ⁢ ( ℝ d ) . Take any positive θ 2 < min ⁡ { q q 1 , 1 - 1 q 1 } < 1 and define q 2 by

Then

Hence q 2 ∈ ( 1 , ∞ ) .

Substituting (3) into (2), we get

where θ := θ 1 ⁢ θ 2 and

Clearly, θ 1 ⁢ ( 1 - θ 2 ) ⁢ ( 1 - θ 1 ⁢ θ 2 ) - 1 > 0 and ( 1 - θ 1 ) ⁢ ( 1 - θ 1 ⁢ θ 2 ) - 1 > 0 . Since

equation (5) can be rewritten in the form

Since r 1 ⁢ ( ⋅ ) belongs to ℬ M ⁢ ( ℝ d ) , it follows from the sufficiency part of the theorem that r ⁢ ( ⋅ ) ∈ ℬ M ⁢ ( ℝ d ) . In view of (4), this completes the proof for any positive Θ p ⁢ ( ⋅ ) , q < θ 1 ⁢ min ⁡ { q q 1 , 1 - 1 q 1 } . ∎