Displaying similar documents to “On boundedness of maximal functions in Sobolev spaces.”

On the boundedness of the maximal operator and singular integral operators in generalized Morrey spaces

Ali Akbulut, Vagif Guliyev, Rza Mustafayev (2012)

Mathematica Bohemica

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In the paper we find conditions on the pair ( ω 1 , ω 2 ) which ensure the boundedness of the maximal operator and the Calderón-Zygmund singular integral operators from one generalized Morrey space ℳ p , ω 1 to another ℳ p , ω 2 , 1 < p < ∞ , and from the space ℳ 1 , ω 1 to the weak space W ℳ 1 , ω 2 . As applications, we get some estimates for uniformly elliptic operators on generalized Morrey spaces.

Commutators of the fractional maximal function on variable exponent Lebesgue spaces

Pu Zhang, Jianglong Wu (2014)

Czechoslovak Mathematical Journal

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Let M β be the fractional maximal function. The commutator generated by M β and a suitable function b is defined by [ M β , b ] f = M β ( b f ) - b M β ( f ) . Denote by 𝒫 ( ℝ n ) the set of all measurable functions p ( · ) : ℝ n → [ 1 , ∞ ) such that 1 < p - : = ess inf x ∈ ℝ n p ( x ) and p + : = ess sup x ∈ ℝ n p ( x ) < ∞ , and by ℬ ( ℝ n ) the set of all p ( · ) ∈ 𝒫 ( ℝ n ) such that the Hardy-Littlewood maximal function M is bounded on L p ( · ) ( ℝ n ) . In this paper, the authors give some characterizations of b for which [ M β , b ] is bounded from L p ( · ) ( ℝ n ) into L q ( · ) ( ℝ n ) , when p ( · ) ∈ 𝒫 ( ℝ n ) , 0 < β < n / p + and 1 / q ( · ) = 1 / p ( · ) - β / n with q ( · ) ( n - β ) / n ∈ ℬ ( ℝ n ) .