Maximal and Potential Operators in Variable Exponent Morrey Spaces

Abstract

We prove the boundedness of the Hardy–Littlewood maximal operator on variable Morrey spaces 𝐿𝑝(·), λ(·)(Ω) over a bounded open set Ω ⊂ ℝ𝑛 and a Sobolev type 𝐿𝑝(·), λ(·) → 𝐿𝑞(·), λ(·)-theorem for potential operators 𝐼 α(·), also of variable order. In the case of constant α, the limiting case is also studied when the potential operator 𝐼 α acts into BMO space.

Boundedness of maximal operator, Hardy operator and Sobolev’s inequalities on non-homogeneous central Herz-Morrey-Musielak-Orlicz spaces

“…For a survey, see [15,19]. The boundedness of the maximal and Riesz potential operators were studied for variable exponent Lebesgue spaces L pðÁÞ (see [16,17,18]), variable exponent Morrey spaces (see [4,22,23,34,39]), Herz spaces with variable exponents (see [3,27,47]), local variable exponent Morrey type spaces (see [23,24]) and non-homogeneous central Morrey spaces of variable exponent (see [38]).…”

Boundedness of Commutators of an Oscillatory Integral Operators in Variable Exponent Morrey Spaces

“…The variable exponent Morrey spaces L p(•),λ(•) (Ω), were introduced and studied in [2] in the Euclidean setting in case of bounded sets. The boundedness of the maximal operator in variable exponent Morrey spaces L p(•),λ(•) (Ω) under the log-condition on p(•), λ(•) was proved in [2].…”

“…The variable exponent Morrey spaces L p(•),λ(•) (Ω), were introduced and studied in [2] in the Euclidean setting in case of bounded sets. The boundedness of the maximal operator in variable exponent Morrey spaces L p(•),λ(•) (Ω) under the log-condition on p(•), λ(•) was proved in [2]. Hästö in [19] used his new "local-to-global" approach to extend the result of [2] on the maximal operator to the case of the whole space R n .…”

“…The boundedness of the maximal operator in variable exponent Morrey spaces L p(•),λ(•) (Ω) under the log-condition on p(•), λ(•) was proved in [2]. Hästö in [19] used his new "local-to-global" approach to extend the result of [2] on the maximal operator to the case of the whole space R n . The boundedness of the maximal operator and the singular integral operator in variable exponent Morrey spaces L p(•),λ(•) in the general setting of metric measure spaces was proved in [21].…”

Vector-valued operators with singular kernel and Triebel–Lizorkin block spaces with variable exponents

“…Recently, there has been a substantial amount of research on generalizing the boundedness of some important operators on Lebesgue spaces to general function spaces. For instance, the results in [3], [9], [12], [13], [18], [28], [29], [35], [43], [51], and [55] give us the boundedness of some important operators on Lebesgue spaces, Morrey spaces, and block spaces in the variable exponent setting.…”