Maximal and Potential Operators in Variable Morrey Spaces Defined on Nondoubling Quasimetric Measure Spaces

ABSTRACT. The boundedness of modified maximal operator and potentials in variable Morrey spaces defined on quasimetric measure spaces, where the doubling condition is not needed, is established. © 2008 Bull. Georg. Natl. Acad. Sci. Key words: maximal operator, potential, non-homogeneous spaces, variable Morrey space, boundedness. Let X: = (X, ρ, μ) be a topological space with a complete measure μ such that the space of compactly supported continuous functions is dense in L1 (X, μ) and there exists a non-negative real-valued function (quasimetric) d on X × X satisfying the conditions: (i) d(x, y) = 0 for all x, y ∈ X; (ii) d(x, y)> 0 for all x ≠ y, x, y ∈ X; (iii) there exists a constant a1>0, such that d(x, y) ≤ a1(d(x, z) + d(z, y)) for all x, y, z ∈ X; (iv) there exists a constant a0>0, such that d(x, y) ≤ a0(d(y, x) for all x, y ∈ X. We assume that the balls B(a, r) : = {x ∈ X: ρ(a, x) < r} are measurable, for all a ∈ X and r> 0, and 0 μ (B(a, r)) <∞; for every neighborhood V of x ∈ X, there exists r> 0, such that B(x, r) ⊂ V. We also suppose that μ (X) = ∞ and μ {a} = 0 for all a ∈ X. The triple (X, ρ, μ) will be called quasimetric measure space. If μ satisfies the doubling condition μ (B(x, 2r)) ≤ cμ (B(x, r)), where the positive constant c does not depend on x ∈ X and r>0, then (X, d, μ) is called a space of homogeneous type (SHT). A quasimetric measure space, where the doubling condition might be failed, is also called a non-homogeneous space. We say that the measure μ satisfies the growth condition (μ ∈ GC) if there is a positive constant b such that for all a ∈ X and r>0, () ( ) brraB ≤,μ (1) The boundedness of maximal and potential operators in Lebesgue spaces on non-homogeneous spaces was established in [1] (for Euclidean spaces), [2-5] (see also [6, Ch. 6]). Suppose that p is a μ-measurable function on X. Denot