Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 45, 2020, 1187–1207
Nihon University, College of Economics
1-3-2 Misaki-cho Kanda Chiyoda-ku Tokyo 101-8360, Japan;
katsu.m 'at' nihon-u.ac.jp
Hiroshima University,
Graduate School of Science,
Department of Mathematics
Higashi-Hiroshima 739-8521, Japan;
yomizuta 'at' hiroshima-u.ac.jp
Hiroshima University,
Graduate School of Education,
Department of Mathematics
Higashi-Hiroshima 739-8524, Japan;
tshimo 'at' hiroshima-u.ac.jp
Abstract. In a metric measure space (X,d,μ), our first aim in this paper is to discuss the weak estimates for the maximal and Riesz potential operators in the non-homogeneous central Morrey type space M1,q,a(X) (about x0 ∈ X) of all measurable functions f on X satisfying
\[ \|f\|_{M^{1,q,a}(X)} = \left( \int_1^{\infty} \left( r^{-a} \|f\|_{L^{1}(B(x_0,r))} \right)^q \frac{dr}{r} \right)^{1/q} < \infty \]
for a ≥ 0 and 0 < q < ∞; when q = ∞, we apply a necessary modification. To do this, we consider the family WMφ,q,a(X) of all measurable functions f ∈ Lloc1(X) such that
\[ \|f \|_{WM^{\varphi,q,a}(X)} =\sup_{\lambda > 0} \lambda \left( \int_1^\infty \left(r^{-a} \varphi^{-1} \left( \int_{B(x_0,r)} \chi_{E_f(\lambda)}(x) \, d\mu(x) \right) \right)^{q} \frac{dr}{r} \right)^{1/q} < \infty, \]
where φ is a general function satisfying certain conditions and χEf(λ) denotes the characteristic function of Ef(λ) = {x &isin X : |f(x)| > λ}. In connection with M1,q,a(X), we treat the complementary space N∞,q,a(X) of all measurable functions f on X satisfying
\[ \|f\|_{N^{\infty,q,a}(X)} = \|f\|_{L^{\infty}(B(x_0,2))}+ \left( \int_1^{\infty} \left( r^{a} \|f\|_{L^{1}(X\setminus B(x_0,r))} \right)^q \frac{dr}{r} \right)^{1/q} < \infty \]
2010 Mathematics Subject Classification: Primary 31B15, 46E35.
Key words: Non-homogeneous central Morrey type space, metric measure space, maximal function, Riesz potentials, Sobolev's inequality, duality.
Reference to this article: K. Matsuoka, Y. Mizuta and T. Shimomura: Weak estimates for the maximal and Riesz potential operators on non-homogeneous central Morrey type spaces in L1 over metric measure spaces. Ann. Acad. Sci. Fenn. Math. 45 (2020), 1187–1207.
https://doi.org/10.5186/aasfm.2020.4561
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