Marcinkiewicz Integrals with Non-Doubling Measures

Abstract

Let µ be a positive Radon measure on R d which may be non doubling. The only condition that µ must satisfy is µ(B(x, r)) ≤ Cr n for all x ∈ R d , r > 0 and some fixed constants C > 0 and n ∈ (0, d]. In this paper, we introduce the Marcinkiewicz integral related to a such measure with kernel satisfying some Hörmander-type condition, and assume that it is bounded on L 2 (µ). We then establish its boundedness, respectively, from the Lebesgue space L 1 (µ) to the weak Lebesgue space L 1,∞ (µ), from the Hardy space H 1 (µ) to L 1 (µ) and from the Lebesgue space L ∞ (µ) to the space RBLO(µ). As a corollary, we obtain the boundedness of the Marcinkiewicz integral in the Lebesgue space L p (µ) with p ∈ (1, ∞). Moreover, we establish the boundedness of the commutator generated by the RBMO(µ) function and the Marcinkiewicz integral with kernel satisfying certain slightly stronger Hörmander-type condition, respectively, from L p (µ) with p ∈ (1, ∞) to itself, from the space L log L(µ) to L 1,∞ (µ) and from H 1 (µ) to L 1,∞ (µ). Some of the results are also new even for the classical Marcinkiewicz integral. Mathematics Subject Classification (2000). Primary 42B25; Secondary 47B47, 42B20, 47A30.

The Hardy Space H1 with Non-doubling Measures and Their Applications

“…/ B; S . Comparing with the corresponding result in [52], this result makes an essential improvement. Moreover, Lin and Yang showed that, if M.f / is bounded form H 1 .X ; / into L 1 .X ; /, then it is also bounded from L 1 .X ; / into the space RBLO.X ; / and, conversely, if the Marcinkiewicz integral is bounded from L 1 b .X ; / (the set of all L 1 .X ; / functions with bounded support) into the space RBMO.X ; /, then it is also bounded from the finite atomic Hardy space H 1; 1 fin .X ; / into L 1 .X ; /.…”

“…• In [52], Hu, Lin and Yang introduced the Marcinkiewicz integral with kernel satisfying the Hörmander-type condition. By assuming that the Marcinkiewicz integral is bounded on L 2 .…”

“…By assuming that the Marcinkiewicz integral is bounded on L 2 . /, the authors in [52] then established its boundedness, respectively, from the Lebesgue space L 1 . / to the weak Lebesgue space L 1; 1 .…”

Query and Analysis of Data on Electric Consumption Based on Hadoop

“…What"s more, Hadoop has a very high parallelism. Therefore Hadoop has a higher efficiency when processing magnanimity data [9][10].…”

Commutators of Littlewood-Paley gκ∗ $g_{\kappa}^{*} $-functions on non-homogeneous metric measure spaces

“…After that, many authors paid much attention to study the properties of the Littlewood-Paley g Ä -functions on various function spaces, for example, see [2][3][4][5][6][7]. With deeper research, the boundedness of Littlewood-Paley operators and their commutators under the cases of non-doubling measures is also widely discussed (see [8][9][10][11][12][13][14]). …”