Estimates for Parameter Littlewood-Paleygκ⁎Functions on Nonhomogeneous Metric Measure Spaces

Abstract

Let(X,d,μ)be a metric measure space which satisfies the geometrically doubling measure and the upper doubling measure conditions. In this paper, the authors prove that, under the assumption that the kernel ofMκ⁎satisfies a certain Hörmander-type condition,Mκ⁎,ρis bounded from Lebesgue spacesLp(μ)to Lebesgue spacesLp(μ)forp≥2and is bounded fromL1(μ)intoL1,∞(μ). As a corollary,Mκ⁎,ρis bounded onLp(μ)for1<p<2. In addition, the authors also obtain thatMκ⁎,ρis bounded from the atomic Hardy spaceH1(μ)into the Lebesgue spaceL1(μ).

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

“….fx 2 X W J 2 .x/ > t 2 g/ Ä C t 1 .j 1 j C j 2 j/ Ä C t With an argument similar to that used in the proof of V 11 in [17 …”

“…So, it is interesting to generalize and improve the known results to the non-homogeneous metric measure spaces, see [16][17][18][19][20][21][22][23][24].…”

Generalized Morrey Spaces Over Nonhomogeneous Metric Measure Spaces

“…Thus, in order to deal with the problem, in 2010, Hytönen in [11] introduced a new class of metric measure spaces satisfying the so-called geometric doubling and the upper doubling conditions, respectively (see Definitions 1.1 and 1.3 below), which are called nonhomogeneous metric measure spaces. Since then, many authors have proved that many known results still hold true if the underlying spaces take the place of the nonhomogeneous metric measure spaces (see [1,6,[12][13][14]).…”