Littlewood–Paley functions on homogeneous groups

Abstract

Abstract. We p r o ve L p estimates for a class of Littlewood-Paley functions on homogeneous groups under a sharp integrability condition of the kernel. The results obtained in the present paper essentially improve s o m e k n o wn results.

Atomic and Littlewood–Paley Characterizations of Anisotropic Mixed-Norm Hardy Spaces and Their Applications

“…To show Lemma 3.11, we need an auxiliary inequality as follows, which is a slight modification of [68, p. 100, (21)], the details being omitted.…”

Hardy Spaces on Homogeneous Groups and Littlewood-Paley Functions

“…where f ∈ S ′ , ϕ ∈ S satisfying H ϕ dx = 0 and ϕ t (x) = t −γ ϕ(A −1 t x). Here S ′ denotes the space of tempered distributions and S the Schwartz space, which are the same as those in the Euclidean case (see [19]); also the convolution F * G for F, G ∈ L 1 is defined by We refer to [4] and [21,13,14] for the study of Littlewood-Paley operators and singular integrals, respectively, on L p spaces for homogeneous groups, 1 ≤ p < ∞.…”

“…We refer to [4] and [21,13,14] for the study of Littlewood-Paley operators and singular integrals, respectively, on L p spaces for homogeneous groups, 1 ≤ p < ∞.…”

Boundedness of Littlewood-Paley operators relative to non-isotropic dilations

“…where f p = f L p (see [16] and [5], [8], [10], [11], [17] for related results; also [6] for a result on the homogeneous groups including the Heisenberg group). We refer to [1] for an earlier result, which requires in addition certain regularity on ψ to get (1.3).…”