Littlewood–Paley Equivalence and Homogeneous Fourier Multipliers

Abstract

Abstract. We consider certain Littlewood-Paley operators and prove characterization of some function spaces in terms of those operators. When treating weighted Lebesgue spaces, a generalization to weighted spaces will be made for Hörmander's theorem on the invertibility of homogeneous Fourier multipliers. Also, applications to the theory of Sobolev spaces will be given. IntroductionWe consider the Littlewood-Paley function on R n defined by, where ψ t (x) = t −n ψ(t −1 x). The following result of Benedek, Calderón and Panzone [2] on the L p boundedness, 1 < p < ∞, of g ψ is well-known.Theorem A. We assume (1.1) for ψ andwhereBy the Plancherel theorem, it follows that g ψ is bounded on L 2 (R n ) if and only if m ∈ L ∞ (R n ), where m(ξ) = ∞ 0 |ψ(tξ)| 2 dt/t, which is a homogeneous function of degree 0. Here the Fourier transform is defined aŝ ψ(ξ) = R n ψ(x)e −2πi x,ξ dx, x, ξ = n k=1x k ξ k .

Hardy Spaces on Homogeneous Groups and Littlewood-Paley Functions

“…See [6] for the Hardy space H p (R n ). Analogous results for L p spaces, 1 < p < ∞, can be found in [1], [10] and [15].…”

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

“…The reverse inequality of (1.3) also holds if a certain non-degeneracy condition on ψ is further assumed (see [13], Theorem 3.8, and also [19]).…”

Vector valued inequalities and Littlewood-Paley operators on Hardy spaces

“…where f p = f L p (see [10] and also [1] for an earlier result). The reverse inequality also holds if a certain non-degeneracy condition on ϕ is assumed in addition (see [7,Theorem 3.8] and also [11]). This is the case for g Q with Q(x) = [(∂/∂t)P (x, t)] t=1 , where P (x, t) is the Poisson kernel associated with the upper half space R n × (0, ∞) defined by decreasing smooth functions on R n ; it is known that any other choice of such Φ gives an equivalent norm (see [4]).…”