Musielak–Orlicz–Hardy Spaces Associated with Operators and Their Applications

Abstract

Let X be a metric space with doubling measure and L a nonnegative selfadjoint operator in L 2 (X ) satisfying the Davies-Gaffney estimates. Let ϕ : X × [0, ∞) → [0, ∞) be a function such that ϕ(x, ·) is an Orlicz function, ϕ(·, t) ∈ A ∞ (X ) (the class of uniformly Muckenhoupt weights), its uniformly critical upper type index I(ϕ) ∈ (0, 1] and it satisfies the uniformly reverse Hölder inequality of order 2/[2 − I(ϕ)]. In this paper, the authors introduce a Musielak-Orlicz-Hardy space H ϕ, L (X ), by the Lusin area function associated with the heat semigroup generated by L, and a Musielak-Orlicz BMO-type space BMO ϕ, L (X ), which is further proved to be the dual space of H ϕ, L (X ) and hence whose ϕ-Carleson measure characterization is deduced. Characterizations of H ϕ, L (X ), including the atom, the molecule and the Lusin area function associated with the Poisson semigroup of L, are presented. Using the atomic characterization, the authors characterize H ϕ, L (X ) in terms of the Littlewood-Paley g * λ -function g * λ, L and establish a Hörmandertype spectral multiplier theorem for L on H ϕ, L (X ). Moreover, for the Musielak-Orlicz-Hardy space H ϕ, L (R n ) associated with the Schrödinger operator L := −∆ + V , where 0 ≤ V ∈ L 1 loc (R n ), the authors obtain its several equivalent characterizations in terms of the non-tangential maximal function, the radial maximal function, the atom and the molecule; finally, the authors show that the Riesz transform ∇L −1/2 is bounded from H ϕ, L (R n ) to the Musielak-Orlicz space L ϕ (R n ) when i(ϕ) ∈ (0, 1], and from H ϕ, L (R n ) to the Musielak-Orlicz-Hardy space H ϕ (R n ) when i(ϕ) ∈ ( n n+1 , 1], where i(ϕ) denotes the uniformly critical lower type index of ϕ.

Riesz Transform Characterizations of Musielak-Orlicz Hardy Spaces

“…Lemma 2.20 is a special case of [44,Lemma 5.6] when the operator L considered therein is the Laplace operator −∆. The only difference is that here we use the (ϕ, q, s)-atoms to replace the operator-adapted atoms therein, the details being omitted.…”

Isomorphisms and several characterizations of Musielak-Orlicz-Hardy spaces associated with some Schrödinger operators

“…This kind of Musielak-Orlicz-Hardy spaces associated with operators generalizes the (Orlicz-)Hardy space and the (weighted) Hardy space associated with operators, which has attracted great interests in recent years. Such function spaces associated with operators play important roles in the study for the boundedness of singular integrals associated with some differential operators, which may not fall within the scope of the classical Calderón-Zygmund theory (see, for example, [3], [5], [11], [12], [13], [17], [16], [18], [29], [30]). …”

“…Observe that i(ϕ) need not be attainable, namely, ϕ need not be of uniformly lower type i(ϕ) (see, for example, [3], [4], [30]). …”

“…Here, A q (R n ) with q ∈ [1, ∞] denotes the class of Muckenhoupt weights (see, for example, [9], [10]). A typical example of such Orlicz function Φ is Φ(t) := t p , with p ∈ (0, 1], for all t ∈ [0, ∞) (see, for example, [31], [30] for more examples of such Φ). Another typical example of a growth function is…”

Herz-Type Hardy Spaces Associated with Operators

“…For example, Auscher, Duong, and McIntosh [7], then Duong and Yan [8,9], introduced the Hardy and BMO spaces adapted to the operator which satisfies the Gaussian heat kernel upper bounds. Yang and his cooperators discussed new Orlicz-Hardy spaces associated with operators [10][11][12][13]. For more results, we refer to [14][15][16][17][18][19] and the references therein.…”