Let be a metric space with doubling measure and L a oneto-one operator of type ω having a bounded H ∞ -functional calculus in L 2 ( ) satisfying the reinforced ( L L ) off-diagonal estimates on balls, where L ∈ [1 2) and L ∈ (2 ∞]. Let:is an Orlicz function, (· ) ∈ A ∞ ( ) (the class of uniformly Muckenhoupt weights), its uniformly critical upper type index I( ) ∈ (0 1] and (· ) satisfies the uniformly reverse Hölder inequality of order ( L /I( )) , where ( L /I( )) denotes the conjugate exponent of L /I( ). In this paper, the authors introduce a Musielak-Orlicz-Hardy space H L ( ), via the Lusin-area function associated with L, and establish its molecular characterization. In particular, when L is nonnegative self-adjoint and satisfies the Davies-Gaffney estimates, the atomic characterization of H L ( ) is also obtained. Furthermore, a sufficient condition for the equivalence between H L (R ) and the classical Musielak-Orlicz-Hardy space H (R ) is given. Moreover, for the Musielak-Orlicz-Hardy space H L (R ) associated with the second order elliptic operator in divergence form on R or the Schrödinger operator L := −∆ + V with 0 ≤ V ∈ L 1 loc (R ), the authors further obtain its several equivalent characterizations in terms of various non-tangential and radial maximal functions; finally, the authors show that the Riesz transform ∇L −1/2 is bounded from H L (R ) to the Musielak-Orlicz space L (R ) when ( ) ∈ (0 1], from H L (R ) to H (R ) when ( ) ∈ ( +1 1], and from H L (R ) to the weak MusielakOrlicz-Hardy space W H (R ) when ( ) = +1 is attainable and (· ) ∈ A 1 ( ), where ( ) denotes the uniformly critical lower type index of .
KeywordsMusielak-Orlicz-Hardy space • molecule • atom • maximal function • Lusin area function • Schrödinger operator • elliptic operator • Riesz transform MSC: 42B35,
Let X be a space of homogeneous type and L be a nonnegative self-adjoint operator on L 2 (X) satisfying Gaussian upper bounds on its heat kernels. In this paper we develop the theory of weighted Besov spacesḂ α,L p,q,w (X) and weighted Triebel-Lizorkin spacesḞ α,L p,q,w (X) associated to the operator L for the full range 0 < p, q ≤ ∞, α ∈ R and w being in the Muckenhoupt weight class A∞. Similarly to the classical case in the Euclidean setting, we prove that our new spaces satisfy important features such as continuous charaterizations in terms of square functions, atomic decompositions and the identifications with some well known function spaces such as Hardy type spaces and Sobolev type spaces. Moreover, with extra assumptions on the operator L, we prove that the new function spaces associated to L coincide with the classical function spaces. Finally we apply our results to prove the boundedness of the fractional power of L and the spectral multiplier of L in our new function spaces.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.