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Weighted Hardy spaces and BMO spaces associated with Schrödinger operators. (English) Zbl 1266.42060

Let :=-Δ+V be a Schrödinger operator on d , d3, where V¬0 is a fixed non-negative potential and belongs to the reverse Hölder class RH s ( d ) for some sd 2; that is, there exists a positive constant C:=C(s,V) such that (1 |B| B [V(x)] s dx) 1 s C |B| B V(x)dx for every ball B d . Moreover, the measure V(x)dx satisfies the doubling condition: there exists a positive constant C such that B(y,2r) V(x)dxC B(y,r) V(x)dx for all y d and r>0. Let {T t } t>0 be the semigroup of linear operators generated by and T t (x,y) be their kernels, that is, for all x d ,

T t f(x):=e -t f(x):= d T t (x,y)f(y)dyforallt>0andfL 2 ( d )·

A weighted Hardy-type space related to is defined by

H 1 (ω):={fL 1 (ω):𝒯 * f(x)L 1 (ω)}andf H 1 (ω) :=𝒯 * f L 1 (ω) ,

where 𝒯 * f(x):=sup t>0 |T t f(x)| for all x d and ω is a Muckenhoupt weight. A function a is called an atom for the weighted Hardy space H 1 (ω) associated to a ball B(x 0 ,r) for some x 0 d and r 0 (0,), if supp aB(x 0 ,r), a L ( d ) 1 ω(B(x 0 ,r)), if x 0 n , then r2 1-n/2 and, if x 0 n and r2 -1-n/2 , then d a(x)dx=0, where n :={x:2 -(n+1)/2 <ρ(x)2 -n/2 } for n and ρ(x):=ρ(x,V):=sup{r>0:1 r d-2 B(x,r) V(y)dy1}. The atomic norm of fH 1 (ω) is defined by f -at(ω) :=inf{|c j |}, where the infimum is taken over all decompositions f=c j a j , {c j } j and {a j } j are H 1 (ω) atoms. The authors establish the following atomic characterization for H 1 (ω): Assume that VRH d/2 ( d ) is a non-negative potential and V¬0, then the norms f H 1 (ω) and f -at(ω) are equivalent. The authors also characterize H 1 (ω) by the Riesz transforms R j := x j -1/2 for j{1,...,d}: if VRH d ( d ) is a non-negative potential, then there exists a positive constant C such that

C -1 f H 1 (ω) f L 1 (ω) + j=1 d R j f L 1 (ω) Cf H 1 (ω) forallfH 1 (ω)·

The authors also prove that the dual space of H 1 (ω) is BMO (ω), where fBMO (ω) means that fBMO(ω) and there exists a positive constant C such that 1 ω(B) B |f(y)|dyC for all B:=B(x,R) with x d and R>ρ(x). A positive measure μ on + d+1 := d ×(0,) is called an ω-Carleson measure if

μ 𝒞 :=sup x d ,r>0 μ(B(x,r)×(0,r)) ω((B(x,r))<·

The authors prove that, if fBMO (ω), V¬0 is a non-negative potential in RH s ( d ) for some sd 2 and ω a weight in A 1 , then dμ f (x,t):=|Q t f(x)| 2 |B(x,r)| ω(B(x,r))dxdt t is an ω-Carleson measure, where

(Q t f)(x):=t 2 dT s ds s=t 2 f(x),(x,t) + d+1 ,

and, conversely, if fL 1 ((1+|x|) -(d+1) dx) and dμ f (x,t) is a ω-Carleson measure, then fBMO (ω). Finally, the authors establish the boundedness of the Hardy-Littlewood maximal operator and the semigroup maximal operator on BMO (ω).

42B35Function spaces arising in harmonic analysis
42B30H p -spaces (Fourier analysis)
42B25Maximal functions, Littlewood-Paley theory
42B20Singular and oscillatory integrals, several variables
42B37Harmonic analysis and PDE