Weighted local Orlicz-Hardy spaces with applications to pseudo-differential operators
Dachun Yang ; Sibei Yang
GDML_Books, (2011), p.

Let Φ be a concave function on (0,∞) of strictly critical lower type index pΦ(0,1] and ωAloc() (the class of local weights introduced by V. S. Rychkov). We introduce the weighted local Orlicz-Hardy space hωΦ() via the local grand maximal function. Let ρ(t)t-1/Φ-1(t-1) for all t ∈ (0,∞). We also introduce the BMO-type space bmoρ,ω() and establish the duality between hωΦ() and bmoρ,ω(). Characterizations of hωΦ(), including the atomic characterization, the local vertical and the local nontangential maximal function characterizations, are presented. Using the atomic characterization, we prove the existence of finite atomic decompositions achieving the norm in some dense subspaces of hωΦ(), from which we further deduce that for a given admissible triplet (ρ,q,s)ω and a β-quasi-Banach space β with β ∈ (0,1], if T is a β-sublinear operator, and maps all (ρ,q,s)ω-atoms and (ρ,q)ω-single-atoms with q < ∞ (or all continuous (ρ,q,s)ω-atoms with q = ∞) into uniformly bounded elements of β, then T uniquely extends to a bounded β-sublinear operator from hωΦ() to β. As applications, we show that the local Riesz transforms are bounded on hωΦ(), the local fractional integrals are bounded from hωpp() to Lωqq() when q > 1 and from hωpp() to hωqq() when q ≤ 1, and some pseudo-differential operators are also bounded on both hωΦ(). All results for any general Φ even when ω ≡ 1 are new.

EUDML-ID : urn:eudml:doc:286013
@book{bwmeta1.element.bwnjournal-rm-doi-10_4064-dm478-0-1,
     author = {Dachun Yang and Sibei Yang},
     title = {Weighted local Orlicz-Hardy spaces with applications to pseudo-differential operators},
     series = {GDML\_Books},
     year = {2011},
     zbl = {1241.46018},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-rm-doi-10_4064-dm478-0-1}
}
Dachun Yang; Sibei Yang. Weighted local Orlicz-Hardy spaces with applications to pseudo-differential operators. GDML_Books (2011),  http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-rm-doi-10_4064-dm478-0-1/