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