Weighted local Orlicz-Hardy spaces with applications to pseudo-differential operators

作者：Yang, Dachun^*; Yang, Sibei

来源：Dissertationes Mathematicae, 2011, (478): 1-78.

DOI：10.4064/dm478-0-1

• local weight
• local Orlicz-Hardy space
• atom
• local grand maximal function
• quasi-Banach space
• BMO-type space
• duality
• local Riesz transform
• local fractional integral
• pseudo-differential operator

Let Phi be a concave function on (0, infinity) of strictly critical lower type index p(Phi) is an element of (0,1) and omega is an element of A(infinity)(loc)(R-n) (the class of local weights introduced by V. S. Rychkov). We introduce the weighted local Orlicz-Hardy space h(omega)(Phi)(R-n) via the local grand maximal function. Let p(t) equivalent to t(-1) / Phi(-1)(t(-1)) for all t is an element of (0, infinity). We also introduce the BMO-type space bmo(p, w)(R-n) and establish the duality between h(omega)(Phi)(R-n) and bmo(p, w)(R-n). Characterizations of h(omega)(Phi)(R-n), 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 h(omega)(Phi)(R-n), from which we further deduce that for a given admissible triplet (p, q, s)(w) and a beta-quasi-Banach space B-beta with beta is an element of (0, 1], if T is a B-beta-sublinear operator, and maps all (p, q, s)(w)-atoms and (p, q)(w)-single-atoms with q < infinity (or all continuous (p, q, s)(w)-atoms with q = infinity) into uniformly bounded elements of B-beta, then T uniquely extends to a bounded B-beta-sublinear operator from h(omega)(Phi)(R-n) to B-beta. As applications, we show that the local Riesz transforms are bounded on h(omega)(Phi)(R-n), the local fractional integrals are bounded from h(wp)(P)(R-n) to L-wq(q)(R-n) when q > 1 and from h(wp)(p)(R-n) to h(wq)(q)(R-n) when q < 1, and some pseudo-differential operators are also bounded on both h(omega)(Phi)(R-n). All results for any general Phi even when omega equivalent to 1 are new.

• 出版日期2011
• 单位北京师范大学