In this paper, the authors establish the anisotropic weak Hardy spaces associated with very general discrete groups of dilations. Moreover, the atomic decomposition theorem of the anisotropic weak Hardy spaces is also given. As some applications of the above results, the authors prove some interpolation theorems and obtain the boundedness of the singular integral operators on these Hardy spaces.
In this paper, we introduce the fractional integral operator T of degree α of order m with respect to a dilation A for 0 < α < 1 and m ∈ N. First we establish the Hardy-Littlewood-Sobolev inequalities for T on anisotropic Hardy spaces associated with dilation A, which show that T is bounded from H p to H q , or from H p to L q , where 0 < p ≤ 1/(1 + α) and 1/q = 1/p − α. Then we give anisotropic Hardy spaces estimates for a class of multilinear operators formed by fractional integrals or Calderón-Zygmund singular integrals. Finally, we apply the above results to give the boundedness of the commutators of T and a BMO function.
Mathematics Subject Classification (2000). 42B20, 47G10.
We present a multiplier theorem on anisotropic Hardy spaces. When m satises the anisotropic, pointwise Mihlin condition, we obtain boundedness of the multiplier operator T m ∶ H p A (R n) → H p A (R n), for the range of p that depends on the eccentricities of the dilation A and the level of regularity of a multiplier symbol m. is extends the classical multiplier theorem of Taibleson and Weiss.
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