Furthermore, we extend their results to multi-dimensional covariates under a weaker smoothness condition on the Gaussian process.
Characteristically for polynomial approximation, one can also construct localized operators which yield approximation commensurate with analyticity of the target function on intervals, rather than the much weaker smoothness conditions studied in the previous section.
In this paper, the authors establish the boundedness of Marcinkiewicz integral (mu _{Omega}) from (H^{varphi}(mathbb{R}^{n})) to (L^{varphi}(mathbb{R}^{n})) under weaker smoothness conditions assumed on Ω.
In 2007, Lin et al. [7] proved that (mu_{Omega}) is bounded from the weighted Hardy space to the weighted Lebesgue space under weaker smoothness conditions assumed on Ω, which is called (L^{q} -Dini type condition of order α (see Section 2 for its definition).
Then we present the boundedness of Marcinkiewicz integral (mu_{Omega}) from (H^{varphi}({{mathbb{R}}^{n}})) to (L^{varphi}({{mathbb{R}}^{n}})) under weaker smoothness conditions assumed on Ω (see Theorem 2.4, Theorem 2.5 and Corollary 2.6), the proofs of which are given in Section 3.
In light of Lin [7] and Ky [13], it is a natural and interesting problem to ask whether (mu_{Omega}) is bounded from (H^{varphi}({{mathbb{R}}^{n}})) to (L^{varphi}({{mathbb{R}}^{n}})) under weaker smoothness conditions assumed on Ω.
What we have seen from the above is the boundedness of Marcinkiewicz integral (mu_{Omega}) from (H^{varphi}) to (L^{varphi}) under weaker smoothness conditions assumed on Ω, which generalizes the corresponding results under the setting of both the weighted Hardy space (see, for example, [14]) and the Orlicz-Hardy space (see, for example, [15, 16]), and hence has a wide generality.
Instead, the residual spatial autocorrelation is likely to be strong in some areas showing smoothness, and weak in some other areas exhibiting abrupt step changes.
It is shown that the smoothness of the weak solution of this problem depends on the structure of the boundary of the domain, the right hand side and the dimension of the space.
