Abstract. In this paper we give estimations of the pointwise scaling exponents of self-similar functions on the n-dimensional Euclidean space R n . These estimations are derived by using a technique based on wavelet analysis. Examples of such self-similar functions include indefinite integrals of self-similar measures on R, and they also include widely oscillatory functions (e.g. the Takagi function, the Weierstrass function and Lévy's function). Pointwise scaling exponents provide an objective description of an irregularity of a function at a point. Our results are applied to compute the scaling exponents of several oscillatory functions.
In this paper we introduce and investigate new 2-microlocal spaces associated with Besov type and Triebel-Lizorkin type spaces. We establish characterizations of these function spaces via the ϕ-transform, the atomic and molecular decomposition and the wavelet decomposition. As applications we consider boundedness of the Calderón-Zygmund operator and the pseudo-differential operator on the function spaces.
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