JLip versus Sobolev Spaces on a Class of Self-Similar Fractal Foliages
Résumé
For a class of self-similar sets $\Gamma^\infty$ in $\R^2$, supplied with a probability measure $\mu$ called the self-similar measure, we investigate if the $B_s^{q,q}(\Gamma^\infty)$ regularity of a function can be characterized using the coefficients of its expansion in the Haar wavelet basis. Using the the Lipschitz spaces with jumps recently introduced by Jonsson, the question can be rephrased: when does $B_s^{q,q}(\Gamma^\infty)$ coincide with $JLip(s,q,q;0;\Gamma^\infty)$? When $\Gamma^\infty$ is totally disconnected, this question has been positively answered by Jonsson for all $s,q$, $00$, $1\le p,q<\infty$, using possibly higher degree Haar wavelets coefficients). Here, we fully answer the question in the case when $0
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