TY - JOUR
T1 - Wave Propagation Speed on Fractals
AU - Ngai, Sze Man
AU - Tang, Wei
AU - Xie, Yuanyuan
N1 - Publisher Copyright:
© 2020, Springer Science+Business Media, LLC, part of Springer Nature.
PY - 2020/4/1
Y1 - 2020/4/1
N2 - We study the wave propagation speed problem on metric measure spaces, emphasizing on self-similar sets that are not post-critically finite. We prove that a sub-Gaussian lower heat kernel estimate leads to infinite propagation speed, extending a result of Lee (Infinite propagation speed for wave solutions on some p.c.f. fractals, https://archive.org/details/arxiv-1111.2938) to include bounded and unbounded generalized Sierpiński carpets, some fractal blowups, and certain iterated function systems with overlaps. We also formulate conditions under which a Gaussian upper heat kernel estimate leads to finite propagation speed, and apply this result to two classes of iterated function systems with overlaps, including those defining the classical infinite Bernoulli convolutions.
AB - We study the wave propagation speed problem on metric measure spaces, emphasizing on self-similar sets that are not post-critically finite. We prove that a sub-Gaussian lower heat kernel estimate leads to infinite propagation speed, extending a result of Lee (Infinite propagation speed for wave solutions on some p.c.f. fractals, https://archive.org/details/arxiv-1111.2938) to include bounded and unbounded generalized Sierpiński carpets, some fractal blowups, and certain iterated function systems with overlaps. We also formulate conditions under which a Gaussian upper heat kernel estimate leads to finite propagation speed, and apply this result to two classes of iterated function systems with overlaps, including those defining the classical infinite Bernoulli convolutions.
KW - Bernoulli convolution
KW - Fractal
KW - Heat-kernel estimate
KW - Laplacian
KW - Wave propagation speed
UR - http://www.scopus.com/inward/record.url?scp=85081324797&partnerID=8YFLogxK
U2 - 10.1007/s00041-019-09716-7
DO - 10.1007/s00041-019-09716-7
M3 - Article
AN - SCOPUS:85081324797
SN - 1069-5869
VL - 26
JO - Journal of Fourier Analysis and Applications
JF - Journal of Fourier Analysis and Applications
IS - 2
M1 - 31
ER -