This paper proves the local Lipschitz property for harmonic (or positive subharmonic) functions on graphs. An example is also obtained to show that the global Lipschitz property of harmonic function on graphs does not hold.

A self-similar measure on R^n is defined to be a probability measure satisfying \mu = \sum_{j=1}^N p_j\mu\circ S_j^{-1} + \sum_{j=1}^M q_j(\mu*\mu)\circ T_j^{-1}, where S_j x = \rho_j R_j x + b_j, T_j x = \eta_j Q_j x + c_j are contractive similarities, 0 < \rho_j < 1, 0 < \eta_j < 1/2, 0 < p_j < 1, 0 < q_j < 1; \sum_{j=1}^N p_j + \sum_{j=1}^M q_j = 1, R_j, Q_j are orthogonal matrix and \mu*\mu is the convolution of two measures. When M = 0, \mu is a linear self-similar measure, we establish the asymptotic behavior of averages of the derivative of the Fourier transform of \mu, such as \int_{|x|\le R} |(\frac{\partial}{\partial x})^\alpha \hat\mu(x)|^2 dx = O(R^{n-\beta}) for any order derivation of \hat\mu(x) as R\to\infty under certain additional hypotheses. When M > 0, \mu is a nonlinear self-similar measure, we get some results of L^p boundedness for maximal operators of l, from the pointwise asymptotic estimate of the Fourier transform of \mu made by Strichartz.

New spaces of Lipschitz type on metric-measure spaces are introduced and they are shown to be just the well-known Besov spaces or Triebel–Lizorkin spaces when the smooth index is less than 1. These theorems also hold in the setting of spaces of homogeneous type, which include Euclidean spaces, Riemannian manifolds and some self-similar fractals. Moreover, the relationships amongst these Lipschitz-type spaces, Hajłasz–Sobolev spaces, Korevaar–Schoen–Sobolev spaces, Newtonian Sobolev space and Cheeger–Sobolev spaces on metric-measure spaces are clarified, showing that they are the same space with equivalence of norms. Furthermore, a Sobolev embedding theorem, namely that the Lipschitz-type spaces with large orders of smoothness can be embedded in Lipschitz spaces, is proved. For metric-measure spaces with heat kernels, a Hardy–Littlewood–Sobolev theorem is establish, and hence it is proved that Lipschitz-type spaces with small orders of smoothness can be embedded in Lebesgue spaces.

The problem is concerned about how large (e.g. the Hausdorff dimension) is Whitney's critical set contained in a given fractal. For this, we prove that the Moran arc, an arc containing a Moran set, is a Whitney's critical set. The excellent open set condition is defined, when the condition holds, the associated self-similar set contains a Whitney's critical subset of full dimension. As its application, the Sierpinski gasket and Koch curve have Whitney's critical subset of full dimension. Finally, we provide a self-similar tree which never contains any Whitney's critical set.

We show that if E is an s-regular set in R^2 for which the triple integral \int_E \int_E \int_E c(x, y, z)^{2s} dH^sx dH^sy dH^sz of the Menger curvature c is finite and if 0 < s ≤ 1/2, then H^s almost all of E can be covered with countably many C^1 curves. We give an example to show that this is false for 1/2 <s< 1.