Let L be a non-negative self-adjoint operator acting on 2 ( X ) , where is space of homogeneous type with dimension n . Suppose that the heat e − t satisfies generalized Gaussian p 0 ′ -estimates order m for some 1 ≤ < It known I + s i bounded ≥ | / and ∈ (see example, [5] [7] [9] [10] [13] [26] ). In this paper we study endpoint case = show weak is, there constant C > independent f so μ { x : ...

given a mapping $f:xto x$ we naturally associate to it a monotonic map $gg_f:exp xto exp x$ from the power set of $x$ into itself, thus inducing a generalized topology on $x$. in this paper we investigate some properties of generalized topologies which are defined by such a procedure.

in this paper, we compute the number of fuzzy subgroups of some classes of non-abeilan groups. explicit formulas are givenfor dihedral groups $d_{2n}$, quasi-dihedral groups $qd_{2^n}$, generalized quaternion groups $q_{4n}$ and modular $p$-groups $m_{p^n}$.