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Perfect dyadic operators were first introduced in [1] , where a local T ( b ) theorem was proved for such operators. In [3] it was shown that for every singular integral operator T with locally bounded kernel on R n × R n there exists a perfect dyadic operator T such that T − T is bounded on L p ( d x ) for all 1 < p < ∞ . In this paper we show a decomposition of perfect dyadic operators on real line into four well known operators: two selfadjoint operators, paraproduct and its adjoint. Based on this decomposition we prove a sharp weighted version of the T ( 1 ) theorem for such operators, which implies A 2 conjecture for such operators with constant which only depends on ‖ T ( 1 ) ‖ BMO d , ‖ T ⁎ ( 1 ) ‖ BMO d and the constant in testing conditions for T . Moreover, the constant depends on these parameters at most linearly. In this paper we also obtain sufficient conditions for the two weight boundedness for a perfect dyadic operator and simplify these conditions under additional assumptions that weights are in the Muckenhoupt class A ∞ d . [ABSTRACT FROM AUTHOR]

This paper studies by means of standard analytic tools the small time behavior of the heat content over a bounded Lebesgue measurable set of finite perimeter by working with the set covariance function and by imposing conditions on the heat kernels. Applications concerning the heat kernels of rotational invariant α -stable processes are given. [ABSTRACT FROM AUTHOR]

In this paper we will give some characterizations for the boundedness of the maximal operators, potential operators and singular integral operators on the vanishing generalized Morrey spaces V L Π p ( ⋅ ) , φ ( Ω ) with variable exponent p ( ⋅ ) and bounded set Ω. Moreover, we also obtain some corresponding results for unbounded set Ω on such these spaces. [ABSTRACT FROM AUTHOR]

A well-known result of W. Ray asserts that if C is an unbounded convex subset of a Hilbert space, then there is a nonexpansive mapping T : C → C that has no fixed point. In this paper we establish some common fixed point properties for a semitopological semigroup S of nonexpansive mappings acting on a closed convex subset C of a Hilbert space, assuming that there is a point c ∈ C with a bounded orbit and assuming that certain subspace of C b ( S ) has a left invariant mean. Left invariant mean (or amenability) is an important notion in harmonic analysis of semigroups and groups introduced by von Neumann in 1929 [28] and formalized by Day in 1957 [5] . In our investigation we use the notion of common attractive points introduced recently by S. Atsushiba and W. Takahashi. [ABSTRACT FROM AUTHOR]

A theorem due to Cartan asserts that every origin-preserving automorphism of bounded circular domains with respect to the origin is linear. In the present paper, by employing the theory of Bergman's representative domain, we prove that under certain circumstances Cartan's assertion remains true for quasi-circular domains in C n . Our main result is applied to obtain some simple criterions for the case n = 3 and to prove that Braun–Kaup–Upmeier's theorem remains true for our class of quasi-circular domains. [ABSTRACT FROM AUTHOR]