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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]

Abstract: In this paper we consider the following Schrödinger–Poisson system where potentials , ( , ), and is asymptotically linear at infinity. Working in a variational setting, we prove the existence and multiplicity of solutions for the system when λ is small and belong to different ranges. We also study the boundedness of the set of the solutions found, as , in the spirit of Ruiz (2006) [21]. [Copyright &y& Elsevier]

Abstract: This paper is dedicated to inverse quasi-variational inequalities which correspond to a mixture of quasi-variational inequalities and inverse variational inequalities. Our aim is to obtain local/global error bounds for inverse quasi-variational inequality problems in terms of different gap functions/merit functions i.e. the residual gap function, the regularized gap function and the -gap function. These bounds provide effective estimated distances between a specific point and the exact solution of the inverse quasi-variational inequality problem. Since the class of inverse quasi-variational inequalities includes the classes of general quasi-variational inequalities and variational inequalities, our results cover and extend similar results for these problems. [Copyright &y& Elsevier]

Abstract: In this paper, we give an example of a Fourier integral operator with a symbol belonging to that cannot be extended as a bounded operator on . [Copyright &y& Elsevier]

Abstract: Let be the triangle in bounded by the lines (the standard simplex in ). Denote by the set of all real bivariate algebraic polynomials of total degree at most two. Let be the unit ball of the space endowed with the supremum norm on . We give a full description of the semidefinite extreme points of . The present paper completes the description of the set of all extreme points of started in Milev and Naidenov (2008) [13] and Milev and Naidenov (2011) [14]. We study, as an application, the path-connectedness of . The conclusion is that consists of two path-connected components. [Copyright &y& Elsevier]

Abstract: Let be an integral self-affine set (not necessarily a self-similar set) satisfying , where is an integer expanding matrix and is a finite set of integer vectors. For “totally disconnected ”, in 1992, Falconer obtained formulas for lower and upper bounds for the Hausdorff dimension of . In order to have such bounds for arbitrary , we consider an extension of Falconer’s formulas to certain graph directed sets and define new bounds. For a very few classes of self-affine sets, the Hausdorff dimension and Falconer’s upper bound are known to be different. In this paper, we present a new such class by using the new upper bound, and show that our upper bound is the box dimension for that class. We also study the computation of those bounds. [Copyright &y& Elsevier]

Abstract: For in , a subset of is called a Furstenberg set of type or -set if for each direction in the unit circle there is a line segment in the direction of such that the Hausdorff dimension of the set is greater than or equal to . In this paper we use generalized Hausdorff measures to give estimates on the size of these sets. Our main result is to obtain a sharp dimension estimate for a whole class of zero-dimensional Furstenberg type sets. Namely, for , , we construct a set of Hausdorff dimension not greater than . Since in a previous work we showed that is a lower bound for the Hausdorff dimension of any , with the present construction, the value is sharp for the whole class of Furstenberg sets associated to the zero dimensional functions . [Copyright &y& Elsevier]

Abstract: In this paper, we give existence, uniqueness and regularity of the solutions of problems whose prototype is where is a bounded open set of , , and . Here is a norm on , is its polar and is the anisotropic Laplacian. [Copyright &y& Elsevier]