In this paper, we consider the Herz-Morrey space H p(·),q,! {x0} (G) of variable exponent consisting of all measurable functions f on a bounded open set G ⊂ R n satisfying kfk H p(·),q,! {x0} (G) = ˆ 2dG H p(·),q,! (G),H p(·),q,! {x0} (G) and ˜ H p(·),q,! (G). Following Fiorenza-Rakotoson (18), Di Fratta-Fiorenza (17) and Gogatishvili-Mustafayev (19), we next discuss the duality properties among these Herz- Morrey spaces.

The paper deals with the invertibility of Toeplitz plus Hankel operators T(a)+H(b) acting on classical Hardy spaces on the unit circle T. It is supposed that the generating functions a and b satisfy the condition a(t)a(1/t) = b(t)b(1/t), t ∈ T. Special attention is paid to the case of piecewise continuous generating functions. In some cases the dimensions of null spaces of the operator T(a) + H(b) and its adjoint are described. Fredholm properties of Toeplitz plus Hankel operatorsT(a)+H(b) with piecewise continuous generating functions a and b have been studied for many years. These operators are considered in various Banach and Hilbert spaces, and the results ob- tained show that the structure of the algebras generated by such operators is much more complicated than the structure of the algebras generated by one-dimensional singular integral operators with piecewise continuous coefficients defined on closed smooth curves. Moreover, calculating the index of Toeplitz plus Hankel operators, one encounters even more difficult problems, and the difficulti es grow if one attempts to study invertibility or one-sided invertibility of such operators. It is worth noting that, in general, Fredholm operators T(a) +H(b) are not one-sided invertible. Nev- ertheless, a few works where one-sided invertibility was discussed, have appeared in literature recently. They are mainly concerned with two special cases of Toeplitz plus Hankel operators—viz. with the operators having the form M(a) := T(a) +H(a) or the form f

We study the relationship between measurable differentiable structures on doubling metric measure spaces and derivations. We prove:

domains in arbitrary complex Banach spaces.

We prove representation theorems, the versions of Smirnov's theorem and Caratheo- dory type theorem for harmonic homeomorphisms of the unit disk onto Jordan surfaces with rectifi- able boundaries. Further we establish the classical isoperimetric inequality and the Riesz-Zygmund inequality for Jordan harmonic surfaces without any smoothness assumptions on the boundary.

This note concerns non-autonomous dynamics of rational functions and, more precisely, the fractal behavior of the Julia sets under perturbation of non-autonomous systems. We provide a necessary and sufficient condition for holomorphic stability which leads to Holder continuity of dimensions of hyperbolic non-autonomous Julia sets with respect to the l 1 -topology on the parameter space. On the other hand we show that, for some particular family, the Hausdorff and packing dimension functions are not differentiable at any point and that these dimensions are not equal on an open dense set of the parameter space still with respect to the l 1 -topology.

We study some criteria for the image of the unit disk under the canonical lift ˜ f of a harmonic mapping f to be a John surface and a linearly connected surface. Growth results of the metric of the minimal surface associated to ˜ f are obtained. Some our results generalize known ones in geometric functions theory.

We prove sharp bounds for the product and the sum of two hyperbolic distances between the opposite sides of hyperbolic Lambert quadrilaterals in the unit disk. Furthermore, we study the images of Lambert quadrilaterals under quasiconformal mappings from the unit disk onto itself and obtain sharp results in this case, too.

We prove that !u( ) ≤ C!f ( ), where u : Ω → R n is the harmonic extension of a continuous map f : @Ω → R n , if u is a K-quasiregular map and Ω is bounded in R n with C 2 boundary. Here C is a constant depending only on n, !f and K and !h denotes the modulus of continuity of h. We also prove a version of this result for Λ!-extension domains with c-uniformly perfect boundary and quasiconformal mappings.

We provide sharp weak estimates for the distribution function of Mwhen onwe impose L 1 , L q and L p;1 restrictions. Here M is the dyadic maximal operator associated to a tree T on a non-atomic probability measure space. As a consequence we produce that the inequality jjMTjjp;1 • jjjjjjp;1 is sharp allowing every possible value for the L 1 and the L q norm for a fixed q such that 1 < q < p, where jjj ¢ jjjp;1 is the integral norm on and jj ¢ jjp;1 the usual quasi norm on L p;1 .