We derive a new result which extends the range of validity of the asymptotic formula for cubefull integers in an interval.

In the present note we characterize the function space $\dot{X}_{r}$, which is the set of pointwise multipliers which map $L^{2}$ into $\dot{H}^{-r}$. To this end, we use wavelets and capacity.

The truncated trigonometric matrix moment problem is a well studied object. In particular, one can find in the literature the extremal value concerning the weight assigned to some point of the unit circle within the solution set of that matrix moment problem for the so-called non-degenerate situation and some special measure which realizes the extremal value. The primary concern of the paper is to give a basic proof for the fact that this distinguished measure is uniquely determined by that extremal feature.

We introduce a Laplace transform for Laplace hyperfunctions valued in a complete locally convex space $X$. In this general case the Laplace transform is a compatible family of holomorphic functions with values in local Banach spaces. Especially interesting is the case where $X=L_b(E,F)$ is the space of operators between locally convex spaces. In the forthcoming paper [6] this will be applied to solve the abstract Cauchy problem for operators in complete ultrabornological locally convex spaces (like spaces of smooth functions and distributions) extending results of Komatsu for operators in Banach spaces.

Define the recurrence set of Gauss transformation $T$ on the field of Laurent series as following $$E(x_0)=\{x\in I: T^n(x)\in I_{t_n}(x_0) for infinitely many $n$\},$$ where $I_{t_n}(x_0)$ denotes $t_n$-th order cylinder of $x_0$. In this paper, the Hausdorff dimension of the set $E(x_0)$ is determined.

We introduce 2-microlocal Besov and Triebel-Lizorkin spaces with variable integrability and give characterizations by decompositions in atoms, molecules and wavelets. These spaces cover the usual Besov and Triebel-Lizorkin spaces as well as spaces of variable smoothness and integrability. We emphasize that the spaces $F^{s(cdot)}_{p(\cdot),q(\cdot)}(\mathbb{R}^n)$ which were defined recently in [12] are included in this approach.

We give the complete solution $(n,a,b,x,y)$ of the title equation when $\gcd(x,y)=1$, except for the case when $xab$ is odd. Our main result is Theorem 1.