Sharp bounds for Hardy operators on product spaces

In this article, we obtain the sharp bounds from L^P$\left({\mathbb{G}}^n\right)$ to the space wL^P$\left({\mathbb{G}}^n\right)$ for Hardy operators on product spaces. More generally, the precise norms of Hardy operators on product spaces from L^P$\left({\mathbb{G}}^n\right)$ to the space L^P_I$\left({\mathbb{G}}^n\right)$ are obtained.     

Numerical ranges of Toeplitz operators with matrix symbols

For Toeplitz operators acting on the vector Hardy space $H^2$ with definite or indefinite metric, the closure of the respective numerical range is completely described. In the definite case, some observations regarding its boundary are also made.     

Scalar extensions of categorical resolutions of singularities

Let X be a quasi-compact, separated scheme over a field k and we can consider the categorical resolution of singularities of X. In this paper let $k^{\prime }/k$ be a field extension and we study the scalar extension of a categorical resolution of singularities of X and we show how it gives a categorical resolution of the base change scheme $X_{k^{\prime }}$. Our construction involves the scalar extension of derived categories of DG-modules over a DG algebra. As an application we use the technique of scalar extension developed in this paper to prove the non-existence of full exceptional collections of categorical resolutions for a projective curve of genus ≥1 over a non-algebraically closed field.     

Non-differentiability and Hölder properties of self-affine functions

We consider the class of self-affine functions. Firstly, we characterize all nowhere differentiable self-affine continuous functions. Secondly, given a self-affine continuous function $?$, we investigate its Hölder properties. We find its best uniform Hölder exponent and when $?$ is $C^1$, we find the best uniform Hölder exponent of ${?}^{\prime }$. Thirdly, we show that the Hölder cut of $?$ takes the same value almost everywhere for the Lebesgue measure. This last result is a consequence of the Borel strong law of large numbers.