Singular stochastic integral operators

In this paper we introduce Calder\'on-Zygmund theory for singular stochastic integrals with operator-valued kernel. In particular, we prove $L^p$-extrapolation results under a H\"ormander condition on the kernel. Sparse domination and sharp weighted bounds are obtained under a Dini condition on the kernel, leading to a stochastic version of the solution to the $A_2$-conjecture. The results are applied to obtain $p$-independence and weighted bounds for stochastic maximal $L^p$-regularity both in the complex and real interpolation scale. As a consequence we obtain several new regularity results for the stochastic heat equation on $\mathbb{R}^d$ and smooth and angular domains.Comment: typos corrected. Accepted for publication in Analysis & PD

Structurally damped plate and wave equations with random point force in arbitrary space dimensions

In this paper we consider structurally damped plate and wave equations with point and distributed random forces. In order to treat space dimensions more than one, we work in the setting of $L^q$--spaces with (possibly small) $q\in(1,2)$. We establish existence, uniqueness and regularity of mild and weak solutions to the stochastic equations employing recent theory for stochastic evolution equations in UMD Banach spaces.Comment: accepted for publication in Differential and Integral Equation

Fourier multiplier theorems involving type and cotype

In this paper we develop the theory of Fourier multiplier operators $T_{m}:L^{p}(\mathbb{R}^{d};X)\to L^{q}(\mathbb{R}^{d};Y)$, for Banach spaces $X$ and $Y$, $1\leq p\leq q\leq \infty$ and $m:\mathbb{R}^d\to \mathcal{L}(X,Y)$ an operator-valued symbol. The case $p=q$ has been studied extensively since the 1980's, but far less is known for $p<q$. In the scalar setting one can deduce results for $p<q$ from the case $p=q$. However, in the vector-valued setting this leads to restrictions both on the smoothness of the multiplier and on the class of Banach spaces. For example, one often needs that $X$ and $Y$ are UMD spaces and that $m$ satisfies a smoothness condition. We show that for $p<q$ other geometric conditions on $X$ and $Y$, such as the notions of type and cotype, can be used to study Fourier multipliers. Moreover, we obtain boundedness results for $T_m$ without any smoothness properties of $m$. Under smoothness conditions the boundedness results can be extrapolated to other values of $p$ and $q$ as long as $\tfrac{1}{p}-\tfrac{1}{q}$ remains constant.Comment: Revised version, to appear in Journal of Fourier Analysis and Applications. 31 pages. The results on Besov spaces and the proof of the extrapolation result have been moved to arXiv:1606.0327