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

Vector-valued extensions of operators through multilinear limited range extrapolation

We give an extension of Rubio de Francia's extrapolation theorem for functions taking values in UMD Banach function spaces to the multilinear limited range setting. In particular we show how boundedness of an $m$-(sub)linear operator $$T:L^{p_1}(w_1^{p_1})\times\cdots\times L^{p_m}(w_m^{p_m})\to L^p(w^p)$$ for a certain class of Muckenhoupt weights yields an extension of the operator to Bochner spaces $L^{p}(w^p;X)$ for a wide class of Banach function spaces $X$, which includes certain Lebesgue, Lorentz and Orlicz spaces. We apply the extrapolation result to various operators, which yields new vector-valued bounds. Our examples include the bilinear Hilbert transform, certain Fourier multipliers and various operators satisfying sparse domination results.Comment: 21 pages. Minor modifications. To appear in Journal of Fourier Analysis and Application

Fourier multipliers in Banach function spaces with UMD concavifications

We prove various extensions of the Coifman-Rubio de Francia-Semmes multiplier theorem to operator-valued multipliers on Banach function spaces. Our results involve a new boundedness condition on sets of operators which we call $\ell^{r}(\ell^{s})$-boundedness, which implies $\mathcal{R}$-boundedness in many cases. The proofs are based on new Littlewood-Paley-Rubio de Francia-type estimates in Banach function spaces which were recently obtained by the authors

On the ℓs\ell^s-boundedness of a family of integral operators

In this paper we prove an $\ell^s$-boundedness result for integral operators with operator-valued kernels. The proofs are based on extrapolation techniques with weights due to Rubio de Francia. The results will be applied by the first and third author in a subsequent paper where a new approach to maximal $L^p$-regularity for parabolic problems with time-dependent generator is developed.Comment: Minor revision. Accepted for publication in Rev. Mat. Iberoamerican

Stein interpolation for the real interpolation method

We prove a complex formulation of the real interpolation method, showing that the real and complex interpolation methods are not inherently real or complex. Using this complex formulation, we prove Stein interpolation for the real interpolation method. We apply this theorem to interpolate weighted Lp-spaces and the sectoriality of closed operators with the real interpolation method

Sparse domination implies vector-valued sparse domination

We prove that scalar-valued sparse domination of a multilinear operator implies vector-valued sparse domination for tuples of quasi-Banach function spaces, for which we introduce a multilinear analogue of the UMD condition. This condition is characterized by the boundedness of the multisublinear Hardy-Littlewood maximal operator and goes beyond examples in which a UMD condition is assumed on each individual space and includes e.g. iterated Lebesgue, Lorentz, and Orlicz spaces. Our method allows us to obtain sharp vector-valued weighted bounds directly from scalar-valued sparse domination, without the use of a Rubio de Francia type extrapolation result. We apply our result to obtain new vector-valued bounds for multilinear Calder\'on-Zygmund operators as well as recover the old ones with a new sharp weighted bound. Moreover, in the Banach function space setting we improve upon recent vector-valued bounds for the bilinear Hilbert transform.Comment: 31 pages. Corrected author nam

Strongly Kreiss Bounded Operators in UMD Banach Spaces

In this paper we give growth estimates for $\|T^n\|$ for $n\to \infty$ in the case $T$ is a strongly Kreiss bounded operator on a UMD Banach space $X$. In several special cases we provide explicit growth rates. This includes known cases such as Hilbert and $L^p$-spaces, but also intermediate UMD spaces such as non-commutative $L^p$-spaces and variable Lebesgue spaces.Comment: 27 page

Is Categorization in Visual Working Memory a Way to Reduce Mental Effort? A Pupillometry Study

Recent studies on visual working memory (VWM) have shown that visual information can be stored in VWM as continuous (e.g., a specific shade of red) as well as categorical representations (e.g., the general category red). It has been widely assumed, yet never directly tested, that continuous representations require more VWM mental effort than categorical representations; given limited VWM capacity, this would mean that fewer continuous, as compared to categorical, representations can be maintained simultaneously. We tested this assumption by measuring pupil size, as a proxy for mental effort, in a delayed estimation task. Participants memorized one to four ambiguous (boundaries between adjacent color categories) or prototypical colors to encourage continuous or categorical representations, respectively; after a delay, a probe indicated the location of the to‐be‐reported color. We found that, for memory load 1, pupil size was larger while maintaining ambiguous as compared to prototypical colors, but without any difference in memory precision; this suggests that participants relied on an effortful continuous representation to maintain a single ambiguous color, thus resulting in pupil dilation while preserving precision. Strikingly, this effect gradually inverted, such that for memory load 4, pupil size was smaller while maintaining ambiguous and prototypical colors, but memory precision was now substantially reduced for ambiguous colors; this suggests that with increased memory load participants increasingly relied on categorical representations for ambiguous colors (which are by definition a poor fit to any category). Taken together, our results suggest that continuous representations are more effortful than categorical representations and that very few continuous representations (perhaps only one) can be maintained simultaneously

Age modulates the effects of mental fatigue on typewriting

In the present study, we examined whether age influences the effects of mental fatigue on task performance, and if we could validate the use of measures based on typing behavior as an index of the effects of mental fatigue on different aspects of cognition. Young (N = 24, 18-30 years) and middle-aged (N = 24, 50-67 years) participants performed a typewriting task and a mouse targeting task for 120 min. At the beginning and at the end of the experiment the level of subjective fatigue was assessed. During task performance measures based on typing behavior and EEG were recorded. Results showed that subjective fatigue increased over the experiment in both the young and the middle-aged group. Typing speed decreased with time-on-task (ToT) in both age groups, reflected in larger general interkey intervals and in an increase in typing time. In addition, typing accuracy decreased with ToT in the young group, however, not in the middle-aged group, reflected by an increase in typing errors. Moreover, the young group used the backspace key more often with ToT due to delayed error-correction, reflected in larger backspace sequences, resulting in larger interkey intervals and increased typing time. This effect was absent in the middle-aged group. In the young group, the P3 brain potential amplitude decreased over the experiment, which was related to an increase in typing time, longer general interkey intervals, and an increase in typing errors, suggesting that decreased task engagement was related to less efficient typewriting, at least in the young group. These results indicate that measures based on typing behavior could give information about the process of mental fatigue, and in addition suggest that age influences the effect of mental fatigue on typewriting. More specifically, younger adults more often adopt a strategy that emphasizes speed, while middle-aged adults act more error-aversive than younger adults.This study, part of SPRINT@Work, is part-financed by the European Regional Development Fund, the province and municipality of Groningen, and the province of Drenthe (Grant No: T-3036, 2013)