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Agresti, A. (author), Veraar, M.C. (author)In this paper we introduce the critical variational setting for parabolic stochastic evolution equations of quasi or semilinear type. Our results improve many of the abstract results in the classical variational setting. In particular, we are able to replace the usual weak or local monotonicity condition by a more flexible local Lipschitz...journal article 2024
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Agresti, A. (author), Veraar, M.C. (author)In this paper we study the stochastic Navier–Stokes equations on the ddimensional torus with transport noise, which arise in the study of turbulent flows. Under very weak smoothness assumptions on the data we prove local wellposedness in the critical case B<sub>q,p</sub><sup>d/q1</sup> for q∈[2,2d) and p large enough. Moreover, we obtain...journal article 2024
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Agresti, Antonio (author), Lindemulder, N. (author), Veraar, M.C. (author)In this paper, we consider traces at initial times for functions with mixed timespace smoothness. Such results are often needed in the theory of evolution equations. Our result extends and unifies many previous results. Our main improvement is that we can allow general interpolation couples. The abstract results are applied to regularity...journal article 2023
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Hytönen, Tuomas (author), van Neerven, J.M.A.M. (author), Veraar, M.C. (author), Weis, Lutz (author)In this chapter we address a couple of topics in the theory of H<sup>∞</sup>calculus centering around the question what can be said about an operator of the form A+B when A and B have certain “good” properties such as being (R)sectorial or admitting a bounded H<sup>∞</sup>calculus.book chapter 2023
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Hytönen, Tuomas (author), van Neerven, J.M.A.M. (author), Veraar, M.C. (author), Weis, Lutz (author)This chapter presents an indepth study of several classes of vectorvalued function spaces defined by smoothness conditions.book chapter 2023
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Hytönen, Tuomas (author), van Neerven, J.M.A.M. (author), Veraar, M.C. (author), Weis, Lutz (author)In this chapter we address two strongly interwoven topics: How to verify the boundedness of the H<sup>∞</sup>calculus of an operator and how to represent and estimate its fractional powers. For concrete operators such as the Laplace operator or elliptic partial differential operators, the fractional domain spaces can often be identifed with...book chapter 2023
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Hytönen, Tuomas (author), van Neerven, J.M.A.M. (author), Veraar, M.C. (author), Weis, Lutz (author)Before addressing this question for the Calderón{Zygmund type operators of the kind studied in Chapter 11, we investigate a number of related objects in a simpler dyadic model. Besides serving as an introduction to some of the key techniques, it turns out that these dyadic operators can be, and will be, also used as building blocks of the...book chapter 2023
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Hytönen, Tuomas (author), van Neerven, J.M.A.M. (author), Veraar, M.C. (author), Weis, Lutz (author)As we have seen in the preceding sections, in the context of inhomogeneous linear evolution equations, maximal regularity enables one to set up an isomorphism between the space of data (initial value and inhomogeneity) and the solution space.book chapter 2023
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Hytönen, Tuomas (author), van Neerven, J.M.A.M. (author), Veraar, M.C. (author), Weis, Lutz (author)The mapping properties of T will of course heavily depend on the assumptions made on the kernel K that we will discuss in more detail in this chapter.book chapter 2023
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Hytönen, Tuomas (author), van Neerven, J.M.A.M. (author), Veraar, M.C. (author), Weis, Lutz (author)In this chapter, we complement the discussion of three major themes of Fourier analysis that we have studied in the previous Volumes.book chapter 2023
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Agresti, A. (author), Veraar, M.C. (author)In this paper we consider a class of stochastic reactiondiffusion equations. We provide local wellposedness, regularity, blowup criteria and positivity of solutions. The key novelties of this work are related to the use transport noise, critical spaces and the proof of higher order regularity of solutions – even in case of nonsmooth...journal article 2023
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Agresti, Antonio (author), Veraar, M.C. (author)In this paper we develop a new approach to nonlinear stochastic partial differential equations with Gaussian noise. Our aim is to provide an abstract framework which is applicable to a large class of SPDEs and includes many important cases of nonlinear parabolic problems which are of quasi or semilinear type. This first part is on local...journal article 2022
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Agresti, Antonio (author), Veraar, M.C. (author)This paper is a continuation of Part I of this project, where we developed a new local wellposedness theory for nonlinear stochastic PDEs with Gaussian noise. In the current Part II we consider blowup criteria and regularization phenomena. As in Part I we can allow nonlinearities with polynomial growth and rough initial values from critical...journal article 2022
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Lorist, E. (author), Veraar, M.C. (author)We introduce CalderónZygmund theory for singular stochastic integrals with operatorvalued kernel. In particular, we prove L <sup>p</sup>extrapolation results under a Hörmander 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...journal article 2021
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van Neerven, J.M.A.M. (author), Veraar, M.C. (author)We prove a new Burkholder–Rosenthal type inequality for discretetime processes taking values in a 2smooth Banach space. As a first application we prove that if (S(t, s)) <sub>⩽</sub><sub>s</sub><sub>≤</sub><sub>t</sub><sub>⩽</sub><sub>T</sub> is a Cevolution family of contractions on a 2smooth Banach space X and (Wt)t∈[0,T] is a...journal article 2021
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Dominguez, Oscar (author), Veraar, M.C. (author)In this paper we study the vectorvalued analogues of several inequalities for the Fourier transform. In particular, we consider the inequalities of Hausdorff–Young, Hardy–Littlewood, Paley, Pitt, Bochkarev and Zygmund. The Pitt inequalities include the Hausdorff–Young and Hardy–Littlewood inequalities and state that the Fourier transform is...journal article 2021
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van Neerven, J.M.A.M. (author), Veraar, M.C. (author)This paper presents a survey of maximal inequalities for stochastic convolutions in 2smooth Banach spaces and their applications to stochastic evolution equations. This article is part of the theme issue 'Semigroup applications everywhere'.review 2020
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 Veraar, M.C. (author) journal article 2020
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Ondrejat, Martin (author), Veraar, M.C. (author)We show that paths of solutions to parabolic stochastic differential equations have the same regularity in time as the Wiener process (as of the current state of art). The temporal regularity is considered in the BesovOrlicz space Bϕ <sup>1/</sup><sub>2</sub><sup>2</sup><sub>∞</sub>(0,T;X)where ϕ<sub>2</sub>(x)= exp(x<sup>2</sup>)− 1 and X...journal article 2020
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Lindemulder, N. (author), Veraar, M.C. (author)In this paper we consider the Laplace operator with Dirichlet boundary conditions on a smooth domain. We prove that it has a bounded H<sup>∞</sup>calculus on weighted L<sup>p</sup>spaces for power weights which fall outside the classical class of A<sub>p</sub>weights. Furthermore, we characterize the domain of the operator and derive...journal article 2020