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A note on optimal probability lower bounds for centered random variables

100%
Veraar M.
Colloquium Mathematicum
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2008
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tom 113
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nr 2
231-240
EN
We obtain lower bounds for ℙ(ξ ≥ 0) and ℙ(ξ > 0) under assumptions on the moments of a centered random variable ξ. The estimates obtained are shown to be optimal and improve results from the literature. They are then applied to obtain probability lower bounds for second order Rademacher chaos.
2
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Sharp embedding results for spaces of smooth functions with power weights

63%
Meyries M., Veraar M.
Studia Mathematica
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2012
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tom 208
|
nr 3
257-293
EN
We consider function spaces of Besov, Triebel-Lizorkin, Bessel-potential and Sobolev type on $ℝ^{d}$, equipped with power weights $w(x) = |x|^{γ}$, γ > -d. We prove two-weight Sobolev embeddings for these spaces. Moreover, we precisely characterize for which parameters the embeddings hold. The proofs are presented in such a way that they also hold for vector-valued functions.
3
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Non-autonomous stochastic Cauchy problems in Banach spaces

63%
Veraar M., Zimmerschied J.
Studia Mathematica
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2008
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tom 185
|
nr 1
1-34
EN
We study the non-autonomous stochastic Cauchy problem on a real Banach space E, $dU(t) = A(t)U(t)dt + B(t)dW_{H}(t)$, t ∈ [0,T], U(0) = u₀. Here, $W_{H}$ is a cylindrical Brownian motion on a real separable Hilbert space H, $(B(t))_{t∈[0,T]}$ are closed and densely defined operators from a constant domain 𝒟(B) ⊂ H into E, $(A(t))_{t∈[0,T]}$ denotes the generator of an evolution family on E, and u₀ ∈ E. In the first part, we study existence of weak and mild solutions by methods of van Neerven and Weis. Then we use a well-known factorisation method in the setting of evolution families to obtain time regularity of the solution. In the second part, we consider the parabolic case in the setting of Acquistapace and Terreni. By means of a factorisation method in the spirit of Da Prato, Kwapień, and Zabczyk we obtain space-time regularity results for parabolic evolution families on Banach spaces. We apply this theory to several examples. In the last part, relying on recent results of Dettweiler, van Neerven, and Weis, we prove a maximal regularity result where the A(t) are as in the setting of Kato and Tanabe.
4
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Estimates for vector-valued holomorphic functions and Littlewood-Paley-Stein theory

63%
Veraar M., Weis L.
Studia Mathematica
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2015
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tom 228
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nr 1
73-99
EN
We consider generalized square function norms of holomorphic functions with values in a Banach space. One of the main results is a characterization of embeddings of the form $L^{p}(X) ⊆ γ(X) ⊆ L^{q}(X)$, in terms of the type p and cotype q of the Banach space X. As an application we prove $L^{p}$-estimates for vector-valued Littlewood-Paley-Stein g-functions and derive an embedding result for real and complex interpolation spaces under type and cotype conditions.
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