Search

In this paper, we consider traces at initial times for functions with mixed time‐space 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 problems for fractional evolution equations and stochastic evolution equations, where uniform trace estimates on the half‐line are shown. [ABSTRACT FROM AUTHOR]

We study traces of weighted Triebel–Lizorkin spaces Fp,qs(Rn,w)$F^s_{p,q}(\mathbb {R}^n,w)$ on hyperplanes Rn−k$\mathbb {R}^{n-k}$, where the weight is of Muckenhoupt type. We concentrate on the example weight wα(x)=|xn|α$w_\alpha (x) = {\big\vert x_n\big\vert }^\alpha$ when |xn|≤1$\big\vert x_n\big\vert \le 1$, x∈Rn$x\in \mathbb {R}^n$, and wα(x)=1$w_\alpha (x)=1$ otherwise, where α>−1$\alpha >-1$. Here we use some refined atomic decomposition argument as well as an appropriate wavelet representation in corresponding (unweighted) Besov spaces. The second main outcome is the description of the real interpolation space (Bp1,p1s1(Rn−k),Bp2,p2s2(Rn−k))θ,r$\big (B^{s_1}_{p_1,p_1}\big (\mathbb {R}^{n-k}\big), B^{s_2}_{p_2,p_2}{\big (\mathbb {R}^{n-k}\big)\big)}_{\theta ,r}$, 00$s>0$ sufficiently large, 0<θ<1$0<\theta <1$, 0

In this paper we investigate the nuclear trace of vector‐valued Fourier multipliers on the torus and its applications to the index theory of periodic pseudo‐differential operators. First we characterise the nuclearity of pseudo‐differential operators acting on Bochner integrable functions. In this regards, we consider the periodic and the discrete cases. We go on to address the problem of finding sharp sufficient conditions for the nuclearity of vector‐valued Fourier multipliers on the torus. We end our investigation with two index formulae. First, we express the index of a vector‐valued Fourier multiplier in terms of its operator‐valued symbol and then we use this formula for expressing the index of certain elliptic operators belonging to periodic Hörmander classes. [ABSTRACT FROM AUTHOR]

We study fractional Sobolev and Besov spaces on noncompact Riemannian manifolds with bounded geometry. Usually, these spaces are defined via geodesic normal coordinates which, depending on the problem at hand, may often not be the best choice. We consider a more general definition subject to different local coordinates and give sufficient conditions on the corresponding coordinates resulting in equivalent norms. Our main application is the computation of traces on submanifolds with the help of Fermi coordinates. Our results also hold for corresponding spaces defined on vector bundles of bounded geometry and, moreover, can be generalized to Triebel-Lizorkin spaces on manifolds, improving [11]. [ABSTRACT FROM AUTHOR]

Under appropriate conditions, we determine the growth envelopes of Besov spaces on so-called h-sets. In the way we clarify what should in general be meant by such Besov spaces and establish some elementary embeddings. Actually, we are even able to give an optimal necessary condition for one of such embeddings as a by-product of the proof of the growth envelope result. [ABSTRACT FROM AUTHOR]

We present a simple approach to the theory of traces on the Marcinkiewicz operator ideal [ABSTRACT FROM AUTHOR]

Dedicated to Professor V. I. Burenkov on the occasion of his 70th birthday We characterize the traces of vector-valued Besov and Lizorkin-Triebel spaces. Therefrom we derive the corresponding assertions for the vector-valued Sobolev spaces \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$W^m_{p}({\mathbf R}^n,E)\,$\end{document}. Here we do not assume the UMD property for the Banach space E. [ABSTRACT FROM AUTHOR]