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In this paper, we consider the trace theorem for modulation spaces, alpha modulation spaces and Besov spaces. For the modulation space, we obtain the sharp results.
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We give sufficient conditions for compactness of localization operators on modulation spaces $textbf{M}^{p,q}_{m_{lambda}}( mathbb{R}^{d})$ of $omega$-tempered distributions whose short-time Fourier transform is in the weighted mixed space $L^{p,q}_{m_lambda}$ for $m_lambda(x)=e^{lambdaomega(x)}$.
Let $X$ be a space of homogeneous type and $L$ be a nonnegative self-adjoint operator on $L^2(X)$ satisfying Gaussian upper bounds on its heat kernels. In this paper we develop the theory of weighted Besov spaces $dot{B}^{alpha,L}_{p,q,w}(X)$ and weighted Triebel--Lizorkin spaces $dot{F}^{alpha,L}_{p,q,w}(X)$ associated to the operator $L$ for the full range $0<p,qle infty$, $alphain mathbb R$ and $w$ being in the Muckenhoupt weight class $A_infty$. Similarly to the classical case in the Euclidean setting, we prove that our new spaces satisfy important features such as continuous charaterizations in terms of square functions, atomic decompositions and the identifications with some well known function spaces such as Hardy type spaces and Sobolev type spaces. Moreover, with extra assumptions on the operator $L$, we prove that the new function spaces associated to $L$ coincide with the classical function spaces. Finally we apply our results to prove the boundedness of the fractional power of $L$ and the spectral multiplier of $L$ in our new function spaces.
We obtain Gabor frame characterisations of modulation spaces defined via a class of translation-modulation invariant Banach spaces of distributions that was recently introduced in $[10]$. We show that these spaces admit an atomic decomposition through Gabor expansions and that they are characterised by summability properties of their Gabor coefficients. Furthermore, we construct a large space of admissible windows. This generalises several fundamental results for the classical modulation spaces $ M^{p,q}_{w}$. Due to the absence of solidity assumptions on the Banach spaces defining these modulation spaces, the methods used for the spaces $M^{p,q}_{w}$ (or, more generally, in coorbit space theory) fail in our setting and we develop here a new approach based on the twisted convolution.
Let $X$ be a space of homogeneous type and let $L$ be a nonnegative self-adjoint operator on $L^2(X)$ which satisfies a Gaussian estimate on its heat kernel. In this paper we prove a Homander type spectral multiplier theorem for $L$ on the Besov and Triebel--Lizorkin spaces associated to $L$. Our work not only recovers the boundedness of the spectral multipliers on $L^p$ spaces and Hardy spaces associated to $L$, but also is the first one which proves the boundedness of a general spectral theorem on Besov and Triebel--Lizorkin spaces.
We study the embeddings of (homogeneous and inhomogeneous) anisotropic Besov spaces associated to an expansive matrix $A$ into Sobolev spaces, with focus on the influence of $A$ on the embedding behaviour. For a large range of parameters, we derive sharp characterizations of embeddings.
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