Fourier-Lebesgue Spaces Research Articles

Sharp convolution and multiplication estimates in weighted spaces

We establish sharp convolution and multiplication estimates in weighted Lebesgue, Fourier Lebesgue and modulation spaces. We cover, especially some results in [L. Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations (Springer, Berlin, 1997); S. Pilipović, N. Teofanov and J. Toft, Micro-local analysis in Fourier Lebesgue and modulation spaces, II, J. Pseudo-Differ. Oper. Appl.1 (2010) 341–376]. The results are also related to some results by Iwabuchi in [T. Iwabuchi, Navier–Stokes equations and nonlinear heat equations in modulation spaces with negative derivative indices, J. Differential Equations248 (2010) 1972–2002].

Almost critical local well-posedness for the space-time monopole equation in Lorenz gauge

Recently, Candy and Bournaveas proved local well-posedness of the space-time monopole equation in Lorenz gauge for initial data in Hs with [Formula: see text]. The equation is L2-critical, and hence a [Formula: see text] derivative gap is left between their result and the scaling prediction. In this paper, we consider initial data in the Fourier–Lebesgue space [Formula: see text] for 1 < p ≤ 2 which coincides with Hs when p = 2 but scales like lower regularity Sobolev spaces for 1 < p < 2. In particular, we will see that as p → 1+, the critical exponent [Formula: see text], in which case [Formula: see text] is the critical space. We shall prove almost optimal local well-posedness to the space-time monopole equation in Lorenz gauge with initial data in the aforementioned spaces that correspond to p close to 1.

On Compactness of Embeddings of Fourier-Lebesgue Spaces into Modulation Spaces

It is shown that, for a certain range of parameters, embeddings of Fourier-Lebesgue Lap∩ℱLbq spaces into modulation spaces Mα,βr, s are compact.

Characterization of Wave Front Sets in Fourier-Lebesgue Spaces and Its Application

In this paper, we characterize the Fourier-Lebesgue type wave front set by using the wave packet transform. We apply this to the propagation of singularities for the first order hyperbolic partial differential equations with constant coefficient.

Invariant weighted Wiener measures and almost sure global well-posedness for the periodic derivative NLS

We construct an invariant weighted Wiener measure associated to the periodic derivative nonlinear Schrödinger equation in one dimension and establish global well-posedness for data living in its support. In particular almost surely for data in a Fourier–Lebesgue space {\mathcal F}L^{s,r}(\mathbb T) with s \ge \frac{1}{2} , 2 < r < 4 , (s-1)r <-1 and scaling like H^{\frac{1}{2}-\epsilon}(\mathbb T), for small \epsilon >0 . We also show the invariance of this measure.

Generalized free time-dependent Schrödinger equation with initial data in Fourier Lebesgue spaces

Consider the solution of the free time-dependent Schro dinger equation with initial data f. It is shown by Sjogren and Sjolin that there exists f in the Sobolev space Hs (Rd), s = d/2 such that tangential convergence can not be widened to convergence regions. The author obtained in a previous paper the corresponding results for a generalized version of the Schrodinger equation, where −Δx is replaced by an operator \({\varphi(D)}\), with special conditions on \({\varphi}\). In this paper we show that similar results may be obtained for initial data in usual and mixed Fourier Lebesgue spaces. We also relax the conditions on \({\varphi}\).

Micro-local analysis in Fourier Lebesgue and modulation spaces: part II

We consider different types of (local) products f1 f2 in Fourier Lebesgue spaces. Furthermore, we prove the existence of such products for other distributions satisfying appropriate wave-front properties. We also consider semi-linear equations of the form $$P(x,D)f = G(x,J_k f),$$ with appropriate polynomials P and G, where Jk denotes the k-jet of f. If the solution locally belongs to appropriate weighted Fourier Lebesgue space \({\fancyscript{F}L^q_{(\omega )} (\mathbf{R}^d)}\) and P is non-characteristic at (x0, ξ0), then we prove that \({(x_0,\xi_0)\not \in {\rm WF}_{\fancyscript{F}L^q_{(\widetilde {\omega })}} (f)}\), where \({\widetilde{\omega }}\) depends on ω, P and G.

Micro-Local Analysis with Fourier Lebesgue Spaces. Part I

Let ω,ω 0 be appropriate weight functions and q∈[1,∞]. We introduce the wave-front set, \(\mathrm{WF}_{\mathcal{F}L^{q}_{(\omega)}}(f)\) of \(f\in \mathcal{S}'\) with respect to weighted Fourier Lebesgue space \(\mathcal{F}L^{q}_{(\omega )}\). We prove that usual mapping properties for pseudo-differential operators Op (a) with symbols a in \(S^{(\omega _{0})}_{\rho ,0}\) hold for such wave-front sets. Especially we prove that $$\begin{array}[b]{lll}\mathrm{WF}_{\mathcal{F}L^q_{(\omega /\omega _0)}}(\operatorname {Op}(a)f)&\subseteq&\mathrm{WF}_{\mathcal{F}L^q_{(\omega )}}(f)\\[6pt]&\subseteq&\mathrm{WF}_{\mathcal{F}L^q_{(\omega/\omega _0)}}(\operatorname {Op}(a)f)\cup \operatorname {Char}(a).\end{array}$$ (*) Here Char (a) is the set of characteristic points of a.

Boundedness of Fourier integral operators on Fourier Lebesgue spaces and affine fibrations

We study Fourier integral operators of Hörmander's type acting on the spaces $\mathcal F L^p(\mathbb R^d)_{\rm comp} $, $1\leq p\leq\infty$, of compactly supported distributions whose Fourier transform is in $L^p$. We show that the sharp loss of derivati

Uncertainty principles as embeddings of modulation spaces

A class of new uncertainty principles is derived in the form of embeddings of Fourier–Lebesgue spaces into modulation spaces. These embeddings provide practical, sufficient conditions for a function to belong to a modulation space. Counterexamples based on the properties of Gabor expansions demonstrate that the embeddings are optimal.