We study the strong ill-posedness (norm inflation with infinite loss of regularity) for the nonlinear wave equation at every initial data in Wiener amalgam and Fourier amalgam spaces with negative regularity. In particular these spaces contain Fourier-Lebesgue, Sobolev and some modulation spaces. The equations are posed on $\mathbb R^d$ and on torus $\mathbb T^d$ and involve a smooth power nonlinearity. Our results are sharp with respect to well-posedness results of Bényi and Okoudjou (2009) and Cordero and Nicola (2009) in the Wiener amalgam and modulation space cases. In particular, we also complement norm inflation result of Christ, Colliander and Tao (2003) and Forlano and Okamoto (2020) by establishing infinite loss of regularity in the aforesaid spaces.
We study the dispersive properties of the Schrödinger equation. Precisely, we look for estimates which give a control of the local regularity and decay at infinity {\it separately}. The Banach spaces that allow such a treatment are the Wiener amalgam spaces, and Strichartz-type estimates are proved in this framework. These estimates improve some of the classical ones in the case of large time.
We consider Benjamin-Bona-Mahony (BBM) equation of the form $$ u_t+u_x+uu_x-u_{xxt}=0, \quad (x, t)\in \mathcal{M}\times \mathbb R $$ where $\mathcal{M}= \mathbb T$ or $\mathbb R.$ We establish norm inflation (NI) with infinite loss of regularity at general initial data in Fourier amalgam and Wiener amalgam spaces with negative regularity. This strengthen several known NI results at zero initial data in $H^s(\mathbb T)$ established by Bona-Dai (2017) and ill-posedness result established by Bona-Tzvetkov (2008) and Panthee (2011) in $H^s(\mathbb R).$ Our result is sharp with respect to local we…
In this paper we study the Strichartz estimates for the Schrödinger propagator in the context of Wiener amalgam spaces which, unlike the Lebesgue spaces, control the local regularity of a function and its decay at infinity separately. This separability makes it possible to perform a finer analysis of the local and global behavior of the propagator. Our results improve some of the classical ones in the case of large time.
In this paper we obtain some new Strichartz estimates for the Dirac flow in the context of Wiener amalgam spaces which control the local regularity of a function and its decay at infinity separately unlike $L^p$ spaces. While it is well understood recently for some flows such as the Schrödinger and wave flows that work in the non-relativistic regime, nothing is known about the Dirac flow which governs a physical system in the case of relativistic fields.
We establish existence of solutions in a scale of classes weaker than the finite energy Leray class and stronger than the infinite energy Lemarié-Rieusset class. The new classes are based on the $L^2$ Wiener amalgam spaces. Solutions in the classes closer to the Leray class are shown to satisfy some properties known in the Leray class but not the Lemarié-Rieusset class, namely eventual regularity and long time estimates on the growth of the local energy. In this sense, these solutions bridge the gap between Leray's original solutions and Lemarié-Rieusset's solutions and help identify scalings …
For a complex function $F$ on $\mathbb C$, we study the associated composition operator $T_{F}(f):=F\circ f= F(f)$ on Wiener amalgam $W^{p,q}(\mathbb R^d) \ (1\leq p< \infty, 1\leq q<2).$ We have shown $T_{F} $ maps $W^{p, 1}(\mathbb R^d)$ to $W^{p,q}(\mathbb R^d)$ if and only if $F$ is real analytic on $\mathbb R^2$ and $F(0)=0.$ Similar result is proved in the case of modulation spaces $M^{p,q}(\mathbb R^d).$ In particular, this gives an affirmative answer to the open question proposed by Bhimani-Ratnakumar (J. Funct. Anal. 270 (2016), p.621-648).
In this paper we obtain some new Strichartz estimates for the wave propagator $e^{it\sqrt{-Δ}}$ in the context of Wiener amalgam spaces. While it is well understood for the Schrödinger case, nothing is known about the wave propagator. This is because there is no such thing as an explicit formula for the integral kernel of the propagator unlike the Schrödinger case. To overcome this lack, we instead approach the kernel by rephrasing it as an oscillatory integral involving Bessel functions and then by carefully making use of cancellation in such integrals based on the asymptotic expansion of Bes…
We prove sharp estimates for the dilation operator $f(x)\longmapsto f(λx)$, when acting on Wiener amalgam spaces $W(L^p,L^q)$. Scaling arguments are also used to prove the sharpness of the known convolution and pointwise relations for modulation spaces $M^{p,q}$, as well as the optimality of an estimate for the Schrödinger propagator on modulation spaces.
We first prove decay estimates and spacetime integral bounds for Stokes flows in amalgam spaces $E^r_q$ which connect the classical Lebesgue spaces to the spaces of uniformly locally $r$-integrable functions. Using these estimates, we construct mild solutions of the Navier-Stokes equations in the amalgam spaces satisfying the corresponding spacetime integral bounds. Time-global solutions are constructed for small data in $E^3_q$, $1\le q \le 3$. Our results provide new bounds for the strong solutions classically constructed by Kato and the more recent solutions in uniformly local spaces constr…
In this paper various properties of global and local changes of variables as well as properties of canonical transforms are investigated on modulation and Wiener amalgam spaces. We establish several relations among localisations of modulation and Wiener amalgam spaces and, as a consequence, we obtain several versions of local and global Beurling-Helson type theorems. We also establish a number of positive results such as local boundedness of canonical transforms on modulation spaces, properties of homogeneous changes of variables, and local continuity of Fourier integral operators on Fourier L…
We give a complete characterization of the continuity of pseudodifferential operators with symbols in modulation spaces $M^{p,q}$, acting on a given Lebesgue space $L^r$. Namely, we find the full range of triples $(p,q,r)$, for which such a boundedness occurs. More generally, we completely characterize the same problem for operators acting on Wiener amalgam space $W(L^r,L^s)$ and even on modulation spaces $M^{r,s}$. Finally the action of pseudodifferential operators with symbols in $W(\Fur L^1,L^\infty)$ is also investigated.
Orthonormal Strichartz estimates for the Schrödinger equations with Hamiltonian on Wiener amalgam spaces
National Institute of Science Education and Research（国家科学教育研究中心）；Homi Bhabha National Institute（霍米·巴巴国立研究所）；Indian Statistical Institute Kolkata（印度统计学院加尔各答）
本文在 Wiener 调和空间上建立薛定谔方程的正交 Strichartz 估计，改进时间可积性指数并推广至正交族，应用于无穷粒子 Hartree 方程的适定性。
We completely characterize the boundedness on $L^p$ spaces and on Wiener amalgam spaces of the short-time Fourier transform (STFT) and of a special class of pseudodifferential operators, called localization operators. Precisely, a well-known STFT boundedness result on $L^p$ spaces is proved to be sharp. Then, sufficient conditions for the STFT to be bounded on the Wiener amalgam spaces $W(L^p,L^q)$ are given and their sharpness is shown. Localization operators are treated similarly. Using different techniques from those employed in the literature, we relax the known sufficient boundedness cond…
We study the local-in-time regularity of the Brownian motion with respect to localized variants of modulation spaces M^{p, q}_s and Wiener amalgam spaces W^{p, q}_s. We show that the periodic Brownian motion belongs locally in time to M^{p, q}_s (T) and W^{p, q}_s (T) for (s-1)q < -1, and the condition on the indices is optimal. Moreover, with the Wiener measure μon T, we show that (M^{p, q}_s (T), μ) and (W^{p, q}_s (T), μ) form abstract Wiener spaces for the same range of indices, yielding large deviation estimates. We also establish the endpoint regularity of the periodic Brownian motion wi…
Exploiting continuity properties of Fourier multipliers on modulation spaces and Wiener amalgam spaces, we study the Cauchy problem for the NLW equation. Local wellposedness for rough data in modulation spaces and Wiener amalgam spaces is shown. The results formulated in the framework of modulation spaces refine those in [3]. The same arguments may apply to obtain local wellposedness for the NLKG equation.
We establish some new properties of the Dunkl-Wiener amalgam spaces defined on the real line. These results allow us to obtain the boundedness of Dunkl-type fractional integral and fractional maximal operators in the Dunkl-Fofana spaces.
The objective of this paper is to report on recent progress on Strichartz estimates for the Schrödinger equation and to present the state-of-the-art. These estimates have been obtained in Lebesgue spaces, Sobolev spaces and, recently, in Wiener amalgam and modulation spaces. We present and compare the different technicalities. Then, we illustrate applications to well-posedness.
The local solvability of the Cauchy problem for the nonlinear vibrating plate equation is showed in the framework of modulation spaces. In the opposite direction, it is proved that there is no local wellposedness in Wiener amalgam spaces even for the solution to the homogeneous vibrating plate equation.
In this note, we study the ill-posedness of nonlinear wave equations (NLW). Namely, we show that NLW experiences norm inflation at every initial data in negative Sobolev spaces. This result covers a gap left open in a paper of Christ, Colliander, and Tao (2003) and extends the result by Oh, Tzvetkov, and the second author (2019) to non-cubic integer nonlinearities. In particular, for some low dimensional cases, we obtain norm inflation above the scaling critical regularity. We also prove ill-posedness for NLW, via norm inflation at general initial data, in negative regularity Fourier-Lebesgue …
The purpose of this paper is to investigate several issues concerning the Dirac equation from a time-frequency analysis perspective. More precisely, we provide estimates in weighted modulation and Wiener amalgam spaces for the solutions of the Dirac equation with rough potentials. We focus in particular on bounded perturbations, arising as the Weyl quantization of suitable time-dependent symbols, as well as on quadratic and sub-quadratic non-smooth functions, hence generalizing the results in a recent paper by Kato and Naumkin. We then prove local well-posedness on the same function spaces for…
We study the continuity on the modulation spaces $M^{p,q}$ of Fourier multipliers with symbols of the type $e^{iμ(ξ)}$, for some real-valued function $μ(ξ)$. A number of results are known, assuming that the derivatives of order $\geq 2$ of the phase $μ(ξ)$ are bounded or, more generally, that its second derivatives belong to the Sjöstrand class $M^{\infty,1}$. Here we extend those results, by assuming that the second derivatives lie in the bigger Wiener amalgam space $W(\mathcal{F} L^1,L^\infty)$; in particular they could have stronger oscillations at infinity such as $\cos |ξ|^2$. Actually ou…
We study the Hermite operator $H=-Δ+|x|^2$ in $\mathbb{R}^d$ and its fractional powers $H^β$, $β>0$ in phase space. Namely, we represent functions $f$ via the so-called short-time Fourier, alias Fourier-Wigner or Bargmann transform $V_g f$ ($g$ being a fixed window function), and we measure their regularity and decay by means of mixed Lebesgue norms in phase space of $V_g f$, that is in terms of membership to modulation spaces $M^{p,q}$, $0< p,q\leq \infty$. We prove the complete range of fixed-time estimates for the semigroup $e^{-tH^β}$ when acting on $M^{p,q}$, for every $0< p,q\leq \infty$…
