Boundedness of Pseudo-Differential Operators onLp, Sobolev and Modulation Spaces

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 is also investigated.

The Quantization of Edge Symbols.- On Rays of Minimal Growth for Elliptic Cone Operators.- Symbolic Calculus of Pseudo-differential Operators and Curvature of Manifolds.- Weyl Transforms, Heat Kernels, Green Functions and Riemann Zeta Functions on Compact Lie Groups.- On the Fourier Analysis of Operators on the Torus.- Wave Kernels of the Twisted Laplacian.- Super-exponential Decay of Solutions to Differential Equations in ?d.- Gevrey Local Solvability for Degenerate Parabolic Operators of Higher Order.- A New Aspect of the L p-extension Problem for Inhomogeneous Differential Equations.- Continuity in Quasi-homogeneous Sobolev Spaces for Pseudo-differential Operators with Besov Symbols.- Continuity and Schatten Properties for Pseudo-differential Operators on Modulation Spaces.- Algebras of Pseudo-differential Operators with Discontinuous Symbols.- A Class of Quadratic Time-frequency Representations Based on the Short-time Fourier Transform.- A Characterization of Stockwell Spectra.- Exact and Numerical Inversion of Pseudo-differential Operators and Applications to Signal Processing.- On the Product of Localization Operators.- Gelfand-Shilov Spaces, Pseudo-differential Operators and Localization Operators.- Continuity and Schatten Properties for Toeplitz Operators on Modulation Spaces.- Microlocalization within Some Classes of Fourier Hyperfunctions.

Pseudodifferential operators on ultra-modulation spaces

Norm estimates for τ-pseudodifferential operators in Wiener amalgam and modulation spaces

In the last twenty years modulation spaces, introduced by H. G. Feichtinger in 1983, have been successfully addressed to the study of signal analysis, PDE's, pseudodifferential operators, quantum mechanics, by hundreds of contributions. In 2011 M. de Gosson showed that the time-frequency representation Short-time Fourier Transform (STFT), which is the tool to define modulation spaces, can be replaced by the Wigner distribution. This idea was further generalized to τ-Wigner representations in [11].In this paper time-frequency representations are viewed as images of symplectic matrices via metaplectic operators. This new perspective highlights that the protagonists of time-frequency analysis are metaplectic operators and symplectic matrices A∈Sp(2d,R). We find conditions on A for which the related symplectic time-frequency representation WA can replace the STFT and give equivalent norms for weighted modulation spaces. In particular, we study the case of covariant matrices A, i.e., their corresponding WA are members of the Cohen class.Finally, we show that symplectic time-frequency representations WA can be efficiently employed in the study of Schrödinger equations. In fact, modulation spaces and WA representations are the frame for a new definition of wave front set, providing a sharp result for propagation of micro-singularities in the case of the quadratic Hamiltonians. This new approach may have further applications in quantum mechanics and PDE's.

Recently, intelligent reflecting surfaces (IRSs) have emerged as potential candidates for overcoming the line-of-sight issue in 5 G/6 G wireless communication. These IRSs can manipulate the direction of reflected beams, enabling efficient beam steering to enhance the performance of wireless communication. Each unit cell (or unit structure) of an IRS commonly consists of electrical elements for phase modulation. However, by employing phase modulation alone, an IRS can steer the reflected electromagnetic waves toward only discrete and specific angles, leaving a wide range of out-of-beam areas. In this work, an IRS that uses both phase modulation and space modulation is presented to improve the beam resolution and continuously cover out-of-beam areas that phase modulation alone cannot address. A positive-intrinsic-negative diode is mounted on a unit cell for phase modulation, and a 4D-printed reconfigured structure is fabricated to demonstrate space modulation. The beam-steering function is achieved by alternating the states of the diodes in the same columns, while the beam resolution is improved by controlling the gaps between the columns. The functions are first theoretically and numerically analyzed and then experimentally verified, demonstrating that additional angles of −46°/+50°, −22°/+14°, and −16°/+12° are achieved with space modulation and −60°/+62°, −30°/+22°, and ±16° are achieved by phase modulation alone. The proposed IRS offers the possibility of functional integration in a variety of indoor applications within the wireless communication field.

In this work, we extend Wigner's original framework to analyze linear operators by examining the relationship between their Wigner and Schwartz kernels. Our approach includes the introduction of (quasi-)algebras of Fourier integral operators (FIOs), which encompass FIOs of type I and II. The symbols of these operators belong to (weighted) modulation spaces, particularly in Sjöstrand's class, known for its favorable properties in time-frequency analysis. One of the significant results of our study is demonstrating the inverse-closedness of these symbol classes.Our analysis includes fundamental examples such as pseudodifferential operators and Fourier integral operators related to Schrödinger-type equations. These examples typically feature classical Hamiltonian flows governed by linear symplectic transformations S∈Sp(d,R). The core idea of our approach is to utilize the Wigner kernel to transform a Fourier integral operator T on Rd into a pseudodifferential operator K on R2d. This transformation involves a symbol σ well-localized around the manifold defined by z=Sw.

On the Hörmander multiplier theorem and modulation spaces

We deduce trace properties for modulation spaces (including certain Wiener-amalgam spaces) of Gelfand–Shilov distributions.We use these results to show that [Formula: see text]dos with amplitudes in suitable modulation spaces, agree with normal type [Formula: see text]dos whose symbols belong to (other) modulation spaces. In particular we extend earlier trace results for modulation spaces, to include quasi-Banach modulation spaces. We also apply our results to extend earlier results on Schatten-von Neumann and nuclear properties for [Formula: see text]dos with amplitudes in modulation spaces.

We consider time-frequency localization operators A_a^{varphi _1,varphi _2} with symbols a in the wide weighted modulation space M^infty _{w}({mathbb {R}^{2d}}), and windows varphi _1, varphi _2 in the Gelfand–Shilov space mathcal {S}^{left( 1right) }(mathbb {R}^d). If the weights under consideration are of ultra-rapid growth, we prove that the eigenfunctions of A_a^{varphi _1,varphi _2} have appropriate subexponential decay in phase space, i.e. that they belong to the Gelfand–Shilov space mathcal {S}^{(gamma )} (mathbb {R^{d}}) , where the parameter gamma ge 1 is related to the growth of the considered weight. An important role is played by tau -pseudodifferential operators Op_{tau } (sigma ). In that direction we show convenient continuity properties of Op_{tau } (sigma )when acting on weighted modulation spaces. Furthermore, we prove subexponential decay and regularity properties of the eigenfunctions of Op_{tau } (sigma ) when the symbol sigma belongs to a modulation space with appropriately chosen weight functions. As an auxiliary result we also prove new convolution relations for (quasi-)Banach weighted modulation spaces.

We prove a Plancherel theorem for a nonlinear Fourier transform in two dimensions arising in the Inverse Scattering method for the defocusing Davey–Stewartson II equation. We then use it to prove global well-posedness and scattering in $$L^2$$ for defocusing DSII. This Plancherel theorem also implies global uniqueness in the inverse boundary value problem of Calderon in dimension 2, for conductivities $$\sigma >0$$ with $$\log \sigma \in \dot{H}^1$$. The proof of the nonlinear Plancherel theorem includes new estimates on classical fractional integrals, as well as a new result on $$L^2$$-boundedness of pseudo-differential operators with non-smooth symbols, valid in all dimensions.

We study the method of layer potentials for manifolds with boundary and cylindrical ends. The fact that the boundary is non-compact prevents us from using the standard characterization of Fredholm or compact pseudo-differential operators between Sobolev spaces, as, for example, in the works of Fabes-Jodeit-Lewis [10] and Kral-Wedland [18]. We first study the layer potentials depending on a parameter on compact manifolds. This then yields the invertibility of the relevant boundary integral operators in the global, non-compact setting. As an application, we prove a well-posedness result for the non-homogeneous Dirichlet problem on manifolds with boundary and cylindrical ends. We also prove the existence of the Dirichlet-to-Neumann map, which we show to be a pseudodifferential operator in the calculus of pseudodifferential operators that are “almost translation invariant at infinity.”

The pseudo-differential Mellin-edge-approach is a tool for studying differential and pseudo-differential operators on manifolds with corners. The Mellin transform, acting on the corner axis ℝ+, is a substitute for the Fourier transform along edge variables in the calculus of wedge pseudo-differential operators. The basic elements of that theory (cf. Schulze [6,8]) are extended to edges like ℝ+ ∋ t with a control of symbols and smoothing operators near the vertext=0. The authors study the weighted Mellin wedge Sobolev spaces, the operator-valued Mellin convention translating Fourier symbols into Mellin ones under preserved smoothness up tot=0, and develop an operator calculus with its characterization on the level of symbols. Throughout the theory, there are involved one-parameter groups of isomorphisms acting on the Banach spaces that are the abstract analogues of the weighted cone Sobolev spaces.

In L2(Rd;Cn), we consider a wide class of matrix elliptic operators A of order 2p (where p2) with periodic rapidly oscillating coefficients (depending on x/). Here 0 is a small parameter. We study the behavior of the operator exponent e-A for 0 and small . We show that the operatore-A converges as 0 in the operator norm in L2(Rd;Cn) to the exponent e-A0 of the effective operator A0. Also we obtain an approximation of the operator exponent e-A in the norm of operators acting from L2(Rd;Cn) to the Sobolev space Hp(Rd; Cn). We derive estimates of errors of these approximations depending on two parameters: and . For a fixed 0 the errors have the exact order O(). We use the results to study the behavior of a solution of the Cauchy problem for the parabolic equation u(x,)= -(A u)(x,)+F(x,) in Rd.

We discuss a sub-algebra ¸ of Pseudo Differential Operators of which is Fredholm-closed and commutative modulo the ideal of compact operators. The algebra ¸ contains in particular all LP - Fourier-multipliers of Mihlin type. is shown to be an algebra with symbol, and its symbol space is determined. As an application we get simple criteria for an operator in ¸to be Fredholm. Furthermore, we regard algebras ¸ on Sobolev spaces , which are isometrically isomorphic to ¸ and have equivalent properties. Finally we conclude Fredholm criteria for mroe general Differential and Pseudo Differential Operators.