Boundedness Of Pseudo-differential Operators Research Articles

$$L^p$$-boundedness of pseudo-differential operators on rank one Riemannian symmetric spaces of noncompact type

$$L^p$$-boundedness of pseudo-differential operators on rank one Riemannian symmetric spaces of noncompact type

$$H^1$$–$$L^1$$ Boundedness of Pseudo-differential Operators with Forbidden Amplitudes

$$H^1$$–$$L^1$$ Boundedness of Pseudo-differential Operators with Forbidden Amplitudes

Large time asymptotics for the fractional modified Korteweg-de Vries equation of order alpha in left[ 4,5right)

We study the large time asymptotics of solutions to the Cauchy problem for the fractional modified Korteweg-de Vries equation ∂tw+1α∂xα-1∂xw=∂xw3,t>0,x∈R,w0,x=w0x,x∈R,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left\\{ \\begin{array}{l} \\partial _{t}w+\\frac{1}{\\alpha }\\left| \\partial _{x}\\right| ^{\\alpha -1}\\partial _{x}w=\\partial _{x}\\left( w^{3}\\right) ,\ ext { }t>0,\\, x\\in {\\mathbb {R}}\ extbf{,}\\\\ w\\left( 0,x\\right) =w_{0}\\left( x\\right) ,\\,x\\in {\\mathbb {R}} \ extbf{,} \\end{array} \\right. \\end{aligned}$$\\end{document}where alpha in left[ 4,5right) , and left| partial _{x}right| ^{alpha }={mathcal {F}}^{-1}left| xi right| ^{alpha }{mathcal {F}} is the fractional derivative. The case of alpha =3 corresponds to the classical modified KdV equation. In the case of alpha =2 it is the modified Benjamin–Ono equation. Our aim is to find the large time asymptotic formulas for the solutions of the Cauchy problem for the fractional modified KdV equation. We develop the method based on the factorization techniques which was started in our previous papers. Also we apply the known results on the {textbf{L}}^{2}—boundedness of pseudodifferential operators.

$L^p$-boundedness of pseudo-differential operators on homogeneous trees

The aim of this article is to study the $L^p$-boundedness of pseudo-differential operators on a homogeneous tree $ \mathfrak X $. For $p\in (1,2)$, we establish a connection between the $L^{p}$-boundedness of the pseudo-differential operators on $ \mathfr

Boundedness and nuclearity of pseudo-differential operators on homogeneous trees

Let \(\mathfrak X\) be the homogeneous tree of degree \(q+1~(q\ge 2)\). In this article, we present symbolic criteria for boundedness of pseudo differential operators on homogeneous tree \(\mathfrak X\). A sufficient condition for weak type (p, q) boundedness is also given. We present necessary and sufficient conditions on the symbols \(\sigma \) such that the corresponding pseudo-differential operators \(T_\sigma \) from \(L^{p_1} (\mathfrak X)\) into \(L^{p_2} (\mathfrak X)\) to be nuclear for \(1\le p_1, p_2<\infty \). Explicit formulas for the symbol of the adjoint and product of the nuclear pseudo differential operators on \(\mathfrak X\) are given.

L2-Lp estimates and Hilbert–Schmidt pseudo differential operators on the Heisenberg motion group

In this paper, we study some operator theoretical properties of pseudo-differential operators with operator-valued symbols on the Heisenberg motion group. Specifically, we investigate – boundedness of pseudo-differential operators on the Heisenberg motion group for the range . We also provide a necessary and sufficient condition on the operator-valued symbols in terms of λ-Weyl transforms such that the corresponding pseudo-differential operators on the Heisenberg motion group are in the class of Hilbert–Schmidt operators. As a consequence, we obtain a characterization of the trace class pseudo-differential operators on the Heisenberg motion group and provide a trace formula for these trace class operators.

A nonlinear Plancherel theorem with applications to global well-posedness for the defocusing Davey–Stewartson equation and to the inverse boundary value problem of Calderón

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.

Pseudodifferential operators with smooth symbols and their commutators on weighted Morrey spaces

In this paper, a local good $$\lambda $$ inequality for a class of new weight functions which include Muckenhoupt weight functions is established. Applying the local good $$\lambda $$ inequality and new sharp maximal functions, the authors give a characterization of weighted Morrey spaces. Furthermore, it is proved that the boundedness of pseudodifferential operators with smooth symbols and their commutators on weighted Morrey spaces.

On the boundedness of pseudo-differential operators on Triebel–Lizorkin and Besov spaces

In this work we show endpoint boundedness properties of pseudo-differential operators of type (ρ,ρ), 0<ρ<1, on Triebel–Lizorkin and Besov spaces. Our results are sharp and they also cover operators defined by compound symbols.

(L p –L q )-boundedness of pseudodifferential operators on the n-dimensional torus

(L p –L q )-boundedness of pseudodifferential operators on the n-dimensional torus

Boundedness of pseudodifferential operators with symbols in Wiener amalgam spaces on modulation spaces

This paper provides sufficient conditions for the boundedness of Weyl operators on modulation spaces. The Weyl symbols belong to Wiener amalgam spaces, or generalized modulation spaces, as recently renamed by their inventor Hans Feichtinger. This is the first result which relates symbols in Wiener amalgam spaces to operators acting on classical modulation spaces.

Boundedness of Pseudodifferential Operators on Realized Homogeneous Besov Spaces

Using the notion of realizations, the realized homogeneous Besov spaces $\dot{\widetilde{B}}{^{s}_{p,q}}(\mathbb{R}^n)$ are subsets of tempered distributions. Then we will study the boundedness of some pseudodifferential operators on $\dot{\widetilde{B}}{^{s}_{p,q}}(\mathbb{R}^n)$, in the cases either $s < n/p$ or $s = n/p$ and $q = 1$.

A study on pseudo-differential operators on $$\mathbb {S}^{1} $$ S 1 and $$\mathbb {Z}$$ Z

In this paper we offer a new sufficient condition for boundedness of pseudo-differential operators on \(L^{p}(\mathbb {S}^{1})\) and \(L^{P}(\mathbb {Z}),\ p\ge 1. \) Also a new sufficient condition for \(L^{2}\)-compactness of these operators is obtained.

Shearlets and pseudo-differential operators

Shearlets on the cone are a multi-scale and multi-directional discrete system that have near-optimal representation of the so-called cartoon-like functions. They form Parseval frames, have better geometrical sensitivity than traditional wavelets and an implementable framework. Recently, it has been proved that some smoothness spaces can be associated to discrete systems of shearlets. Moreover, there exist embeddings between the classical isotropic dyadic spaces and the shearlet generated spaces. We prove boundedness of pseudo-differential operators (PDO’s) with non regular symbols on the shear anisotropic inhomogeneous Besov spaces and on the shear anisotropic inhomogeneous Triebel–Lizorkin spaces (which are up to now the only Triebel–Lizorkin-type spaces generated by either shearlets or curvelets and more generally by any parabolic molecule, as far as we know). The type of PDO’s that we study includes the classical Hormander definition with x-dependent parameter $$\delta $$ for a range limited by the anisotropy associated to the class. One of the advantages is that the anisotropy of the shearlet spaces is not adapted to that of the PDO.

On boundedness of pseudo-differential operators in Hölder-Zygmund spaces with variable order of smoothness

We consider the Holder-Zygmund spaces of functions with variable index of smoothness defined on finite-dimensional real space. The index of smoothness depends on a point in this space, and can be negative. We prove the boundedness theorem for pseudo-differential operators with “exotic” symbols from the Hormander classes in these spaces.

Pseudo-differential operators of infinite order on $$W^{\Omega }_{M}(\mathbb C^{n})$$ W M Ω ( C n ) -space involving fractional Fourier transform

The mapping properties of pseudo-differential operators associated with symbol \(\theta (z,\xi ), z= x+iy\) and \(\xi =u+it\) on \(W^{\Omega }_{M}(\mathbb C^{n})\)-space are investigated. Motivated from Wong (An introduction to pseudo-differential operators. World Scientific, Singapore, 2014), \(L^{p}(\mathbb R^{n})\)-boundedness of pseudo-differential operators on \(W^{\Omega }_{M}(\mathbb C^{n})\)-space is studied by using the fractional Fourier transform.

Schrödinger equations with rough Hamiltonians

We consider a class of linear Schrödinger equations in $\mathbb{R}^d$ with rough Hamiltonian, namely with certain derivatives in the Sjöostrand class $M^{\infty,1}$. We prove that the corresponding propagator is bounded on modulation spaces. The present results improve several contributions recently appeared in the literature and can be regarded as the evolution counterpart of the fundamental result of Sjöstrand about the boundedness of pseudodifferential operators with symbols in that class.&nbsp; &nbspFinally we consider nonlinear perturbations of real-analytic type and we prove local wellposedness of the corresponding initial value problem in certain modulation spaces.

Boundedness of pseudodifferential operators in Hölder-Zygmund spaces of variable order

We introduce the Holder-Zygmund spaces with a variable smoothness exponent, define equivalent norms, and prove a theorem on boundedness of pseudodifferential operators with Hormander symbols in these spaces.

Pseudodifferential operators on localized Besov spaces

We study the boundedness of pseudodifferential operators σ(x,D) of order m and symbols σ(x,ξ) which satisfy a condition of Dini-type, on localized Besov spaces \((B_{{p},{q}}^{s}({\mathbb{R}}^{n}))_{\ell^{r}}\). In the case s>n/p and p≤q, we deduce the boundedness of such an operator σ(x,D) on pointwise multipliers Besov space \(M(B_{{p},{q}}^{s}({\mathbb{R}}^{n}))\).

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

We introduce new classes of modulation spaces over phase space. By means of the Kohn-Nirenberg correspondence, these spaces induce norms on pseudo-differential operators that bound their operator norms on Lp–spaces, Sobolev spaces, and modulation spaces.