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We present a basis of p-adic wavelets for Sobolev-type spaces consisting of eigenvectors of certain pseudodifferential operators. Our result extends a well-known result due to S. Kozyrev.

The Cauchy problem for a class of hyperbolic operators with triple characteristics is analyzed. Some a priori estimates in Sobolev spaces with negative indexes are proved. Subsequently, an existence result for the Cauchy problem is obtained.

Considering compact pseudodifferential operators with symbols whose smoothness in $ x $ vanishes on a prescribed set, we obtain some validity conditions for the Weyl spectral asymptotics of singular values. These results are applied to the symbols whose decay order as $ |\xi|\to\infty $ is a nonsmooth function of $ x $ .

In this paper we prove sufficient conditions for the Fredholm property of a non-smooth pseudodifferential operator $P$ which symbol is in a Holder space with respect to the spatial variable. As a main ingredient for the proof we use a suitable symbol-smoothing.

We prove that the Moyal multiplier algebras of the generalized Gelfand–Shilov spaces of type S contain Palamodov spaces of type $${\mathscr {E}}$$ and the inclusion maps are continuous. We also give a direct proof that the Palamodov spaces are algebraically and topologically isomorphic to the strong duals of the spaces of convolutors for the corresponding spaces of type S. The obtained results provide a general and efficient way to describe the algebraic and continuity properties of pseudodifferential operators with symbols having an exponential or super-exponential growth at infinity.

In this paper, a new class of degenerate pseudo-differential operators is investigated, with a variable symbol depending on the complex parameter. Pseudodifferential operators are constructed by a special integral transform. Theorems on the composition and boundedness of these operators in special weighted spaces are proved. The behavior of these operators on hyperplanes of degeneration is investigated. The theorems on the commutation of these operators with differentiation operators are established. A adjoint operator is constructed and an analogue of Goring inequality for degenerate pseudodifferential operators is proved.

Compact pseudodifferential operators whose symbol fails to be smooth with respect to x on a given set are considered. Conditions under which Weyl’s law of spectral asymptotics remains valid for such operators are obtained. The results are applied to operators with symbols such that their order of decay as |ξ| → ∞ is a nonsmooth function of x.

We develop a set of Lp estimates for functions u that are joint quasi-modes (approximate eigenfunctions) of r semiclassical pseudodifferential operators p1(x, hD),...,pr(x, hD). This work extends Sarnak [10] and Marshall’s [8] work on symmetric space to cover a more general class of manifolds/operators.

Boundedness properties for bilinear pseudodifferential operators that go beyond the Calderon–Zygmund theory and whose symbols belong to the bilinear Hormander classes of critical order are established. Such operators are shown to satisfy Leibniz-type rules in the setting of Besov and local Hardy spaces.

Using the matrix representation of Fourier integral operators with respect to a Gabor frame, we study their compactness on weighted modulation spaces. As a consequence, we recover and improve some compactness results for pseudodifferential operators.

In this article a class of Markov processes on the ring of finite adeles of the rational numbers are introduced. A class of non-Archimedean metrics on $\mathbb{A}_{f}$ are chosen in order to describe this ring as a general polyadic ring and to introduce a family of pseudodifferential operators and parabolic-type equations on ${L^2}(\mathbb{A}_{f})$ . The fundamental solutions of these parabolic equations determine transition functions of time and space homogeneous Markov processes on $\mathbb{A}_{f}$ which are invariant under multiplication by units. Considering the infinite place ℝ, we extend these results to the complete ring of adeles.

All of the integral operators with nonintegrable kernels are given in terms of computable Hadamard's partie finie, i.e. finite part integrals [168], which can be applied to problems in applications (Guiggiani [163], Schwab et al [380]).

This chapter concerns the relation between the boundary integral operators and classical pseudodifferential operators. A large class of boundary integral operators including those presented in the previous chapters belong to the special class of classical pseudodifferential operators on compact manifolds. We are particularly interested in strongly elliptic systems of pseudodifferential operators providing Garding's inequality, see Theorem 9.1.4.

The class of pseudodifferential operators is essentially the smallest algebra of operators which contains all differential operators, all fundamental solutions of elliptic differential operators and all integral operators with pseudohomogeneous kernel expansions. The linear pseudodifferential operators can be characterized by generalized Fourier multipliers, known as symbols. The development of the theory of pseudodifferential operators has made it possible to provide a unified treatment of differential and integral operators.

We prove that the theorem of Egorov, on the canonical transformation of symbols of pseudodifferential operators conjugated by Fourier integral operators, can be sharpened. The main result is that the statement of Egorov's theorem remains true if, instead of just considering the principal symbols in $S^m/S^{m-1}$ for the pseudodifferential operators, one uses refined principal symbols in $S^m/S^{m-2}$, which for classical operators correspond simply to the principal plus the subprincipal symbol, and can generally be regarded as the first two terms of its Weyl symbol expansion: we call it the principal Weyl symbol of the pseudodifferential operator. Particular unitary Fourier integral operators, associated to the graph of the canonical transformation, have to be used in the conjugation for the higher accuracy to hold, leading to microlocal representations by oscillatory integrals with specific symbols that are given explicitly in terms of the generating function that locally describes the graph of the transformation. The motivation for the result is based on the optimal symplectic invariance properties of the Weyl correspondence in ${\mathbb R}^n$ and its symmetry for real symbols.

The pseudodifferential operators provide a unified treatment of differential and integral operators. They are based on the intensive use of the Fourier transformation \(\mathcal{F}\) (3.1.12) and its inverse \(\mathcal{F}\) –1 \(=\mathcal{F}\)*(3.1.14). The linear pseudodifferential operators can be characterized by generalized Fourier multipliers, called symbols.

We define an important microlocal tool – H-distributions, a generalization of H-measures, on Hormander B p , k spaces. We also consider applications to linear partial differential equations with non-smooth coefficients and to some semilinear equations.

The paper concerns the study of the Cauchy–Dirichlet problem for a class of hyperbolic second-order operators with double characteristics in presence of transition in a domain of $${\mathbb {R}}^3$$ R 3 . Firstly, we establish some a priori local and global estimates. Then, we obtain some existence results.

We introduce a new class of ultradifferentiable pseudodifferential operators on the torus whose calculus allows us to show that global hypoellipticity, in ultradifferentiable classes, with a finite loss of derivatives of a system of pseudodifferential operators, is stable under perturbations by lower order pseudodifferential operators whose order depends on the loss of derivatives. The key point in our study is our definition of loss of derivatives. We also give an easy proof of the fact that if a system of pseudodifferential operators is globally $${\mathcal {M}}$$ -hypoelliptic then its transpose is globally solvable in $$D'_{\mathcal {M}}\left( {\mathbb {T}}^N\right) $$ . Finally we present an application of our results.

In this paper we establish uniqueness theorems for the noncommutative residue and the canonical trace on pseudodifferential operators on noncommutative tori of arbitrary dimension. The former is the unique trace up to constant multiple on integer order pseudodifferential operators.The latter is the unique trace up to constant multiple on non-integer order pseudodifferential operators. This improves previous uniqueness results by Fathizadeh-Khalkhali, Fathizadeh-Wong, and L\'evy-Neira-Paycha.

© 2019, Springer-Verlag GmbH Austria, part of Springer Nature. Mixed-norm Lebesgue spaces found their place in the study of some questions in the theory of partial differential equations, as it can be seen from recent interest in continuity of certain classes of pseudodifferential operators on these spaces. We present a general framework for dealing with continuity of linear operators on these spaces. This allows us to prove the boundedness of a large class of pseudodifferential operators and also the boundedness of integral operators on mixed-norm Lebesgue spaces. In some cases, the generalisations to mixed-norm Sobolev spaces are obtained as well, together with applications to some interpolation and compactness results.

In this paper we show the invariance of the Fredholm index of non-smooth pseudodifferential operators with coefficients in H\"older spaces. By means of this invariance we improve previous spectral invariance results for non-smooth pseudodifferential operators $P$ with coefficients in H\"older spaces. For this purpose we approximate $P$ with smooth pseudodifferential operators and use a spectral invariance result of smooth pseudodifferential operators. Then we get the spectral invariance result in analogy to a proof of the spectral invariance result for non-smooth differential operators by Rabier., Comment: 37 pages

We consider pseudodifferential operators of variable order acting on Besov spaces of variable smoothness. We prove the boundedness and compactness of such operators and study the Fredholm property of pseudodifferential operators of variable order with symbols slowly oscillating at infinity on weighted Besov spaces with variable smoothness.

We consider a simplest elliptic pseudodifferential equation in a multi-dimensional cone (multi-dimensional angle) and describe all possible structures of its solutions related to the wave factorization of the elliptic symbol. Depending on the index of wave factorization, we consider various statements of well-posed boundary-value problems. The existence of solutions is studied in Sobolev–Slobodetskii spaces.

Let W be a three-dimensional wedge, and K be the double layer potential operator associated to W and the Laplacian. We show that 1 2 ± K are isomorphisms between suitable weighted Sobolev spaces, which implies a solvability result in weighted Sobolev spaces for the Dirichlet problem on W . Furthermore, we show that the double layer potential operator K is an element in C ⁎ ( G ) ⊗ M 2 ( C ) , where G is the action (transformation) groupoid M ⋊ G , with G = { ( 1 0 a b ) : a ∈ R , b ∈ R + } , which is a Lie group, and M is a kind of compactification of G. This result can be used to prove the Fredholmness of 1 2 + K Ω , where Ω is “a domain with edge singularities” and K Ω the double layer potential operator associated to the Laplacian and Ω.

We prove a criterion for a “magnetic” Weyl operator to be trace-class by extending a method developed by Cordes, Kato and Arsu. Using the Calderon–Vaillancourt type Theorem for magnetic Weyl operat...

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.

We study the Hermite operator $H=-\Delta+|x|^2$ in $\mathbb{R}^d$ and its fractional powers $H^\beta$, $\beta>0$ 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 $e^{-tH^\beta}$ when acting on $M^{p,q}$, for every $0< p,q\leq \infty$, exhibiting the optimal global-in-time decay as well as phase-space smoothing. As an application, we establish global well-posedness for the nonlinear heat equation for $H^{\beta}$ with power-type nonlinearity (focusing or defocusing), with small initial data in modulation spaces or in Wiener amalgam spaces. We show that such a global solution exhibits the same optimal decay $e^{-c t}$ as the solution of the corresponding linear equation, where $c=d^\beta$ is the bottom of the spectrum of $H^\beta$. This is in sharp contrast to what happens for the nonlinear focusing heat equation without potential, where blow-up in finite time always occurs for (even small) constant initial data - hence in $M^{\infty,1}$., Comment: 18 pages

© 2018, Institute of Mathematics. Polish Academy of Sciences. All rights reserved. We prove that the complex powers of a class of infinite order hypoelliptic pseudodifferential operators can always be represented as hypoelliptic pseudodifferential operators modulo ultra-smoothing operators. We apply this result to the study of semigroups generated by square roots of non-negative hypoelliptic infinite order operators. For this purpose, we derive precise estimates of the corresponding heat kernel.

Let $T^n$ denote the n-dimensional torus. The class of the bounded operators on $L^2(T^n)$ with analytic orbit under the action of $T^n$ by conjugation with the translation operators is shown to coincide with the class of the zero-order pseudodifferential operators on $T^n$ whose discrete symbol $(a_j)_{j\in Z^n}$ is uniformly analytic, in the sense that there exists $C>1$ such that the derivatives of $a_j$ satisfy $|\partial^\alpha a_j(x)|\leq C^{1+|\alpha|}\alpha!$ for all $x\in T^n$, all $j\in Z^n$ and all $\alpha\in N^n$. This implies that this class of pseudodifferential operators is a spectrally invariant *-subalgebra of the algebra of all bounded operators on $L^2(T^n)$.

Author(s): Krupchyk, Katya; Uhlmann, Gunther | Abstract: We obtain sharp uniform bounds on the low lying eigenfunctions for a class of semiclassical pseudodifferential operators with double characteristics and complex valued symbols, under the assumption that the quadratic approximations along the double characteristics are elliptic.

We present a unifying approach to establish mapping properties for bilinear pseudodifferential operators with homogeneous symbols in the settings of function spaces that admit a discrete transform and molecular decompositions in the sense of Frazier and Jawerth. As an application, we obtain related Kato-Ponce inequalities.

In this paper, we consider pseudodifferential operators on the torus with operator-valued symbols and prove continuity properties on vector-valued toroidal Besov spaces, without assumptions on the underlying Banach spaces. The symbols are of limited smoothness with respect to $x$ and satisfy a finite number of estimates on the discrete derivatives. The proof of the main result is based on a description of the operator as a convolution operator with a kernel representation which is related to the dyadic decomposition appearing in the definition of the Besov space.

Jacobi-like forms for a discrete subgroup $$\Gamma $$ of $$SL(2, \mathbb R)$$ are formal power series which generalize Jacobi forms, and they are in one-to-one correspondence with automorphic pseudodifferential operators for $$\Gamma $$ . The well-known Cohen–Kuznetsov lifting of a modular form f provides a Jacobi-like form and therefore an automorphic pseudodifferential operator associated to f. Given a pair $$(\lambda , \mu )$$ of integers, automorphic pseudodifferential operators can be extended to those of mixed weight. We show that each coefficient of an automorphic pseudodifferential operator of mixed weight is a quasimodular form and prove the existence of a lifting of Cohen–Kuznetsov type for each quasimodular form.

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., 11 pages; added reference [1]

This is a survey on recent progress concerning maximal regularity of non-autonomous equations governed by time-dependent forms on a Hilbert space. It also contains two new results showing the limits of the theory., 11 pages, 1 figure

Smooth pseudodifferential operators on $\mathbb{R}^n$ can be characterized by their mapping properties between $L^p-$Sobolev spaces due to Beals and Ueberberg. In applications such a characterization would also be useful in the non-smooth case, for example to show the regularity of solutions of a partial differential equation. Therefore, we will show that every linear operator $P$, which satisfies some specific continuity assumptions, is a non-smooth pseudodifferential operator of the symbol-class $C^{\tau} S^m_{1,0}(\mathbb{R}^n \times \mathbb{R}^n)$. The main new difficulties are the limited mapping properties of pseudodifferential operators with non-smooth symbols., Comment: 42 pages

We compare three different ways of checking the Jacobi identity for weakly nonlocal Poisson brackets using the theory of distributions, of pseudodifferential operators and of Poisson vertex algebras, respectively. We show that the three approaches lead to similar computations and same results., 47 pages

We have by now defined both the weighted and unweighted versions of the modulation spaces and indicated in Chaps. 4 and 5 their place in the time-frequency analysis of pseudodifferential operators and their good properties regarding certain nonlinear operations stemming from PDEs. Hidden in the seemingly simple definition of modulation spaces lies the practical problem of computing the STFT of a function or distribution and then further calculating its appropriate mixed Lebesgue norm. Therefore, a natural question arising in this context is whether any embeddings between such modulation spaces and other classical function spaces of analysis exist, something that we have alluded to already in Sects. 2.3 and 2.4. Let us recall also immediately that Sjostrand’s class, \(\mathcal {M}^{\infty , 1}\), contains the Hormander class \(S^0_{0,0}\), a fact that played a relevant role throughout Chap. 4; see again the comments after Corollary 4.22 and Sjostrand (Math Res Lett 1(2):185–192, 1994).

We study a class of quadratic time-frequency representations that, roughly speaking, are obtained by linear perturbations of the Wigner transform. They satisfy Moyal’s formula by default and share many other properties with the Wigner transform, but in general they do not belong to Cohen’s class. We provide a characterization of the intersection of the two classes. To any such time-frequency representation, we associate a pseudodifferential calculus. We investigate the related quantization procedure, study the properties of the pseudodifferential operators, and compare the formalism with that of the Weyl calculus.

We consider the initial value problem for the plate equation with (t, x) −depending complex valued lower order terms. Under suitable decay conditions as |x|→∞ on the imaginary part of the subprincipal term we prove energy estimates in weighted Sobolev spaces. This provides also well posedness of the Cauchy problem in the Schwartz space \(\mathcal {S}(\mathbb R^n)\) and in \(\mathcal {S}^\prime (\mathbb R^n)\).

We study the continuity in weighted Fourier–Lebesgue spaces for a class of pseudodifferential operators, whose symbol has finite Fourier–Lebesgue regularity with respect to x and satisfies a quasi-homogeneous decay of derivatives with respect to the $$\xi $$ variable. Applications to Fourier–Lebesgue microlocal regularity of linear and nonlinear partial differential equations are given.

We study two sufficient conditions for the boundedness of a class of pseudodifferential operators T with symbols in the Hölmander class Sρ,δmℝn on weighted Hardy spaces Hω1ℝn, where ω belongs to Muckenhoupt class Ap. The first one is an estimate from Hω1ℝn into Lω1ℝn. We get a better range of admissible p and m. The second one is a weighted version bounded for the operators T on Hω1ℝn, and it is an addition to the literature.

In this paper, we analyze the Friedrichs part of an operator with polynomially bounded symbol. Namely, we derive a precise expression of its asymptotic expansion. In the case of symbols satisfying Gevrey estimates, we also estimate precisely the regularity of the terms in the asymptotic expansion. These results allow new and refined applications of the sharp G&aring, rding inequality in the study of the Cauchy problem for p-evolution equations.

The mixed Cauchy–Neumann and Cauchy–Robin problems for a class of hyperbolic operators with double characteristics in presence of transition is investigated. Some a priori estimates in Sobolev spaces with negative indexes are proved. Subsequently, existence and uniqueness results for the mixed problems are obtained.