Mapping Properties Of Pseudodifferential Operators Research Articles

Characterization of Pseudo-Differential Operators Associated with the Coupled Fractional Fourier Transform

The main aim of this article is to derive certain continuity and boundedness properties of the coupled fractional Fourier transform on Schwartz-like spaces. We extend the domain of the coupled fractional Fourier transform to the space of tempered distributions and then study the mapping properties of pseudo-differential operators associated with the coupled fractional Fourier transform on a Schwartz-like space. We conclude the article by applying some of the results to obtain an analytical solution of a generalized heat equation.

Rough pseudodifferential operators on Hardy spaces for Fourier integral operators

We prove mapping properties of pseudodifferential operators with rough symbols on Hardy spaces for Fourier integral operators. The symbols a(x,η) are elements of C r* S m1,δ classes that have limited regularity in the x variable. We show that the associated pseudodifferential operator a(x, D) maps between Sobolev spaces ℌ s,pFIO (ℝn) and ℌ t,pFIO (ℝn) over the Hardy space for Fourier integral operator ℌ pFIO (ℝn). Our main result implies that for m = 0, δ =l/2 and r > n − 1, a(x, D) acts boundedly on ℌ pFIO (ℝn) for all p ∈ (1, ∞).

Extrapolation to weighted Morrey spaces with variable exponents and applications

AbstractThis paper establishes the mapping properties of pseudo-differential operators and the Fourier integral operators on the weighted Morrey spaces with variable exponents and the weighted Triebel–Lizorkin–Morrey spaces with variable exponents. We obtain these results by extending the extrapolation theory to the weighted Morrey spaces with variable exponents. This extension also gives the mapping properties of Calderón–Zygmund operators on the weighted Hardy–Morrey spaces with variable exponents and the wavelet characterizations of the weighted Hardy–Morrey spaces with variable exponents.

Mapping properties of operator-valued pseudo-differential operators

In this paper, we investigate the mapping properties of pseudo-differential operators with operator-valued symbols. We prove the boundedness of regular symbols on Sobolev spaces H2α(Rd;L2(M)) and Besov spaces Bp,qα(Rd;Lp(M)) for α∈R and 1≤p,q≤∞, as well as the boundedness of forbidden symbols on H2α(Rd;L2(M)) and Bp,qα(Rd;Lp(M)) for α>0 and 1≤p,q≤∞. Thanks to the smooth atomic decomposition of the operator-valued Triebel-Lizorkin spaces F1α,c(Rd,M) obtained in our previous paper, we establish the F1α,c-regularity of regular symbols for every α∈R, and the F1α,c-regularity of forbidden symbols for α>0. As applications, we obtain the same results on the usual and quantum tori.

Operator-Valued Triebel–Lizorkin Spaces

This paper is devoted to the study of operator-valued Triebel–Lizorkin spaces. We develop some Fourier multiplier theorems for square functions as our main tool, and then study the operator-valued Triebel–Lizorkin spaces on $$\mathbb {R}^d$$ . As in the classical case, we connect these spaces with operator-valued local Hardy spaces via Bessel potentials. We show the lifting theorem, and get interpolation results for these spaces. We obtain Littlewood–Paley type, as well as the Lusin type square function characterizations in the general way. Finally, we establish smooth atomic decompositions for the operator-valued Triebel–Lizorkin spaces. These atomic decompositions play a key role in our recent study of mapping properties of pseudo-differential operators with operator-valued symbols.

Characterization of Non-Smooth Pseudodifferential Operators

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.

Spectral Invariance of Non-Smooth Pseudo-Differential Operators

We discuss some spectral invariance results for non-smooth pseudodifferential operators with coefficients in Holder spaces in this paper. In analogy to the proof in the smooth case of Beals and Ueberberg, c.f. (Duke Math J 44(1):45–57, 1977; Manuscripta Math 61(4):459–475, 1988), we use the characterization of non-smooth pseudodifferential operators to get such a result. The main new difficulties are the limited mapping properties of pseudodifferential operators with non-smooth symbols and the fact, that in general the composition of two non-smooth pseudodifferential operators is not a pseudodifferential operator. In order to improve these spectral invariance results for certain subsets of non-smooth pseudodifferential operators with coefficients in Holder spaces, we improve the characterization of non-smooth pseudodifferential operators of A. and P., c.f. (Abels and Pfeuffer, Characterization of non-smooth pseudodifferential operators. arXiv:1512.01127 , 2015).

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.

On multipliers on compact Lie groups

In this note we announce Lp multiplier theorems for invariant and non-invariant operators on compact Lie groups in the spirit of the well-known Hormander-Mikhlin theorem on Rn and its variants on tori Tn. Applications are given to the mapping properties of pseudo-differential operators on Lp-spaces and to a-priori estimates for non-hypoelliptic operators.

Sobolev spaces on Lie manifolds and regularity for polyhedral domains

We study some basic analytic questions related to differential operators on Lie manifolds, which are manifolds whose large scale geometry can be described by a a Lie algebra of vector fields on a compactification. We extend to Lie manifolds several classical results on Sobolev spaces, elliptic regularity, and mapping properties of pseudodifferential operators. A tubular neighborhood theorem for Lie submanifolds allows us also to extend to regular open subsets of Lie manifolds the classical results on traces of functions in suitable Sobolev spaces. Our main application is a regularity result on polyhedral domains $\PP \subset \RR^3$ using the weighted Sobolev spaces $\Kond{m}a(\PP)$. In particular, we show that there is no loss of $\Kond{m}a$--regularity for solutions of strongly elliptic systems with smooth coefficients. For the proof, we identify $\Kond{m}a(\PP)$ with the Sobolev spaces on $\PP$ associated to the metric $r_{\PP}^{-2} g_E$, where $g_E$ is the Euclidean metric and $r_{\PP}(x)$ is a smoothing of the Euclidean distance from $x$ to the set of singular points of $\PP$. A suitable compactification of the interior of $\PP$ then becomes a regular open subset of a Lie manifold. We also obtain the well-posedness of a non-standard boundary value problem on a smooth, bounded domain with boundary $\maO \subset \RR^n$ using weighted Sobolev spaces, where the weight is the distance to the boundary.

Mapping Properties of Pseudodifferential Operators on Manifolds with Bounded Geometry

We investigate classes of uniform pseudodifferential operators on Riemannian manifolds with bounded geometry. We prove that operators belonging to the classes are bounded in function spaces of Hardy–Sobolev–Besov type defined on the manifold. The classes contain the powers of the Laplace–Beltrami operators so we get the fractional lift property for the function spaces.