Boundedness Of Pseudo-differential Operators Research Articles

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.

On a Class of Bilinear Pseudodifferential Operators

We provide a direct proof for the boundedness of pseudodifferential operators with symbols in the bilinear Hörmander classBS1,δ0,0≤δ<1. The proof uses a reduction to bilinear elementary symbols and Littlewood-Paley theory.

Bilinear estimates in dyadic BMO and the Navier-Stokes equations

We establish bilinear estimates in dyadic BMO as an extension of Kozono and Taniuchi's result on the usual BMO. To establish the bilinear estimates we use sharp maximal functions, while they used the boundedness of pseudo-differential operators by Coifman and Meyer. By this extension we prove that the dyadic BMO norm of the velocity controls the blow-up phenomena of smooth solutions to the Navier-Stokes equations. Moreover, we give an odd function with type II singularity which clarifies the difference between BMO and dyadic BMO.

Decompositions of Besov–Hausdorff and Triebel–Lizorkin–Hausdorff spaces and their applications

Let p ∈ ( 1 , ∞ ) , q ∈ [ 1 , ∞ ) , s ∈ R and τ ∈ [ 0 , 1 − 1 max { p , q } ] . In this paper, the authors establish the φ-transform characterizations of Besov–Hausdorff spaces B H ˙ p , q s , τ ( R n ) and Triebel–Lizorkin–Hausdorff spaces F H ˙ p , q s , τ ( R n ) ( q > 1 ); as applications, the authors then establish their embedding properties (which on B H ˙ p , q s , τ ( R n ) is also sharp), smooth atomic and molecular decomposition characterizations for suitable τ. Moreover, using their atomic and molecular decomposition characterizations, the authors investigate the trace properties and the boundedness of pseudo-differential operators with homogeneous symbols in B H ˙ p , q s , τ ( R n ) and F H ˙ p , q s , τ ( R n ) ( q > 1 ), which generalize the corresponding classical results on homogeneous Besov and Triebel–Lizorkin spaces when p ∈ ( 1 , ∞ ) and q ∈ [ 1 , ∞ ) by taking τ = 0 .

Pseudo-differential operators on Sobolev and Lipschitz spaces

In this paper, by discovering a new fact that the Lebesgue boundedness of a class of pseudo-differential operators implies the Sobolev boundedness of another related class of pseudo-differential operators, the authors establish the boundedness of pseudo-differential operators with symbols in Sρ,δm, on Sobolev spaces, where m ∈ ℝ, ρ ≤ 1 and δ ≤ 1. As its applications, the boundedness of commutators generated by pseudo-differential operators on Sobolev and Bessel potential spaces is deduced. Moreover, the boundedness of pseudo-differential operators on Lipschitz spaces is also obtained.

A note on Besov-Morrey spaces and Triebel-Lizorkin-Morrey spaces

In the present paper, we obtain three independent results on the Besov-Morrey spaces and the Triebel-Lizorkin-Morrey spaces. That is, we are going to obtain the characterization of local means, the boundedness of pseudo-differential operators and the characterization of the Hardy-Morrey spaces. By using the maximal estimate and the molecular decomposition, we shall integrate and extend the known results on these spaces.

On the [formula omitted]-boundedness of pseudo-differential operators and their commutators with symbols in α-modulation spaces

The results of Sjöstrand [J. Sjöstrand, An algebra of pseudodifferential operators, Math. Res. Lett. 1 (1994) 185–192] and Sugimoto [M. Sugimoto, L p -boundedness of pseudo-differential operators satisfying Besov estimates, I, J. Math. Soc. Japan 40 (1988) 105–122] on a mapping property of pseudo-differential operators are two different kinds of extensions of the pioneering work by Calderón and Vaillancourt [A.P. Calderón, R. Vaillancourt, On the boundedness of pseudo-differential operators, J. Math. Soc. Japan 23 (1971) 374–378]. The objective of this paper is to show that these two results, which appeared to be independent ones, can be proved based on the same principle. For the purpose, we use the α-modulation spaces, a parameterized family of function spaces, which include Besov spaces and modulation spaces as special cases. As an application, we also discuss the L 2 -boundedness of the commutator [ σ ( X , D ) , a ] , where a ( x ) is a Lipschitz function and σ belongs to an α-modulation space.

Boundedness of pseudodifferential operators of a C*-algebra-valued symbol

Let us consider the set SA(Rn) of rapidly decreasing functions G: Rn → A, where A is a separable C*-algebra. We prove a version of the Calderón–Vaillancourt theorem for pseudodifferential operators acting on SA(Rn) whose symbol is A-valued. Given a skew-symmetric matrix, J, we prove that a pseudodifferential operator that commutes with G(x + JD), G ∈ SA(Rn), is of the form F(x − JD), for F a C∞-function with bounded derivatives of all orders.

Pseudodifferential operators on ultra-modulation spaces

The boundedness of pseudodifferential operators on modulation spaces defined by the means of almost exponential weights is studied. The results are applied to symbol class with almost exponential bounds including polynomial and ultra-polynomial symbols. The Weyl correspondence is used and it is noted that the results can be transferred to the operators with appropriate anti-Wick symbols. It is proved that a class of elliptic pseudodifferential operators can be almost diagonalized by the elements of Wilson bases, and estimates for their eigenvalues are given. Furthermore, it is shown that the same can be done by using Gabor frames.

Boundedness of pseudodifferential operators on modulation spaces

We study classes of pseudodifferential operators which are bounded on large collections of modulation spaces. The conditions on the operators are stated in terms of the Lp,q estimates for the continuous Gabor transforms of their symbols. In particular, we show how these classes are related to the class of operators of Gröchenig and Heil, which is bounded on all modulation spaces.

The boundedness of certain operators on Holder and Sobolev spaces

The main topic in this note is to discuss the boundedness of pseudo-differential operators and para-product operators on the Holder and Sobolev spaces, respectively. It is the preparation of the theory of Gibbs-Butzer defferential operators and differential equations.

Pseudodifferential operators on manifolds with edges

We define pseudodifferential operators on manifolds with singularities of smooth edge type and construct a calculus of such operators. We prove that a pseudodifferential operator is invariant with respect to a certain natural class of diffeomorphisms of the manifold. We introduce a scale of function spaces (weighted analogs of the Sobolev classes) and establish theorems on boundedness of pseudodifferential operators in this scale. Bibliography: 8 titles.

L p -Boundedness of Pseudo-Differential Operators of Class S 0,0

L p -Boundedness of Pseudo-Differential Operators of Class S 0,0

The Boundedness of Pseudodifferential Operators on Modulation Spaces

AbstractWe give a characterization of d‐dimensional modulation spaces with moderate weights by means of the d‐dimensional Wilson basis. As an application we prove that pseudodifferential operators with generalized Weyl symbols are bounded on these modulation spaces.

On the $L^2$-boundedness of pseudo-differential operators

On the $L^2$-boundedness of pseudo-differential operators

The L 2 -Boundedness of Pseudodifferential Operators

The L 2 -Boundedness of Pseudodifferential Operators

The $L\sp 2$-boundedness of pseudodifferential operators

We give a new proof of the Calderon-Vaillancourt theorem. We also obtain the ${L^2}$-continuity of $a(x,D)$ if its symbol $a(x,\xi )$ satisfies some suitable conditions.

On the boundedness of pseudodifferential operators

On the boundedness of pseudodifferential operators

The Lp-boundedness of pseudo-differential operators with non-regular symbols

(1977). The Lp-boundedness of pseudo-differential operators with non-regular symbols. Communications in Partial Differential Equations: Vol. 2, No. 10, pp. 1045-1061.