THE QUASI-GEOSTROPHIC EQUATION AND ITS TWO REGULARIZATIONS

Abstract

A symbolic calculus for the transposes of a class of bilinear pseudodifferential operators is developed. The calculus is used to obtain boundedness results on products of Lebesgue spaces. A larger class of pseudodifferential operators that does not admit a calculus is also considered. Such a class is the bilinear analog of the so-called exotic class of linear pseudodifferential operators and fail to produce bounded operators on products of Lebesgue spaces. Nevertheless, the operators are shown to be bounded on products of Sobolev spaces with positive smoothness, generalizing the Leibniz rule estimates for products of functions.     

Boundedness of Multilinear Pseudo-differential Operators on Modulation Spaces

Boundedness results for multilinear pseudodifferential operators on products of modulation spaces are derived based on ordered integrability conditions on the short-time Fourier transform of the operators’ symbols. The flexibility and strength of the introduced methods is demonstrated by their application to the bilinear and trilinear Hilbert transform.     

Weighted Norm Inequalities for Paraproducts and Bilinear Pseudodifferential Operators with Mild Regularity

We establish boundedness properties on products of weighted Lebesgue, Hardy, and amalgam spaces of certain paraproducts and bilinear pseudodifferential operators with mild regularity. We do so by showing that these operators can be realized as generalized bilinear Calderón–Zygmund operators.      

Bilinear pseudodifferential operators with forbidden symbols on Lipschitz and Besov spaces

We show that bilinear pseudodifferential operators with symbols in the forbidden class are bounded on products of Lipschitz and Besov spaces.     

Sobolev space estimates and symbolic calculus for bilinear pseudodifferential operators

Bilinear operators are investigated in the context of Sobolev spaces and various techniques useful in the study of their boundedness properties are developed. In particular, several classes of symbols for bilinear operators beyond the so-called Coifman-Meyer class are considered. Some of the Sobolev space estimates obtained apply to both the bilinear Hilbert transform and its singular multipliers generalizations as well as to operators with variable dependent symbols. A symbolic calculus for the transposes of bilinear pseudodifferential operators and for the composition of linear and bilinear pseudodifferential operators is presented too.     

Multilinear localization operators

In this paper we introduce a notion of multilinear localization operators. By reinterpreting these operators as multilinear Kohn-Nirenberg pseudodifferential operators, we give sufficient conditions for their boundedness on products of modulation spaces. Moreover, we prove that these conditions are also necessary for the boundedness of the operators. Finally, as application, we construct various examples of bounded multilinear pseudodifferential operators.     

On the Hörmander Classes of Bilinear Pseudodifferential Operators

Bilinear pseudodifferential operators with symbols in the bilinear analog of all the Hörmander classes are considered and the possibility of a symbolic calculus for the transposes of the operators in such classes is investigated. Precise results about which classes are closed under transposition and can be characterized in terms of asymptotic expansions are presented. This work extends the results for more limited classes studied before in the literature and, hence, allows the use of the symbolic calculus (when it exists) as an alternative way to recover the boundedness on products of Lebesgue spaces for the classes that yield operators with bilinear Calderón–Zygmund kernels. Some boundedness properties for other classes with estimates in the form of Leibniz’ rule are presented as well.     

Commutators of Bilinear Pseudodifferential Operators and Lipschitz Functions

Commutators of bilinear pseudodifferential operators and the operation of multiplication by a Lipschitz function are studied. The bilinear symbols of the pseudodifferential operators considered belong to classes that are shown to properly contain certain bilinear Hörmander classes of symbols of order one. The corresponding commutators are proved to be bilinear Calderón–Zygmund operators.     

Multidimensional ultrametric pseudodifferential equations

We develop an analysis of wavelets and pseudodifferential operators on multidimensional ultrametric spaces which are defined as products of locally compact ultrametric spaces. We introduce bases of wavelets, spaces of generalized functions and the space D′₀(X) of generalized functions on a multidimensional ultrametric space. We also consider some family of pseudodifferential operators on multidimensional ultrametric spaces. The notions of Cauchy problem for ultrametric pseudodifferential equations and of ultrametric characteristics are introduced. We prove an existence theorem and describe all solutions for the Cauchy problem (an analog of the Kovalevskaya theorem).     

On the Functional Characterization of the Analytic Wave Front Set of an Hyperfunction

We 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.     

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 L^p,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.     

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.     

Composition and spectral invariance of pseudodifferential Operators on Modulation Spaces

We introduce new classes of Banach algebras of pseudodifferential operators with symbols in certain modulation spaces and investigate their composition and the functional calculus. Operators in these algebras possess the spectral invariance property on the associated family of modulation spaces. These results extend and contain Sjöstrand's theory, and they are obtained with new phase-space methods instead of “hard analysis”.     

Generalized Anti-Wick Operators with Symbols in Distributional Sobolev spaces

Generalized Anti-Wick operators are introduced as a class of pseudodifferential operators which depend on a symbol and two different window functions. Using symbols in Sobolev spaces with negative smoothness and windows in so-called modulation spaces, we derive new conditions for the boundedness on L² of such operators and for their membership in the Schatten classes. These results extend and refine results contained in [20], [10], [5], [4], and [14].     

Anti-Wick Symbols for Infinite Products in K-Homology

We consider infinite products in K-homology. We study these products in relation with operators on filtered Hilbert spaces, and infinite iterations of universal constructions on C*-algebras. In particular, infinite tensor power of extensions of pseudodifferential operators on R are considered. We extend anti-Wick pseudodifferential operators to infinite tensor products of spaces of the type L ²(R), and compare our infinite tensor power construction with an extension of pseudodifferential operators on R . We show that the K-theory connecting maps coincide. We propose a natural definition of ellipticity for anti-Wick operators on R, compute the corresponding index, and draw some consequences concerning these operators.     

Discrete decompositions for bilinear operators and almost diagonal conditions

Using discrete decomposition techniques, bilinear operators are naturally associated with trilinear tensors. An intrinsic size condition on the entries of such tensors is introduced and is used to prove boundedness for the corresponding bilinear operators on several products of function spaces. This condition should be considered as the direct analogue of an almost diagonal condition for linear operators of Calderón-Zygmund type. Applications include a reduced theorem for bilinear pseudodifferential operators and the extension of an multiplier result of Coifman and Meyer to the full range of spaces. The results of this article rely on decomposition techniques developed by Frazier and Jawerth and on the vector valued maximal function estimate of Fefferman and Stein.

     

Pseudodifferential operators on mixed‐norm Besov and Triebel–Lizorkin spaces

We prove boundedness of pseudodifferential operators on anisotropic mixed‐norm Besov and Triebel–Lizorkin spaces. Our proof relies only on general maximal function estimates and provides a new perspective even in the case of spaces without mixed norms. Moreover, we cover the case of Fourier multipliers on the above mentioned spaces. As application we establish boundedness of pseudodifferential operators and Fourier multipliers on anisotropic mixed‐norm Sobolev spaces.     

Truncations of multilinear Hankel operators

We extend to multilinear Hankel operators the fact that some truncations of bounded Hankel operators are still bounded. We prove and use a continuity property of bilinear Hilbert transforms on products of Lipschitz spaces and Hardy spaces.

     

On interpolation of bilinear operators

In this paper we study interpolation of bilinear operators between products of Banach spaces generated by abstract methods of interpolation in the sense of Aronszajn and Gagliardo. A variant of bilinear interpolation theorem is proved for bilinear operators from corresponding weighted c₀ spaces into Banach spaces of non-trivial the periodic Fourier cotype. This result is then extended to the spaces generated by the well-known minimal and maximal methods of interpolation determined by quasi-concave functions. In the case when a maximal construction is generated by Hilbert spaces, we obtain a general variant of bilinear interpolation theorem. Combining this result with the abstract Grothendieck theorem of Pisier yields further results. The results are applied in deriving a bilinear interpolation theorem for Calderón-Lozanovsky, for Orlicz spaces and an embedding interpolation formula for (E,p)-summing operators.     

Elliptic pseudodifferential operators in the improved scale of spaces on a closed manifold

We study linear elliptic pseudodifferential operators in the improved scale of functional Hilbert spaces on a smooth closed manifold. Elements of this scale are isotropic Hörmander-Volevich-Paneyakh spaces. We investigate the local smoothness of a solution of an elliptic equation in the improved scale. We also study elliptic pseudodifferential operators with parameter.