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1.
Compressions of Toeplitz operators to coinvariant subspaces of H2 are called truncated Toeplitz operators. We study two questions related to these operators. The first, raised by Sarason, is whether boundedness of the operator implies the existence of a bounded symbol; the second is the Reproducing Kernel Thesis. We show that in general the answer to the first question is negative, and we exhibit some classes of spaces for which the answers to both questions are positive.  相似文献   

2.
It is known that Fourier integral operators arising when solving Schrödinger-type operators are bounded on the modulation spaces ? p,q , for 1≤p= q≤∞, provided their symbols belong to the Sjöstrand class M ∞,1. However, they generally fail to be bounded on ? p,q for pq. In this paper we study several additional conditions, to be imposed on the phase or on the symbol, which guarantee the boundedness on ? p,q for pq, and between ? p,q →? q,p , 1≤q<p≤∞. We also study similar problems for operators acting on Wiener amalgam spaces, recapturing, in particular, some recent results for metaplectic operators. Our arguments make heavily use of the uncertainty principle.  相似文献   

3.
We investigate the boundedness of unimodular Fourier multipliers on modulation spaces. Surprisingly, the multipliers with general symbol eiα|ξ|, where α∈[0,2], are bounded on all modulation spaces, but, in general, fail to be bounded on the usual Lp-spaces. As a consequence, the phase-space concentration of the solutions to the free Schrödinger and wave equations are preserved. As a byproduct, we also obtain boundedness results on modulation spaces for singular multipliers |ξ|δsin(α|ξ|) for 0?δ?α.  相似文献   

4.
We use the characterizations of the classes of all infinite matrices that map the spaces of sequences which are strongly summable or bounded by the Cesàro method of order 1 into the spaces of null or convergent sequences given by Ba?ar, Malkowsky and Altay [Matrix transformations on the matrix domains of triangles in the spaces of strongly C1-summable and bounded sequences, Publ. Math. Debrecen 73 (1-2) (2008), 193-213] and the Hausdorff measure of noncompactness to characterize the classes of all compact operators between those spaces.  相似文献   

5.
We study Toeplitz operators between analytic Bloch-type spaces of the unit disk. We construct suitable classes of distributions that generate bounded Toeplitz operators between these spaces. The classes are naturally connected to the corresponding results in the reflexive Bergman space setting and previously known results on A 1 and the Bloch space. We also study distributional symbols satisfying logarithmic BMO-condition. In addition, sufficient compactness criteria are provided.  相似文献   

6.
We completely characterize the boundedness on Wiener amalgam spaces of the short-time Fourier transform (STFT), and on both L p and Wiener amalgam spaces of a special class of pseudodifferential operators, called localization operators. Precisely, sufficient conditions for the STFT to be bounded on the Wiener amalgam spaces W(L p , L q ) are given and their sharpness is shown. Localization operators are treated similarly: using different techniques from those employed in the literature, we relax the known sufficient boundedness conditions for these operators to be bounded on L p spaces and prove the optimality of our results. Next, we exhibit sufficient and necessary conditions for such operators to be bounded on Wiener amalgam spaces.  相似文献   

7.
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].  相似文献   

8.
We study boundary value problems for some differential operators on Euclidean space and the Heisenberg group which are invariant under the conformal group of a Euclidean subspace, respectively, Heisenberg subgroup. These operators are shown to be self-adjoint in certain Sobolev type spaces and the related boundary value problems are proven to have unique solutions in these spaces. We further find the corresponding Poisson transforms explicitly in terms of their integral kernels and show that they are isometric between Sobolev spaces and extend to bounded operators between certain Lp-spaces.

The conformal invariance of the differential operators allows us to apply unitary representation theory of reductive Lie groups, in particular recently developed methods for restriction problems.  相似文献   

9.
We study a class of pseudodifferential operators known as time-frequency localization operators, Anti-Wick operators, Gabor-Toeplitz operators or wave packets. Given a symbol a and two windows ?1,?2, we investigate the multilinear mapping from to the localization operator Aa?1,?2 and we give sufficient and necessary conditions for Aa?1,?2 to be bounded or to belong to a Schatten class. Our results are formulated in terms of time-frequency analysis, in particular we use modulation spaces as appropriate classes for symbols and windows.  相似文献   

10.
We develop our earlier approach to the Weyl calculus for representations of infinite-dimensional Lie groups by establishing continuity properties of the Moyal product for symbols belonging to various modulation spaces. For instance, we prove that the modulation space of symbols M ∞,1 is an associative Banach algebra and the corresponding operators are bounded. We then apply the abstract results to two classes of representations, namely the unitary irreducible representations of nilpotent Lie groups, and the natural representations of the semidirect product groups that govern the magnetic Weyl calculus. The classical Weyl–Hörmander calculus is obtained for the Schrödinger representations of the finite-dimensional Heisenberg groups, and in this case we recover the results obtained by J. Sjöstrand (Math Res Lett 1(2):185–192, 1994).  相似文献   

11.
We study the commutator of the multiplication and harmonic Bergman projection, Hankel and Toeplitz operators on the harmonic Bergman spaces. The same type operators have been well studied on the analytic Bergman spaces. The main difficulty of this study is that the bounded harmonic function space is not an algebra! In this paper, we characterize theL p boundedness and compactness of these operators with harmonic symbols. Results about operators in Schatten classes, the cut-off phenomenon and general symbols are also included.Partially supported by a grant from the Research Grants Committee of the University of Alabama.  相似文献   

12.
This article deals with linear operators T on a complex Hilbert space ?, which are bounded with respect to the seminorm induced by a positive operator A on ?. The A-adjoint and A 1/2-adjoint of T are considered to obtain some ergodic conditions for T with respect to A. These operators are also employed to investigate the class of orthogonally mean ergodic operators as well as that of A-power bounded operators. Some classes of orthogonally mean ergodic or A-ergodic operators, which come from the theory of generalized Toeplitz operators are considered. In particular, we give an example of an A-ergodic operator (with an injective A) which is not Cesàro ergodic, such that T ?* is not a quasiaffine transform of an orthogonally mean ergodic operator.  相似文献   

13.
The paper considers Bergman type operators introduced by Shields and Williams depending on a normal pair of weight functions. We prove that there exist values of parameter ß for which these operators are bounded on mixed norm spaces L(p, q, ß) on the unit ball in Cn.  相似文献   

14.
In this paper, we study some new kinds of Hausdorff operators on n-dimensional product spaces. We obtain their power weights from L p to L q boundedness and characterize the necessary and sufficient conditions for the operators being bounded on power weight L p spaces. Moreover, we get the sharp constants for the case p = q.  相似文献   

15.
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.  相似文献   

16.
We completely characterize the boundedness on Wiener amalgam spaces of the short-time Fourier transform (STFT), and on both L p and Wiener amalgam spaces of a special class of pseudodifferential operators, called localization operators. Precisely, sufficient conditions for the STFT to be bounded on the Wiener amalgam spaces W(L p , L q ) are given and their sharpness is shown. Localization operators are treated similarly: using different techniques from those employed in the literature, we relax the known sufficient boundedness conditions for these operators to be bounded on L p spaces and prove the optimality of our results. Next, we exhibit sufficient and necessary conditions for such operators to be bounded on Wiener amalgam spaces.  相似文献   

17.
Continuing some investigations started in previous papers, we introduce and study a sequence of multidimensional positive integral operators which generalize the Gauss-Weierstrass operators. We show that this sequence is an approximation process in some classes of weighted L p spaces on ? N , N ≥ 1. Estimates of the rate of convergence are also obtained. Our mean tool is a Korovkin-type theorem which we establish in the context of L p (X, µ) spaces, X being a locally compact Hausdorff space and µ a regular positive Borel measure on X. Several examples are explicitly indicated as well.  相似文献   

18.
We give embedding theorems for weighted Bergman–Orlicz spaces on the ball and then apply our results to the study of the boundedness and the compactness of composition operators in this context. As one of the motivations of this work, we show that there exist some weighted Bergman–Orlicz spaces, different from H , on which every composition operator is bounded.  相似文献   

19.
A well-known result going back to the 1930s states that all bounded linear operators mapping scalar-valued L 1-spaces into L -spaces are kernel operators and that in fact this relation induces an isometric isomorphism between the space of such operators and the space of all bounded kernels. We extend this result to the case of spaces of vector-valued functions. A recent result due to Arendt and Thomaschewski states that the local operators acting on L p -spaces of functions with values in separable Banach spaces are precisely the multiplication operators. We extend this result to non-separable dual spaces. Moreover, we relate positivity and other order properties of the operators to corresponding properties of the representations.  相似文献   

20.
Considering the measurable and nonnegative functions ? on the half-axis [0, ∞) such that ?(0) = 0 and ?(t) → ∞ as t → ∞, we study the operators of weak type (?, ?) that map the classes of ?-Lebesgue integrable functions to the space of Lebesgue measurable real functions on ?n. We prove interpolation theorems for the subadditive operators of weak type (?0, ?0) bounded in L (?n) and subadditive operators of weak types (?0, ?0) and (?1, ?1) in L ?(? n ) under some assumptions on the nonnegative and increasing functions ?(x) on [0, ∞). We also obtain some interpolation theorems for the linear operators of weak type (?0, ?0) bounded from L (?n) to BMO(? n). For the restrictions of these operators to the set of characteristic functions of Lebesgue measurable sets, we establish some estimates for rearrangements of moduli of their values; deriving a consequence, we obtain a theorem on the boundedness of operators in rearrangement-invariant spaces.  相似文献   

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