Annals of Functional Analysis最新文献

We integrate in the framework of the semilocal trace formula two recent discoveries on the spectral realization of the zeros of the Riemann zeta function by introducing a semilocal analogue of the prolate wave operator. The latter plays a key role both in the spectral realization of the low lying zeros of zeta—using the positive part of its spectrum—and of their ultraviolet behavior—using the Sonin space which corresponds to the negative part of the spectrum. In the archimedean case the prolate operator is the sum of the square of the scaling operator with the grading of orthogonal polynomials, and we show that this formulation extends to the semilocal case. We prove the stability of the semilocal Sonin space under the increase of the finite set of places which govern the semilocal framework and describe their relation with Hilbert spaces of entire functions. Finally, we relate the prolate operator to the metaplectic representation of the double cover of ({text {SL}}(2,mathbb {R})) with the goal of obtaining (in a forthcoming paper) a second candidate for the semilocal prolate operator.

For two given symmetric sequence spaces E and F we study the action of the main triangular projection T on B(E, F), the space of all bounded linear operators from E to F, and give a lower estimate for the norm of T in terms of the fundamental functions of E and F and the fundamental function of the generalized dual space F : E. In addition, we give a condition for the boundedness of the operator T in terms of F-absolutely summing operators. Furthermore, we apply our results to some concrete symmetric sequence spaces. In particular, we study the question of the boundedness or unboundedness of T on the spaces (B(ell _{p_1,q_1},ell _{p}),) (B(ell _{p},ell _{p_1,q_1})) and (B(ell _{p_1,q_1},ell _{p_2,q_2})).

Constructing multivariate tight framelets is a challenging problem in wavelet and framelet theory. The problem is intrinsically related to the Hermitian sum of squares decomposition of multivariate trigonometric polynomials and the spectral factorization of multivariate trigonometric polynomial matrices. To circumvent the relevant difficulties, the notion of a quasi-tight framelet has been introduced in recent years, which generalizes the concept of tight framelets. On one hand, quasi-tight framelets behave similarly to tight framelets. On the other hand, compared to tight framelets, quasi-tight framelets have much more flexibility and advantages. Motivated by several recent studies of multivariate quasi-tight and tight framelets, we work on quincunx quasi-tight and tight framelets with the interpolatory properties in this paper. We first show that from any interpolatory quincunx refinement filter, one can always construct an interpolatory quasi-tight framelet with three generators. Next, we shall present a way to construct interpolatory quincunx quasi-tight framelets with high-order vanishing moments. Finally, we will establish an algorithm to construct interpolatory quincunx tight framelets from any interpolatory quincunx refinement filter that satisfies the so-called sum-of-squares (SOS) condition. All our proofs are constructive, and several examples in dimension (d=2) will be provided to illustrate our main results.

In this paper, we are devoted to studying some sharp bounds for Hardy-type operators on mixed radial-angular type function spaces. In addition, we will establish the sharp weak-type estimates for the fractional Hardy operator and its conjugate operator, respectively.

Let (mu _{M,D}) be a self-affine measure generated by an iterated function systems ({phi _d(x)=M^{-1}(x+d) (xin mathbb {R}^2)}_{din D}), where (Min M_2(mathbb {Z})) is an expanding integer matrix and (D = {(0,0)^t,(1,0)^t,(0,1)^t}). In this paper, we study the spectral eigenmatrix problem of (mu _{M,D}), i.e., we characterize the matrix R which (RLambda ) is also a spectrum of (mu _{M,D}) for some spectrum (Lambda ). Some necessary and sufficient conditions for R to be a spectral eigenmatrix are given, which extends some results of An et al. (Indiana Univ Math J, 7(1): 913–952, 2022). Moreover, we also find some irrational spectral eigenmatrices of (mu _{M,D}), which is different from the known results that spectral eigenmatrices are rational.

For (mathbb {B}^n) the n-dimensional unit ball and (D_n) its Siegel unbounded realization, we consider Toeplitz operators acting on weighted Bergman spaces with symbols invariant under the actions of the maximal Abelian subgroups of biholomorphisms (mathbb {T}^n) (quasi-elliptic) and (mathbb {T}^n times mathbb {R}_+) (quasi-hyperbolic). Using geometric symplectic tools (Hamiltonian actions and moment maps) we obtain simple diagonalizing spectral integral formulas for such kinds of operators. Some consequences show how powerful the use of our differential geometric methods are.

The aim of this work is to describe subsets of Banach limits in terms of a certain functional characteristic. We compute radii and cardinalities for some of these subsets.

In this paper, we first obtain Fourier multiplier theorem, the approximation characterization and embedding for (B^u_omega ) type Morrey–Triebel–Lizorkin spaces with variable smoothness and integrability. Then, we characterize these spaces via Peetre’s maximal functions, the Lusin area function, and the Littlewood–Paley (g^*_lambda )-function. Finally, we obtain the boundedness of the pseudo-differential operators on these spaces.

We give applications of the higher Lefschetz theorems for foliations of Benameur and Heitsch (J. Funct. Anal. 259:131–173, 2010), primarily involving Haefliger cohomology. These results show that the transverse structures of foliations carry important topological and geometric information. This is in the spirit of the passage from the Atiyah–Singer index theorem for a single compact manifold to their families index theorem, involving a compact fiber bundle over a compact base. For foliations, Haefliger cohomology plays the role that the cohomology of the base space plays in the families index theorem. We obtain highly useful numerical invariants by paring with closed holonomy invariant currents. In particular, we prove that the non-triviality of the higher (widehat{A}) class of the foliation in Haefliger cohomology can be an obstruction to the existence of non-trivial leaf-preserving compact connected group actions. We then construct a large collection of examples for which no such actions exist. Finally, we relate our results to Connes’ spectral triples, and prove useful integrality results.

A semibounded operator or relation S in a Hilbert space with lower bound (gamma in {{mathbb {R}}}) has a symmetric extension (S_textrm{f}=S , widehat{+} ,({0} times mathrm{mul,}S^*)), the weak Friedrichs extension of S, and a selfadjoint extension (S_{textrm{F}}), the Friedrichs extension of S, that satisfy (S subset S_{textrm{f}} subset S_textrm{F}). The Friedrichs extension (S_{textrm{F}}) has lower bound (gamma ) and it is the largest semibounded selfadjoint extension of S. Likewise, for each (c le gamma ), the relation S has a weak Kreĭn type extension (S_{textrm{k},c}=S , widehat{+} ,(mathrm{ker,}(S^*-c) times {0})) and Kreĭn type extension (S_{textrm{K},c}) of S, that satisfy (S subset S_{textrm{k},c} subset S_{textrm{K},c}). The Kreĭn type extension (S_{textrm{K},c}) has lower bound c and it is the smallest semibounded selfadjoint extension of S which is bounded below by c. In this paper these special extensions and, more generally, all extremal extensions of S are constructed via the semibounded sesquilinear form ({{mathfrak {t}}}(S)) that is associated with S; the representing map for the form ({{mathfrak {t}}}(S)-c) plays an essential role here.