Mathematics

Following the well-known theory of Beurling and Roumieu ultradistributions, we investigate new spaces of ultradistributions as dual spaces of test functions which correspond to associated functions of logarithmic-type growth at infinity. In the given framework we prove that boundary values of analytic functions with the corresponding logarithmic growth rate towards the real domain are ultradistributions. The essential condition for that purpose, condition (M.2) in the classical ultradistribution theory, is replaced by the new one, (M.2) ˜ . For that reason, new techniques were performed in the proofs. As an application, we discuss the corresponding wave front sets.

We show that an infinite Toeplitz+Hankel matrix T(φ)+H(ψ) generates a bounded (compact) operator on ℓ p ( N 0 ) with 1≤p≤∞ if and only if both T(φ) and H(ψ) are bounded (compact). We also give analogous characterizations for Toeplitz+Hankel operators acting on the reflexive Hardy spaces. In both cases, we provide an intrinsic characterization of bounded operators of Toeplitz+Hankel form similar to the Brown-Halmos theorem. In addition, we establish estimates for the norm and the essential norm of such operators.

We use Cramér-Chernoff type estimates in order to study the Calderón-Zygmund structure of the kernels ??I?�D a I (?) ? I (x) ? I (y) where a I are subgaussian independent random variables and { ? I :I?�D} is a wavelet basis where D are the dyadic intervals in R . We consider both, the cases of standard smooth wavelets and the case of the Haar wavelet.

The aim of this note is to give the boundedness conditions for Hausdorff operators on Hardy spaces H 1 with the norm defined via (1,q) atoms over homogeneous spaces of Lie groups with doubling property and to apply results we obtain to generalized Delsarte operators and to Hausdorff operators over multidimensional spheres.

In this study, we investigate the boundedness of composition operators acting on Morrey spaces and weak Morrey spaces. The primary aim of this study is to investigate a necessary and sufficient condition on the boundedness of the composition operator induced by a diffeomorphism on Morrey spaces. In particular, detailed information is derived from the boundedness, i.e., the bi-Lipschitz continuity of the mapping that induces the composition operator follows from the continuity of the composition mapping. The idea of the proof is to determine the Morrey norm of the characteristic functions, and employ a specific function composed of a characteristic function. As the specific function belongs to Morrey spaces but not to Lebesgue spaces, the result reveals a new phenomenon not observed in Lebesgue spaces. Subsequently, we prove the boundedness of the composition operator induced by a mapping that satisfies a suitable volume estimate on general weak-type spaces generated by normed spaces. As a corollary, a necessary and sufficient condition for the boundedness of the composition operator on weak Morrey spaces is provided.

Boundedness of the maximal function and the Caldeón-Zygmund singular integrals in central Morrey-Orlicz spaces were proved in papers by the second and third authors. The weak-type estimates have also been proven. Here we show boundedness of the Riesz potential in central Morrey-Orlicz spaces and the corresponding weak-type version.

The paper is devoted to the problem of exact calculation of the norms in ideal spaces for monotone operators on the cones of functions with monotonicity properties. We implement a general approach to this problem that covers many concrete variants of monotone operators in ideal spaces and different monotonicity conditions for functions. As applications, we calculate the norms of some integral operators on the cones, associate norms over some cones in Lebesgue spaces, the norms of the dilation operator and embedding operators on weighted Lorentz spaces with general weights. Under some more general conditions, we present order sharp estimates for Hardy-type operators on the cones.

Let Φ be a C ∗ -subalgebra of L ∞ (R) and S O ⋄ X(R) be the Banach algebra of slowly oscillating Fourier multipliers on a Banach function space X(R) . We show that the intersection of the Calkin image of the algebra generated by the operators of multiplication aI by functions a∈Φ and the Calkin image of the algebra generated by the Fourier convolution operators W 0 (b) with symbols in S O ⋄ X(R) coincides with the Calkin image of the algebra generated by the operators of multiplication by constants.

For a compact Hausdorff space K , we give descriptions of the dual of C(K ) δ , the Dedekind completion of the Banach lattice C(K) of continuous, real-valued functions on K . We characterize those functionals which are ? -order continuous and order continuous, respectively, in terms of Oxtoby's category measures. This leads to a purely topological characterization of hyper-Stonean spaces.

We obtain descriptions of central operator-valued Schur and Herz-Schur multipliers, akin to a classical characterisation due to Grothendieck, that reveals a close link between central (linear) multipliers and bilinear multipliers into the trace class. Restricting to dynamical systems where a locally compact group acts on itself by translation, we identify their convolution multipliers as the right completely bounded multipliers, in the sense of Junge-Neufang-Ruan, of a canonical quantum group associated with the underlying group. We provide characterisations of contractive idempotent operator-valued Schur and Herz-Schur multipliers. Exploiting the link between Herz-Schur multipliers and multipliers on transformation groupoids, we provide a combinatorial characterisation of groupoid multipliers that are contractive and idempotent.