Uncertainty product for Vilenkin groups

We study a localization of functions defined on Vilenkin groups. To measure the localization, we introduce two uncertainty products [Formula: see text] and [Formula: see text] that are similar to the Heisenberg uncertainty product. [Formula: see text] and [Formula: see text] differ from each other by the metric used for the Vilenkin group [Formula: see text]. We discuss analogs of a quantitative uncertainty principle. Representations for [Formula: see text] and [Formula: see text] in terms of Walsh and Haar basis are given.

N-Valid trees in wavelet theory on Vilenkin groups

We consider a class of (N, M)-elementary step functions on the p-adic Vilenkin group. We prove that (N, M)-elementary step function generates a MRA on p-adic Vilenkin group if and only if it is generated by a special N-valid rooted tree on the set of vertices {0,1,…p - 1} with the vector (0,…,0) ∈ ℤN as a root.

Modulus of Continuity and Best Approximation with Respect to Vilenkin-Like Systems in Some Function Spaces

We rephrase Fridli's result [Fridli, Acta Math. Hungar. 45: 393–396, 1985] on the modulus of continuity with respect to a Vilenkin group in the Lebesgue space. We show that this result is valid in the logarithm space and for Vilenkin-like systems. In addition, we prove that there is a strong connection between the best approximation of Fourier series and the modulus of continuity, not only in the Lebesgue space [Gát, Acta Math. Acad. Paedagog. Nyhzi. (N.S.) 17: 161–169, 2001] but in the logarithm space too. We formulate two variable generalizations of the obtained results, which have not been known till now even in the Walsh case.

The boundedness of multilinear commutators on locally compact Vilenkin groups

LetGbe a locally compact Vilenkin group. In this paper, the authors investigate the boundedness of multilinear commutators of fractional integral operator on Lebesgue spaces onG. Furthermore, the boundedness on Hardy spaces are also obtained in this paper.

The boundedness of commutators on locally compact Vilenkin groups

LetGbe a locally compact Vilenkin group. In this paper, the authors investigate the boundedness of commutators of singluar integral operator on Triebel-Lizorkin spaces onG. Furthermore, the boundedness on the Herz-type Triebel-Lizorkin spaces are also studied.

Lp-multipliers of mixed-norm type on locally compact Vilenkin groups

Let G be a locally compact Vilenkin group with dual group Γ. We prove Littlewood-Paley type inequalities corresponding to arbitrary coset decompositions of Γ. These inequalities are then applied to obtain new Lp(G) multiplier theorems. The sharpness of some of these results is also discussed.

Potential operators and multipliers on locally compact Vilenkin groups

We study, under the setting of a locally compact Vilenkin group G, a weighted norm inequality for the potential operators of Riesz type and its applications to multipliers on G. We also consider the maximal operators of fractional type.