Vilenkin Groups Research Articles

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

AbstractLet 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.

A Note on H1 Multipliers for Locally Compact Vilenkin Groups

AbstractKitada and then Onneweer and Quek have investigated multiplier operators on Hardy spaces over locally compact Vilenkin groups. In this note, we provide an improvement to their results for the Hardy space H1 and provide examples showing that our result applies to a significantly larger group of multipliers.

Boundedness of Sublinear Operators in Herz Spaces on Vilenkin Groups and its Application

The authors establish the boundedness of some sublinear operators in weighted Herz spaces on Vilenkin groups under certain weak local hypotheses on the size of these operators at the identity. This class of operators includes most of the important operators in harmonic analysis on Vilenkin groups. The main theorems are best possible under the conditions of the theorems. As applications, the authors establish the Littlewood-Paley function characterization of some Herz spaces and the relations between Herz spaces and Herz-type Hardy spaces.

Some Remarks on Singular Integrals and Power Weights on Vilenkin Groups

Let G be a Vilenkin group and T a sublinear operator. The author investigates under what weak conditions on the size of T, the Lp(G)-boundedness of T implies its Lp|x|α (G)-boundedness. For those sublinear operators which map Lp(G) to Lq(G) with p < q the author also obtains some similar results.

Littlewood–Paley and Multiplier Theorems on Weighted Spaces over Locally Compact Vilenkin Groups

We extend the Littlewood–Paley theorem toLpw(G), whereGis a locally compact Vilenkin group andware weights satisfying the MuckenhouptApcondition. As an application we obtain a mixed-norm type multiplier result onLpw(G) and prove the sharpness of our result. We also obtain a sufficient condition for φ∈L∞(Γ) to be a multiplier on the power weightedLpα(G) in terms of its smoothness condition.

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.

Some non-homogeneous Hardy spaces on locally compact Vilenkin groups

Some non-homogeneous Hardy spaces on locally compact Vilenkin groups

Multipliers of weak type on locally compact Vilenkin groups

Let G G be a locally compact Vilenkin group with dual group Γ \Gamma . We give a sufficient condition for f ∈ L ∞ ( Γ ) f\in L^\infty (\Gamma ) to be a multiplier of weak type ( p , p ) (p,p) on G G . Some applications of our result are given. We also prove that our result is sharp.

Para-product operators and para-linearization on locally compact Vilenkin groups

The concept of para-product operators over locally compact Vilenkin groups is established and the applications to the para-linearization in nonlinear problems are studied. This kind of operators plays a special role in dealing with those functions which do not have the classical derivatives.

Sequences of connected spectrum and the Vilenkin group

Sequences of connected spectrum and the Vilenkin group

On Lp-Multipliers of Mixed-Norm Type

Given a sequence (gn) of Fourier multipliers for Lp(R), 1 < p < ∞, let g ≔ Σ∞−∞gnχn, where χn denotes the characteristic function of the interval [2n, 2n+1] in R. Assuming (gn) ∈ ℓs(M(p)) for some s with 0 < s ≤ ∞, we determine the values of s for which g is, or is not, a multiplier of Lp(R). Our results sharpen a result of Littman et al. who, in 1968, considered the case when s = ∞. The same problem is also considered for multipliers in Lp-spaces defined on a locally compact Vilenkin group.

Weighted Triebel‐Lizorkin Spaces on Locally Compact Vilenkin Groups

AbstractLet G be a locally compact Vilenkin group. We consider the atomic and molecular decomposition of the weighted Triebel‐Lizorkin spaces on G. As an application of it we prove boundedness results for certain pseudodifferential operators on such spaces.

On 𝐻^{𝑝}(𝐑ⁿ)-multipliers of mixed-norm type

For a function m in L ∞ ( R n ) {L^\infty }({{\mathbf {R}}^n}) , an appropriately chosen function η \eta in C ∞ ( R n ) {C^\infty }({{\mathbf {R}}^n}) and δ &gt; 0 \delta &gt; 0 we define m δ {m_\delta } by m δ ( ξ ) = m ( δ ξ ) η ( ξ ) {m_\delta }(\xi ) = m(\delta \xi )\eta (\xi ) . We show that if 0 &gt; p ≤ 1 0 &gt; p \leq 1 and if the sequence ( ( m 2 n ) ∅ ^ ) ((m_{2^n})\hat \emptyset ) belongs to a certain mixed-norm space, depending on p, then m is a Fourier multiplier for the corresponding Hardy space H p ( R n ) {H^p}({{\mathbf {R}}^n}) . Moreover, we prove the sharpness of our multiplier theorem. Comparable results had been proved earlier for multipliers for Hardy spaces defined on a locally compact Vilenkin group.

Multipliers for weighted Hardy spaces on locally compact Vilenkin groups

Let G be a locally compact Vilenkin group. We study multipliers which satisfy a generalised Hörmander condition from power-weighted Hardy space (G) to (G) with 0 &lt; p ≤ q &lt; ∞, 0 &lt; p ≤ 1, −1 &lt; β, β′.

Multipliers for Hardy spaces on locally compact Vilenkin groups

AbstractIn a recent paper In a recent paper the authors proved a multiplier theorem for Hardy spaces Hp (G), 0 &lt; p ≤ 1, defined on a locally compact Vilenkin group G. The assumptions on the multiplier were expressed in terms of the “norms” of certain Herz spaces K (1/p − 1/?r, r, p) with r restricted to 1 ≤ r &lt; ∞ and p &lt; r. In the present paper we show how this restriction on r may be weakened to p ≤ r ∞. Furthermore, we present two modifications of our main theorem and compare these with certain results for multipliers on LP (Rn)-spaces, 1 &lt; p &lt; ∞, due to Seeger and to Cowling, Fendler and Foumier. We also discuss the sharpness of some of our results.

Growth Conditions for Thin Sets in Vilenkin Groups of Bounded Order

Growth Conditions for Thin Sets in Vilenkin Groups of Bounded Order

An Integrability Theorem on Unbounded Vilenkin Groups

An analogue of the classical Fomin′s theorem holds in general Vilenkin systems. We prove this using F. Riesz′s general version of the Hausdorff-Young inequality on the cosets of a subgroup of a given Vilenkin group. The class of sequences involved and the class of quasiconvex sequences are incomparable.

Dirichlet sets in Vilenkin groups

For the purposes of this paper, G win denote a compact, totally disconnected, abe]Jan, metric group, i.e. a Vilenkin group, and F will denote the Pontryagin dual of G. We set A(G) to be the algebra of absolutely convergent Fourier series on G and PM(G) the Banach space dual of A(G). Elements of PM(G) are called pseudomeasures. If S is a pseudomeasure, write S ( 7 ) = = (S, 7) where ( , } denotes the dual pairing of PM(G) and A(G). We call S a pseudofunction if S(7) ~ 0 as 7 ~ co in F. The collection of all pseudofunctions is called PF(G)o A pseudomeasure is said to be supported on a closed set E if it is suppor ted as a distribution with test functions from A(G). A closed set is called a U-set if it supports no nontrivial pseudofunction. The set E is called a strong U-set if ]Jm sup )S(7)I = sup 1~'(7)1 7---*o0 "yEF

Growth conditions for thin sets in Vilenkin groups of bounded order

Let G G be a Vilenkin group of bounded order and H n {H_n} a sequence of clopen subgroups of G G forming a base at the identity. If E E is a subset of G G , let N n ( E ) {N_n}(E) denote the number of cosets of H n {H_n} which intersect E E . If \[ lim _ N n ( E ) log ⁡ [ G : H n ] &gt; ∞ , \underline {\lim } \frac {{{N_n}(E)}} {{\log [G:{H_n}]}} &gt; \infty , \] then E E is a U-set in the group G G . It is also shown that for G G satisfying a growth condition and φ ( n ) → ∞ \varphi (n) \to \infty , there is an M-set, E E , with \[ N n ( E ) = O ( φ ( n ) log ⁡ [ G : H n ] ) . {N_n}(E) = O(\varphi (n)\,\log [G:{H_n}]). \]

Some new Hardy spaces on locally compact Vilenkin groups

Some new Hardy spaces on locally compact Vilenkin groups