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Walsh-Marcinkiewicz means and Hardy spaces

Károly Nagy, George Tephnadze (2014)

Open Mathematics

The main aim of this paper is to investigate the Walsh-Marcinkiewicz means on the Hardy space H p, when 0 < p < 2/3. We define a weighted maximal operator of Walsh-Marcinkiewicz means and establish some of its properties. With its aid we provide a necessary and sufficient condition for convergence of the Walsh-Marcinkiewicz means in terms of modulus of continuity on the Hardy space H p, and prove a strong convergence theorem for the Walsh-Marcinkiewicz means.

Wave equation and multiplier estimates on ax + b groups

Detlef Müller, Christoph Thiele (2007)

Studia Mathematica

Let L be the distinguished Laplacian on certain semidirect products of ℝ by ℝⁿ which are of ax + b type. We prove pointwise estimates for the convolution kernels of spectrally localized wave operators of the form e i t L ψ ( L / λ ) for arbitrary time t and arbitrary λ > 0, where ψ is a smooth bump function supported in [-2,2] if λ ≤ 1 and in [1,2] if λ ≥ 1. As a corollary, we reprove a basic multiplier estimate of Hebisch and Steger [Math. Z. 245 (2003)] for this particular class of groups, and derive Sobolev...

Wave front set for positive operators and for positive elements in non-commutative convolution algebras

Joachim Toft (2007)

Studia Mathematica

Let WF⁎ be the wave front set with respect to C , quasi analyticity or analyticity, and let K be the kernel of a positive operator from C to ’. We prove that if ξ ≠ 0 and (x,x,ξ,-ξ) ∉ WF⁎(K), then (x,y,ξ,-η) ∉ WF⁎(K) and (y,x,η,-ξ) ∉ WF⁎(K) for any y,η. We apply this property to positive elements with respect to the weighted convolution u B φ ( x ) = u ( x - y ) φ ( y ) B ( x , y ) d y , where B C is appropriate, and prove that if ( u B φ , φ ) 0 for every φ C and (0,ξ) ∉ WF⁎(u), then (x,ξ) ∉ WF⁎(u) for any x.

Weak amenability of general measure algebras

Javad Laali, Mina Ettefagh (2008)

Colloquium Mathematicae

We study the weak amenability of a general measure algebra M(X) on a locally compact space X. First we show that not all general measure multiplications are separately weak* continuous; moreover, under certain conditions, weak amenability of M(X)** implies weak amenability of M(X). The main result of this paper states that there is a general measure algebra M(X) such that M(X) and M(X)** are weakly amenable without X being a discrete topological space.

Weak amenability of the second dual of a Banach algebra

M. Eshaghi Gordji, M. Filali (2007)

Studia Mathematica

It is known that a Banach algebra inherits amenability from its second Banach dual **. No example is yet known whether this fails if one considers the weak amenability instead, but the property is known to hold for the group algebra L¹(G), the Fourier algebra A(G) when G is amenable, the Banach algebras which are left ideals in **, the dual Banach algebras, and the Banach algebras which are Arens regular and have every derivation from into * weakly compact. In this paper, we extend this class of...

Weak amenability of weighted group algebras on some discrete groups

Varvara Shepelska (2015)

Studia Mathematica

Weak amenability of ℓ¹(G,ω) for commutative groups G was completely characterized by N. Gronbaek in 1989. In this paper, we study weak amenability of ℓ¹(G,ω) for two important non-commutative locally compact groups G: the free group ₂, which is non-amenable, and the amenable (ax + b)-group. We show that the condition that characterizes weak amenability of ℓ¹(G,ω) for commutative groups G remains necessary for the non-commutative case, but it is sufficient neither for ℓ¹(₂,ω) nor for ℓ¹((ax + b),ω)...

Weak* properties of weighted convolution algebras II

Sandy Grabiner (2010)

Studia Mathematica

We show that if ϕ is a continuous homomorphism between weighted convolution algebras on ℝ⁺, then its extension to the corresponding measure algebras is always weak* continuous. A key step in the proof is showing that our earlier result that normalized powers of functions in a convolution algebra on ℝ⁺ go to zero weak* is also true for most measures in the corresponding measure algebra. For some algebras, we can determine precisely which measures have normalized powers converging to zero weak*. We...

Weak spectral synthesis in Fourier algebras of coset spaces

Eberhard Kaniuth (2010)

Studia Mathematica

Let G be a locally compact group, K a compact subgroup of G and A(G/K) the Fourier algebra of the coset space G/K. Applying results from [E. Kaniuth, Weak spectral synthesis in commutative Banach algebras, J. Funct. Anal. 254 (2008), 987-1002], we establish injection and localization theorems relating weak spectral sets and weak Ditkin sets for A(G/K) to such sets for A(H/H ∩ K), where H is a closed subgroup of G. We also prove some results towards the analogue of Malliavin's theorem for weak spectral...

Weak type (1,1) multipliers on LCA groups

José Raposo (1997)

Studia Mathematica

In [ABB] Asmar, Berkson and Bourgain prove that for a sequence ϕ j j = 1 of weak type (1, 1) multipliers in n and a function k L 1 ( n ) the weak type (1,1) constant of the maximal operator associated with k ϕ j j is controlled by that of the maximal operator associated with ϕ j j . In [ABG] this theorem is extended to LCA groups with an extra hypothesis: the multipliers must be continuous. In this paper we prove a more general version of this last result without assuming the continuity of the multipliers. The proof arises...

Weak type radial convolution operators on free groups

Tadeusz Pytlik, Ryszard Szwarc (2008)

Studia Mathematica

Radial convolution operators on free groups with nonnegative kernel of weak type (2,2) and of restricted weak type (2,2) are characterized. Estimates of weak type (p,p) are obtained as well for 1 < p < 2.

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