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Convolution-dominated integral operators

Gero Fendler, Karlheinz Gröchenig, Michael Leinert (2010)

Banach Center Publications

For a locally compact group G we consider the algebra CD(G) of convolution-dominated operators on L²(G), where an operator A: L²(G) → L²(G) is called convolution-dominated if there exists a ∈ L¹(G) such that for all f ∈ L²(G) |Af(x)| ≤ a⋆|f|(x), for almost all x ∈ G. (1) The case of discrete groups was treated in previous publications [fgl08a, fgl08]. For non-discrete groups we investigate a subalgebra of regular convolution-dominated operators generated by product convolution operators, where the...

Convolutions on compact groups and Fourier algebras of coset spaces

Brian E. Forrest, Ebrahim Samei, Nico Spronk (2010)

Studia Mathematica

We study two related questions. (1) For a compact group G, what are the ranges of the convolution maps on A(G × G) given for u,v in A(G) by u × v ↦ u*v̌ (v̌(s) = v(s^-1)) and u × v ↦ u*v? (2) For a locally compact group G and a compact subgroup K, what are the amenability properties of the Fourier algebra of the coset space A(G/K)? The algebra A(G/K) was defined and studied by the first named author. In answering the first question, we obtain, for compact groups which do not...

Copies of l p n ’s uniformly in the spaces Π 2 ( C [ 0 , 1 ] , X ) and Π 1 ( C [ 0 , 1 ] , X )

Dumitru Popa (2017)

Czechoslovak Mathematical Journal

We study the presence of copies of l p n ’s uniformly in the spaces Π 2 ( C [ 0 , 1 ] , X ) and Π 1 ( C [ 0 , 1 ] , X ) . By using Dvoretzky’s theorem we deduce that if X is an infinite-dimensional Banach space, then Π 2 ( C [ 0 , 1 ] , X ) contains λ 2 -uniformly copies of l n ’s and Π 1 ( C [ 0 , 1 ] , X ) contains λ -uniformly copies of l 2 n ’s for all λ > 1 . As an application, we show that if X is an infinite-dimensional Banach space then the spaces Π 2 ( C [ 0 , 1 ] , X ) and Π 1 ( C [ 0 , 1 ] , X ) are distinct, extending the well-known result that the spaces Π 2 ( C [ 0 , 1 ] , X ) and 𝒩 ( C [ 0 , 1 ] , X ) are distinct.

Corrections to the paper “The boundedness of certain sublinear operator in the weighted variable Lebesgue spaces“

Rovshan A. Bandaliev (2013)

Czechoslovak Mathematical Journal

In this paper the author proved the boundedness of the multidimensional Hardy type operator in weighted Lebesgue spaces with variable exponent. As an application he proved the boundedness of certain sublinear operators on the weighted variable Lebesgue space. The proof of the boundedness of the multidimensional Hardy type operator in weighted Lebesgue spaces with a variable exponent does not contain any mistakes. But in the proof of the boundedness of certain sublinear operators on the weighted...

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