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Necessary conditions for the L p -convergence ( 0 < p < 1 ) of single and double trigonometric series

Xhevat Z. Krasniqi, Péter Kórus, Ferenc Móricz (2014)

Mathematica Bohemica

We give necessary conditions in terms of the coefficients for the convergence of a double trigonometric series in the L p -metric, where 0 < p < 1 . The results and their proofs have been motivated by the recent papers of A. S. Belov (2008) and F. Móricz (2010). Our basic tools in the proofs are the Hardy-Littlewood inequality for functions in H p and the Bernstein-Zygmund inequalities for the derivatives of trigonometric polynomials and their conjugates in the L p -metric, where 0 < p < 1 .

Nonconvolution transforms with oscillating kernels that map 1 0 , 1 into itself

G. Sampson (1993)

Studia Mathematica

We consider operators of the form ( Ω f ) ( y ) = ʃ - Ω ( y , u ) f ( u ) d u with Ω(y,u) = K(y,u)h(y-u), where K is a Calderón-Zygmund kernel and h L (see (0.1) and (0.2)). We give necessary and sufficient conditions for such operators to map the Besov space 1 0 , 1 (= B) into itself. In particular, all operators with h ( y ) = e i | y | a , a > 0, a ≠ 1, map B into itself.

Norm convergence of some power series of operators in L p with applications in ergodic theory

Christophe Cuny (2010)

Studia Mathematica

Let X be a closed subspace of L p ( μ ) , where μ is an arbitrary measure and 1 < p < ∞. Let U be an invertible operator on X such that s u p n | | U | | < . Motivated by applications in ergodic theory, we obtain (optimal) conditions for the convergence of series like n 1 ( U f ) / n 1 - α , 0 ≤ α < 1, in terms of | | f + + U n - 1 f | | p , generalizing results for unitary (or normal) operators in L²(μ). The proofs make use of the spectral integration initiated by Berkson and Gillespie and, more particularly, of results from a paper by Berkson-Bourgain-Gillespie....

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