Pointwise convergence of noncommutative Fourier series

Abstract

This paper is devoted to the study of convergence of Fourier series for nonabelian groups and quantum groups. It is well-known that a number of approximation properties of groups can be interpreted as some summation methods and mean convergence of associated noncommutative Fourier series. Based on this framework, this work studies the refined counterpart of pointwise convergence of these Fourier series. We establish a general criterion of maximal inequalities for approximative identities of noncommutative Fourier multipliers. As a result we prove that for any countable discrete amenable group, there exists a sequence of finitely supported positive definite functions tending to 1 pointwise, so that the associated Fourier multipliers on noncommutative Lp-spaces satisfy the pointwise convergence for all 1 < p < ∞. In a similar fashion, we also obtain results for a large subclass of groups (as well as discrete quantum groups) with the Haagerup property and weak amenability. We also consider the analogues of Fejér means and Bochner-Riesz means in the noncommutative setting. Our results in particular apply to the almost everywhere convergence of Fourier series of Lp-functions on non-abelian compact groups. On the other hand, we obtain as a byproduct the dimension free bounds of noncommutative Hardy-Littlewood maximal inequalities associated with convex bodies. As an ingredient, our proof also provides a refined version of Junge-Le Merdy-Xu's square function estimates Hp(M) ≃ L • p (M) when p → 1.

Maximal singular integral operators acting on noncommutative $L_p$-spaces

“…As far as we know, except some non-sharp results for Bochner-Riesz means in [4,20], there does not exist in the literature any other non-trivial noncommutative maximal inequalities for families of non-positive linear operators, such as the truncated Calderón-Zygmund operators and Dirichlet means.…”

“…Motivated by quantum mechanics, operator algebra, noncommutative geometry and quantum probability, noncommutative harmonic analysis gains rapid development recently and there are many fundamental works appeared (see e.g. [22,32,35,4,45,44,25,26,11,12,23,24,27,38,20]). Due to the noncommutativity, many real variable tools or methods such as maximal functions, stopping times etc are not available, which impose numerous difficulties in developing noncommutative theory.…”

“…Even though the noncommutative maximal inequalities for ergodic averages and conditional expectations have now been established, there appear a lot of difficulties to obtain maximal inequalities for other families of linear operators. For instance, Mei [32] has to invent an ingenious idea to show the noncommutative Hardy-Littlewood maximal inequalities based on Junge's noncommutative Doob's inequalities; in [18], Hong et al exploited the group structure or probabilistic method to show the noncommutative maximal ergodic inequalities associated with the action of groups of polynomial growth; since the lack of estimates of Fourier multipliers on group algebras, the first author and his collaborators [20] invented some quantum semigroup and analyze carefully its difference with Fourier multiplier to establish the maximal inequalities and show the pointwise convergence of noncommutative Fourier series. We refer the reader to the above mentioned papers and references therein for more results on noncommutative maximal inequalities.…”

Sharp estimates of noncommutative Bochner-Riesz means on two-dimensional quantum tori

“…On the other hand, many useful theories in harmonic analysis, such as Littlewood-Paley-Stein square functions, Hardy-Littlewood maximal operators, duality of H 1 -BMO, Calderón-Zygmund operators and multiplier operators, have been successfully transferred to the noncommutative setting (see e.g. [4,21,23,35,36,39,63]). Motivated by the development of this noncommutative harmonic analysis, the noncommutative Bochner-Riesz means on quantum tori have been investigated partially by Z. Chen, Q. Xu and Z. Yin [7] with limited indexes of λ and p. Due to the lack of commutativity, the study of noncommutative Bochner-Riesz means seems to be more challenging.…”

Lower bounds in $L^p$-transference for crossed-products

“…We will denote their associated L p -spaces by L p (LΓ). Asking whether, given a symbol m, their associate Fourier and Herz-Schur multipliers are bounded over L p (LΓ) and S p (ℓ 2 Γ) respectively its an extremely difficult question that has received attention for its connections with approximations properties [LdlS11], the theory of Markovian semigroups [GPJP17,JMP18] as well as the convergence of Fourier series over non-Abelian groups [HWW20] among other problems. For noncommutative L p -spaces, the analogues of points (B) and (C) are widely open.…”