If $x$ is a bounded function on $\mathbf{Z}$, the multiplier with symbol $x$ (denoted by $M_x)$ is defined by $(M_{x}f)\widehat{}=x\widehat{f}$, $f\in L^2\left(\mathbf{T}\right)$. We give some conditions on $x$ ensuring the “interpolation inequality” $\parallel M_{x}f{\parallel }_{L^p}\le C\parallel f{\parallel }_{L^1}^{\alpha }\parallel M_{x}f{\parallel }_{L^q}^{1-\alpha }$ (here $1\<p\<q$ and $\alpha =\alpha (p,q,x)$ is between 0 and 1). In most cases considered $M_x$ fails to have stronger $L^1$-regularity properties (e.g. fails to be of weak type (1,1)). The results are applied to prove that for many sets $E\subset \mathbf{Z}$ every positive sequence in ${\ell }^2\left(E\right)$ can be majorized by the sequence $\{$$...$

Two operator-valued Fourier multiplier theorems for Hölder spaces are proved, one periodic, the other on the line. In contrast to the $L^p$-situation they hold for arbitrary Banach spaces. As a consequence, maximal regularity in the sense of Hölder can be characterized by simple resolvent estimates of the underlying operator.

Let $f\in V_r\left(\right){\cup }_r\left(\right)$, where, for 1 ≤ r < ∞, $V_r\left(\right)$ (resp., ${}_r\left(\right)$) denotes the class of functions (resp., bounded functions) g: → ℂ such that g has bounded r-variation (resp., uniformly bounded r-variations) on (resp., on the dyadic arcs of ). In the author’s recent article [New York J. Math. 17 (2011)] it was shown that if is a super-reflexive space, and E(·): ℝ → () is the spectral decomposition of a trigonometrically well-bounded operator U ∈ (), then over a suitable non-void open interval of r-values, the condition...

We consider the maximal regularity problem for the discrete time evolution equation $u_{n+1}-Tuₙ=fₙ$ for all n ∈ ℕ₀, u₀ = 0, where T is a bounded operator on a UMD space X. We characterize the discrete maximal regularity of T by two types of conditions: firstly by R-boundedness properties of the discrete time semigroup ${\left(Tⁿ\right)}_{n\in \mathbb{N}₀}$ and of the resolvent R(λ,T), secondly by the maximal regularity of the continuous time evolution equation u’(t) - Au(t) = f(t) for all t > 0, u(0) = 0, where A:= T - I. By recent results of...

Modeling of repulsive forces is essential to the understanding of certain bio-physical processes, especially for the motion of DNA molecules. These kinds of phenomena seem to be driven by some sort of “energy” which especially prevents the molecules from strongly bending and forming self-intersections. Inspired by a physical toy model, numerous functionals have been defined during the past twenty-five years that aim at modeling self-avoidance. The general idea is to produce “detangled” curves having...

We present a new criterion for the weighted $L^p-L^q$ boundedness of multiplier operators for Laguerre and Hermite expansions that arise from a Laplace-Stieltjes transform. As a special case, we recover known results on weighted estimates for Laguerre and Hermite fractional integrals with a unified and simpler approach.

2000 Mathematics Subject Classification: 42A45.For a Hilbert space H ⊂ L1loc(R) of functions on R we obtain a representation theorem for the multipliers M commuting with the shift operator S. This generalizes the classical result for multipliers in L2(R) as well as our previous result for multipliers in weighted space L2ω(R). Moreover, we obtain a description of the spectrum of S.

Let X be a closed subspace of $L^p\left(\mu \right)$, where μ is an arbitrary measure and 1 < p < ∞. Let U be an invertible operator on X such that $su{p}_{n\in \mathbb{Z}}\left|\right|Uⁿ\left|\right|<\infty$. Motivated by applications in ergodic theory, we obtain (optimal) conditions for the convergence of series like ${\sum }_{n\ge 1}\left(Uⁿf\right)/n^{1-\alpha }$, 0 ≤ α < 1, in terms of $\left|\right|f+\cdots +U^{n-1}f{\left|\right|}_p$, 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....