Carleson's Theorem from 1965 states that the partial Fourier sums of a square integrable function converge pointwise. We prove an equivalent statement on the real line, following the method developed by the author and C. Thiele. This theorem, and the proof presented, is at the center of an emerging theory which complements the statement and proof of Carleson's theorem. An outline of these variations is also given.
It is shown that the operator below maps into itself for 1 < p < ∞. . The supremum over b alone gives the famous theorem of L. Carleson [2] on the pointwise convergence of Fourier series. The supremum over a alone is an observation of E. M. Stein [12]. The method of proof builds upon Stein’s observation and an approach to Carleson’s theorem jointly developed by the author and C. M. Thiele [7].
Let the coefficients of a lacunary cosine series be bounded and not square-summable. Then the partial sums of the series are recurrent.
We introduce p-quasilocal operators and prove that if a sublinear operator T is p-quasilocal and bounded from to then it is also bounded from the classical Hardy space to (0 < p ≤ 1). As an application it is shown that the maximal operator of the one-parameter Cesàro means of a distribution is bounded from to (3/4 < p ≤ ∞) and is of weak type . We define the two-dimensional dyadic hybrid Hardy space and verify that the maximal operator of the Cesàro means of a two-dimensional...
Let be a finite subset of an abelian group and let be a closed half-plane of the complex plane, containing zero. We show that (unless possesses a special, explicitly indicated structure) there exists a non-trivial Fourier coefficient of the indicator function of which belongs to . In other words, there exists a non-trivial character such that .
We characterize associate spaces of generalized weighted weak-Lorentz spaces and use this characterization to study embeddings between these spaces.
Let H be a separable Hilbert space, L(H) be the algebra of all bounded linear operators of H and Bess(H) be the set of all Bessel sequences of H. Fixed an orthonormal basis E = {ek}k∈N of H, a bijection αE: Bess(H) → L(H) can be defined. The aim of this paper is to characterize α-1E (A) for different classes of operators A ⊆ L(H). In particular, we characterize the Bessel sequences associated to injective operators, compact operators and Schatten p-classes.