Elementary estimates for the Riesz kernel and for its derivative are given. Using these we show that the maximal operator of the Riesz means of a tempered distribution is bounded from $H_p\left(ℝ\right)$ to $L_p\left(ℝ\right)$ (1/(α+1) < p < ∞) and is of weak type (1,1), where $H_p\left(ℝ\right)$ is the classical Hardy space. As a consequence we deduce that the Riesz means of a function $⨍\in L_1\left(ℝ\right)$ converge a.e. to ⨍. Moreover, we prove that the Riesz means are uniformly bounded on $H_p\left(ℝ\right)$ whenever 1/(α+1) < p < ∞. Thus, in case $⨍\in H_p\left(ℝ\right)$, the Riesz means converge...

We give a method for constructing functions $\phi$ and $\psi$ for which $H(x,y)=\phi \left(x\right)-\psi \left(y\right)$ has a specified subharmonic minorant $h(x,y)$. By a theorem of B. Cole, this procedure establishes integral mean inequalities for conjugate functions. We apply this method to deduce sharp inequalities for conjugates of functions in the class $L{log}^{\alpha }L$, for $-1\le \alpha \&lt;\infty$. In particular, the case $\alpha =1$ yields an improvement of Pichorides’ form of Zygmund’s classical inequality for the conjugates of functions in $LlogL$. We also apply the method to produce a new proof of the...

We study the “Fourier symmetry” of measures and distributions on the circle, in relation with the size of their supports. The main results of this paper are:(i) A one-side extension of Frostman’s theorem, which connects the rate of decay of Fourier transform of a distribution with the Hausdorff dimension of its support;(ii) A construction of compacts of “critical” size, which support distributions (even pseudo-functions) with anti-analytic part belonging to $l^2$.We also give examples of non-symmetry...

Let U be a trigonometrically well-bounded operator on a Banach space , and denote by ${ₙ\left(U\right)}_{n=1}^{\infty }$ the sequence of (C,2) weighted discrete ergodic averages of U, that is, $ₙ\left(U\right)=1/n{\sum }_{0<\left|k\right|\le n}(1-|k|/(n+1))U^k$. We show that this sequence ${ₙ\left(U\right)}_{n=1}^{\infty }$ of weighted ergodic averages converges in the strong operator topology to an idempotent operator whose range is x ∈ : Ux = x, and whose null space is the closure of (I - U). This result expands the scope of the traditional Ergodic Theorem, and thereby serves as a link between Banach space spectral theory and...

This paper gives a survey of some recent developments in the spectral theory of linear operators on Banach spaces in which the Hilbert transform and its abstract analogues play a fundamental role.

This paper addresses the recovery of piecewise smooth functions from their discrete data. Reconstruction methods using both pseudo-spectral coefficients and physical space interpolants have been discussed extensively in the literature, and it is clear that an a priori knowledge of the jump discontinuity location is essential for any reconstruction technique to yield spectrally accurate results with high resolution near the discontinuities. Hence detection of the jump discontinuities is critical...

This paper addresses the recovery of piecewise smooth functions from their discrete data. Reconstruction methods using both pseudo-spectral coefficients and physical space interpolants have been discussed extensively in the literature, and it is clear that an a priori knowledge of the jump discontinuity location is essential for any reconstruction technique to yield spectrally accurate results with high resolution near the discontinuities. Hence detection of the jump discontinuities is critical...

It is shown that the Muckenhoupt structure constants for f and f* on the real line are the same.

The Stein-Weiss theorem that the distribution function of the Hilbert transform of the characteristic function of E depends only on the measure of E is generalized for the ergodic Hilbert transform in the case of a one-parameter flow of measure-preserving transformations on a σ-finite measure space.

The transplantation operators for the Hankel transform are considered. We prove that the transplantation operator maps an integrable function under certain conditions to an integrable function. As an application, we obtain the L¹-boundedness and H¹-boundedness of Cesàro operators for the Hankel transform.

In this paper we study integral operators of the form $$Tf\left(x\right)=\int |x-a_1{y|}^{-{\alpha }_1}\cdots {|x-a_{m}y|}^{-{\alpha }_m}f\left(y\right)\mathrm{d}y,$$${\alpha }_1+\cdots +{\alpha }_{}$