Integral estimates for convergent positive series

Uniqueness for spherically convergent multiple trigonometric series

In 1870 Cantor proved that representation of a function of one variable by a trigonometric series can be done in only one way. In 1996 Bourgain proved the same thing for spherical convergence and multiple trigonometric series. His proof involves injecting a lot of new ideas into the theory of uniqueness. We give here an exposition of Bourgain’s proof, specialized to the case of dimension 2. Our...

New estimates for the maximal singular integral

In this paper we pursue the study of the problem of controlling the maximal singular integral T ∗f by the singular integral Tf . Here T is a smooth homogeneous Calderón-Zygmund singular integral of convolution type. We consider two forms of control, namely, in the L2(Rn) norm and via pointwise estimates of T ∗f by M(Tf) or M2(Tf) , where M is the Hardy-Littlewood maximal operator and M2 = M ◦M ...

Weighted Estimates for the Averaging Integral Operator

Let 1 < p ≤ q < +∞ and let v, w be weights on (0,+∞) satisfying: (?) v(x)xis equivalent to a non-decreasing function on (0,+∞) for some ρ ≥ 0; [w(x)x] ≈ [v(x)x] for all x ∈ (0,+∞). We prove that if the averaging operator (Af)(x) := 1 x R x 0 f(t) dt, x ∈ (0,+∞), is bounded from the weighted Lebesgue space Lp((0,+∞); v) into the weighted Lebesgue space Lq((0,+∞);w), then there exists ε0 ∈ (0, p−...

On Absolutely Convergent Dirichlet Series

satisfying zZ\ \^A < °°, if and only if f is) is bounded away from zero in the half-plane 0-3:0. This discovery amounts to a determination of the spectrum of the Banach-algebra element associated with fis), and it thus makes available for the theory of ordinary Dirichlet series the well-known theorem of Gelfand on analytic functions of Banach-algebra elements (see §24D of [7]) and its generaliz...

On Cantor's Series with Convergent