We determine extremal entire functions for the problem of majorizing, minorizing, and approximating the Gaussian function e −πλx 2 by entire functions of exponential type. The combination of the Gaussian and a general distribution approach provides the solution of the extremal problem for a wide class of even functions that includes most of the previously known examples (for instance [3], [4], [10] and [17]), plus a variety of new interesting functions such as |x| α for −1 < α; log (x 2 + α 2 )/(x 2 + β 2 ) , for 0 ≤ α < β; log x 2 +α 2 ; and x 2n log x 2 , for n ∈ N. Further applications to number theory include optimal approximations of theta functions by trigonometric polynomials and optimal bounds for certain Hilbert-type inequalities related to the discrete Hardy-Littlewood-Sobolev inequality in dimension one.

Let u : R × R n → C be the solution of the linear Schrödinger equationIn the first part of this paper we obtain a sharp inequality for the Strichartz norm u(t, x) L 2k t L 2kx (R×R n ) , where k ∈ Z, k ≥ 2 and (n, k) = (1, 2), that admits only Gaussian maximizers. As corollaries we obtain sharp forms of the classical Strichartz inequalities in low dimensions (works of Foschi [4] and Hundertmark -Zharnitsky [6]) and also sharp forms of some Sobolev-Strichartz inequalities. In the second part of the paper we express Foschi's [4] sharp inequalities for the Schrödinger and wave equations in the broader setting of sharp restriction/extension estimates for the paraboloid and the cone.

Let S(t) = 1 π arg ζ 1 2 + it be the argument of the Riemann zeta-function at the point 1 2 + it. For n ≥ 1 and t > 0 define its iterateswhere δn is a specific constant depending on n and S 0 (t) := S(t). In 1924, J. E. Littlewood proved, under the Riemann hypothesis (RH), that Sn(t) = O(log t/(log log t) n+1 ). The order of magnitude of this estimate was never improved up to this date. The best bounds for S(t) and S 1 (t) are currently due to Carneiro, Chandee and Milinovich. In this paper we establish, under RH, an explicit form of this estimate 2010 Mathematics Subject Classification. 11M06, 11M26, 41A30.

In this paper we study the regularity properties of fractional maximal operators acting on BVfunctions. We establish new bounds for the derivative of the fractional maximal function, both in the continuous and in the discrete settings.Question A. (Haj lasz and Onninen [10]) Is the operator f → |∇M f | bounded from W 1,1 (R d ) to L 1 (R d )?A standard dilation argument reveals the true nature of this question: whether the variation of the maximal function is controlled by the variation of the original function, i.e. if we haveProgress on this problem has been restricted to dimension d = 1. For the uncentered maximal operator (defined similarly as in (1.1), with the supremum taken over all balls containing the point x in its closure), which we denote here by M , Tanaka [22] showed that if f ∈ W 1,1 (R) then M f is weakly differentiable andIn the case 1 ≤ β < d, Question B admits a positive answer, which follows from the main result of Kinnunen and Saksman in their aforementioned work [13]. In fact, [13, Theorem 3.1] states the following regularizing effect: if f ∈ L r (R d ) with 1 < r < d and 1 ≤ β < d/r, then M β f is weakly differentiable and C(d, β) is a universal constant. In our case, given 1 ≤ β < d and f ∈ W 1,1 (R d ), by the Sobolev embedding we have f ∈ L p * (R d ), where p * = d/ (d − 1), and hence f ∈ L r (R d ) for any 1 ≤ r ≤ p * . We may choose r with 1 < r < d such that 1 ≤ β < d/r and hence (1.5) holds.ThenQuestion B (in the case 0 ≤ β < 1) is the main motivation for this work. The presence of the fractional part introduces additional difficulties as we shall see in the course of the paper (e.g. one does not necessarily have M β (f )(x) ≥ |f (x)| a.e.). Here we give a positive answer to Question B in dimension d = 1 for the uncentered

Abstract. In this work we obtain optimal majorants and minorants of exponential type for a wide class of radial functions on R N . These extremal functions minimize the L 1 (R N , |x| 2ν+2−N dx)-distance to the original function, where ν > −1 is a free parameter. To achieve this result we develop new interpolation tools to solve an associated extremal problem for the exponential function F λ (x) = e −λ|x| , where λ > 0, in the general framework of de Branges spaces of entire functions. We then specialize the construction to a particular family of homogeneous de Branges spaces to approach the multidimensional Euclidean case. Finally, we extend the result from the exponential function to a class of subordinated radial functions via integration on the parameter λ > 0 against suitable measures. Applications of the results presented here include multidimensional versions of Hilbert-type inequalities, extremal one-sided approximations by trigonometric polynomials for a class of even periodic functions and extremal one-sided approximations by polynomials for a class of functions on the sphere S N−1 with an axis of symmetry.