In this paper we establish that several maximal operators of convolution type, associated to elliptic and parabolic equations, are variation-diminishing. Our study considers maximal operators on the Euclidean space R d , on the torus T d and on the sphere S d . The crucial regularity property that these maximal functions share is that they are subharmonic in the corresponding detachment sets.where B 1 is the unit ball centered at the origin and m(B 1 ) is its ddimensional Lebesgue measure. In this case, due to the work of Kurka [13], the one-dimensional estimate (1.2) is known to hold with constant C = 240, 004, but the problem with C = 1 remains open. For the onedimensional right (or left) Hardy-Littlewood maximal operator, i.e. when ϕ(x, t) = 1 t χ [0,1] (x/t), estimate (1.2) holds with C = 1 due to the work of Tanaka [21]. The sharp bound (1.2) with constant C = 1 also Date: October= 4, 2018. 2010 Mathematics Subject Classification. 42B25, 46E35, 35B50, 31B05, 35J05, 35K08.holds for the one-dimensional uncentered version of this operator, as proved by Aldaz and Pérez Lázaro [1].Higher dimensional analogues of (1.2) for the Hardy-Littlewood maximal operator, centered or uncentered, are open problems (see, for instance, the work of Haj lasz and Onninen [9]). Other interesting works related to the regularity of the Hardy-Littlewood maximal operator and its variants, when applied to Sobolev and BV functions,are [2,3,4,5,8,10,11,12,14,15,20,22].In the precursor of this work [6, Theorems 1 and 2], Carneiro and Svaiter proved the variation-diminishing property, i.e. inequality (1.2) with C = 1, for the maximal operators associated to the Poisson kernel(1.3) and the Gauss kernelTheir proof is based on an interplay between the analysis of the maximal functions and the structure of the underlying partial differential equations (Laplace's equation and heat equation). The aforementioned examples are the only maximal operators of convolution type for which inequality (1.2) has been established (even allowing a constant C > 1). Maximal operators associated to elliptic equations. A question that derives from our precursor[6] is whether the variation-diminishing property is a peculiarity of the smooth kernels (1.3) and (1.4) or if these can be seen as particular cases of a general family. One could, for example, look at the semigroup structure via the Fourier transforms 1 (in space) of these kernels:A reasonable way to connect these kernels would be to consider the one-parameter family ϕ α (ξ, t) = e −t(2π|ξ|) α , for 1 ≤ α ≤ 2. However, in this case, the function u(x, t) = ϕ α (·, t) * u 0 (x) solves an evolution equation related to the fractional Laplacian u t + (−∆) α/2 u = 0 , for which we do not have a local maximum principle, essential to run the argument of Carneiro and Svaiter in [6]. The problem of proving that the corresponding maximal operator is variation-diminishing seems more delicate and it is currently open.
For L(·, π) in a large class of L-functions, assuming the generalized Riemann hypothesis, we show an explicit bound for the function S 1 (t, π) = 1 π ∞ 1/2 log |L(σ + it, π)| dσ, expressed in terms of its analytic conductor. This enables us to give an alternative proof of the most recent (conditional) bound for S(t, π) = 1 π arg L 1 2 + it, π , which is the derivative of S 1 (·, π) at t.
In this article, we establish polynomial-growth bound for the sequence of Fourier coefficients associated to even integer weight vector-valued automorphic forms of Fuchsian groups of the first kind. At the end, their L-functions and exponential sums have been discussed.
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