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Register a new userBeyond the spherical sup-norm problem
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We solve the sup-norm problem for non-spherical Maass forms on an arithmetic quotient of G=SL_2(C) with maximal compact K=SU_2(C) when the dimension of the associated K-type gets large. Our results cover the case of vector-valued Maass forms as well as all the individual scalar-valued Maass forms of the Wigner basis. They establish the first subconvex bounds for the sup-norm problem in the K-aspect in a non-abelian situation and yield sub-Weyl exponents in some cases. On the way, we develop theory of independent interest for the group G, including localization estimates for generalized spherical functions of high K-type and a Paley-Wiener theorem for the corresponding spherical transform acting on the space of rapidly decreasing functions.
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We solve the sup-norm problem for spherical Hecke-Maass newforms of square-free level for the group GL(2) over a number field, with a power saving over the local geometric bound simultaneously in the eigenvalue and the level aspect. Our bounds feature a Weyl-type exponent in the level aspect, they reproduce or improve upon all known special cases, and over totally real fields they are as strong as the best known hybrid result over the rationals.
In this paper we prove a hybrid subconvexity bound for class group $L$-functions associated to a quadratic extension $K/mathbb{Q}$ (real or imaginary). Our proof relies on relating the class group $L$-functions to Eisenstein series evaluated at Heegner points using formulas due to Hecke. The main technical contribution is the following uniform sup norm bound for Eisenstein series $E(z,1/2+it)ll_varepsilon y^{1/2} (|t|+1)^{1/3+varepsilon}, ygg 1$, extending work of Blomer and Titchmarsh. Finally, we propose a uniform version of the sup norm conjecture for Eisenstein series.
In the Gaussian white noise model, we study the estimation of an unknown multidimensional function $f$ in the uniform norm by using kernel methods. The performances of procedures are measured by using the maxiset point of view: we determine the set of functions which are well estimated (at a prescribed rate) by each procedure. So, in this paper, we determine the maxisets associated to kernel estimators and to the Lepski procedure for the rate of convergence of the form $(log n/n)^{be/(2be+d)}$. We characterize the maxisets in terms of Besov and Holder spaces of regularity $beta$.
A function F:R^2->R is sup-measurable if F_f:R->R given by F_f(x)=F(x,f(x)), x in R, is measurable for each measurable function f:R->R. It is known that under different set theoretical assumptions, including CH, there are sup-measurable non-measurable functions, as well as their category analog. In this paper we will show that the existence of category analog of sup-measurable non-measurable functions is independent of ZFC. A similar result for the original measurable case is a subject of a work in prepartion by Roslanowski and Shelah.
We consider parabolic systems with nonlinear dynamic boundary conditions, for which we give a rigorous derivation. Then, we give them several physical interpretations which includes an interpretation for the porous-medium equation, and for certain reaction-diffusion systems that occur in mathematical biology and ecology. We devise several strategies which imply (uniform)}$L^{p} and}$L^{infty}$ estimates on the solutions for the initial value problems considered.
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