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Register a new userTauberian constants associated to centered translation invariant density bases
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This paper provides a necessary and sufficient condition on Tauberian constants associated to a centered translation invariant differentiation basis so that the basis is a density basis. More precisely, given $x in mathbb{R}^n$, let $mathcal{B} = cup_{x in mathbb{R}^n} mathcal{B}(x)$ be a collection of bounded open sets in $mathbb{R}^n$ containing $x$. Suppose moreover that these collections are translation invariant in the sense that, for any two points $x$ and $y$ in $mathbb{R}^n$ we have that $mathcal{B}(x + y) = {R + y : R in mathcal{B}(x)}.$ Associated to these collections is a maximal operator $M_{mathcal{B}}$ given by $$M_{mathcal{B}}f(x) :=sup_{R in mathcal{B}(x)} frac{1}{|R|} int_R |f|.$$ The Tauberian constants $C_{mathcal{B}}(alpha)$ associated to $M_{mathcal{B}}$ are given by $$C_{mathcal{B}}(alpha) :=sup_{E subset mathbb{R}^n atop 0 < |E| < infty} frac{1}{|E|}|{x in mathbb{R}^n :, M_{mathcal{B}}chi_E(x) > alpha}|.$$ Given $0 < r < infty$, we set $mathcal{B}_r(x) :={R in mathcal{B}(x) : mathrm{diam } R < r}$, and let $mathcal{B}_r :=cup_{x in mathbb{R}^n} mathcal{B}_r (x).$ We prove that $mathcal{B}$ is a density basis if and only if, given $0 < alpha < infty$, there exists $ r = r(alpha) >0$ such that $C_{mathcal{B}_r}(alpha) < infty$. Subsequently, we construct a centered translation invariant density basis $mathcal{B} = cup_{x in mathbb{R}^n} mathcal{B}(x)$ such that there does not exist any $0 < r$ satisfying $C_{mathcal{B}_{r}}(alpha) < infty$ for all $0 < alpha < 1$.
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This paper concerns the smoothness of Tauberian constants of maximal operators in the discrete and ergodic settings. In particular, we define the discrete strong maximal operator $tilde{M}_S$ on $mathbb{Z}^n$ by [ tilde{M}_S f(m) := sup_{0 in R subset mathbb{R}^n}frac{1}{#(R cap mathbb{Z}^n)}sum_{ jin R cap mathbb{Z}^n} |f(m+j)|,qquad min mathbb{Z}^n, ] where the supremum is taken over all open rectangles in $mathbb{R}^n$ containing the origin whose sides are parallel to the coordinate axes. We show that the associated Tauberian constant $tilde{C}_S(alpha)$, defined by [ tilde{C}_S(alpha) := sup_{substack{E subset mathbb{Z}^n 0 < #E < infty} } frac{1}{#E}#{m in mathbb{Z}^n:, tilde{M}_Schi_E(m) > alpha}, ] is Holder continuous of order $1/n$. Moreover, letting $U_1, ldots, U_n$ denote a non-periodic collection of commuting invertible transformations on the non-atomic probability space $(Omega, Sigma, mu)$ we define the associated maximal operator $M_S^ast$ by [ M^ast_{S}f(omega) := sup_{0 in R subset mathbb{R}^n}frac{1}{#(R cap mathbb{Z}^n)}sum_{(j_1, ldots, j_n)in R}|f(U_1^{j_1}cdots U_n^{j_n}omega)|,qquad omegainOmega. ] Then the corresponding Tauberian constant $C^ast_S(alpha)$, defined by [ C^ast_S(alpha) := sup_{substack{E subset Omega mu(E) > 0}} frac{1}{mu(E)}mu({omega in Omega :, M^ast_Schi_E(omega) > alpha}), ] also satisfies $C^ast_S in C^{1/n}(0,1).$ We will also see that, in the case $n=1$, that is in the case of a single invertible, measure preserving transformation, the smoothness of the corresponding Tauberian constant is characterized by the operator enabling arbitrarily long orbits of sets of positive measure.
Let $mathsf M$ and $mathsf M _{mathsf S}$ respectively denote the Hardy-Littlewood maximal operator with respect to cubes and the strong maximal operator on $mathbb{R}^n$, and let $w$ be a nonnegative locally integrable function on $mathbb{R}^n$. We define the associated Tauberian functions $mathsf{C}_{mathsf{HL},w}(alpha)$ and $mathsf{C}_{mathsf{S},w}(alpha)$ on $(0,1)$ by [ mathsf{C}_{mathsf{HL},w}(alpha) :=sup_{substack{E subset mathbb{R}^n 0 < w(E) < infty}} frac{1}{w(E)}w({x in mathbb{R}^n : mathsf M chi_E(x) > alpha}) ] and [ mathsf{C}_{mathsf{S},w}(alpha) := sup_{substack{E subset mathbb{R}^n 0 < w(E) < infty}} frac{1}{w(E)}w({x in mathbb{R}^n : mathsf M _{mathsf S}chi_E(x) > alpha}). ] Utilizing weighted Solyanik estimates for $mathsf M$ and $mathsf M_{mathsf S}$, we show that the function $mathsf{C}_{mathsf{HL},w} $ lies in the local Holder class $C^{(c_n[w]_{A_{infty}})^{-1}}(0,1)$ and $mathsf{C}_{mathsf{S},w} $ lies in the local Holder class $C^{(c_n[w]_{A_{infty}^ast})^{-1}}(0,1)$, where the constant $c_n>1$ depends only on the dimension $n$.
In this paper we continue to study {it quasi associated homogeneous distributions rm{(}generalized functionsrm{)}} which were introduced in the paper by V.M. Shelkovich, Associated and quasi associated homogeneous distributions (generalized functions), J. Math. An. Appl., {bf 338}, (2008), 48-70. [arXiv:math/0608669]. For the multidimensional case we give the characterization of these distributions in the terms of the dilatation operator $U_{a}$ (defined as $U_{a}f(x)=f(ax)$, $xin bR^n$, $a >0$) and its generator $sum_{j=1}^{n}x_jfrac{partial}{partial x_j}$. It is proved that $f_kin {cD}(bR^n)$ is a quasi associated homogeneous distribution of degree $lambda$ and of order $k$ if and only if $bigl(sum_{j=1}^{n}x_jfrac{partial}{partial x_j}-lambdabigr)^{k+1}f_{k}(x)=0$, or if and only if $bigl(U_a-a^lambda Ibigr)^{k+1}f_k(x)=0$, $forall , a>0$, where $I$ is a unit operator. The structure of a quasi associated homogeneous distribution is described.
For Riesz-like kernels $K(x,y)=f(|x-y|)$ on $Atimes A$, where $A$ is a compact $d$-regular set $Asubset mathbb{R}^p$, we prove a minimum principle for potentials $U_K^mu=int K(x,y)dmu(x)$, where $mu$ is a Borel measure supported on $A$. Setting $P_K(mu)=inf_{yin A}U^mu(y)$, the $K$-polarization of $mu$, the principle is used to show that if ${ u_N}$ is a sequence of measures on $A$ that converges in the weak-star sense to the measure $ u$, then $P_K( u_N)to P_K( u)$ as $Nto infty$. The continuous Chebyshev (polarization) problem concerns maximizing $P_K(mu)$ over all probability measures $mu$ supported on $A$, while the $N$-point discrete Chebyshev problem maximizes $P_K(mu)$ only over normalized counting measures for $N$-point multisets on $A$. We prove for such kernels and sets $A$, that if ${ u_N}$ is a sequence of $N$-point measures solving the discrete problem, then every weak-star limit measure of $ u_N$ as $N to infty$ is a solution to the continuous problem.
We bring a precision to our cited work concerning the notion of Borel measures, as the choice among different existing definitions impacts on the validity of the results.
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