Related papers: Sharp total variation results for maximal function…
We prove that if $f:I\subset \Bbb R\to \Bbb R$ is of bounded variation, then the noncentered maximal function $Mf$ is absolutely continuous, and its derivative satisfies the sharp inequality $\|DMf\|_1\le |Df|(I)$. This allows us obtain,…
In this paper we establish new optimal bounds for the derivative of some discrete maximal functions, both in the centered and uncentered versions. In particular, we solve a question originally posed by Bober, Carneiro, Hughes and Pierce.
We prove sharp local and global variation bounds for the centred Hardy--Littlewood maximal functions of indicator functions in one dimension. We characterise maximisers, treat both the continuous and discrete settings and extend our results…
A well known conjecture states that constant functions are extremizers of the $L^2 \to L^6$ Tomas-Stein extension inequality for the circle. We prove that functions supported in a $\sqrt{6}/80$-neighbourhood of a pair of antipodal points on…
We study sharp $p$-variational inequalities for the Hardy-Littlewood maximal operator on complete graphs, answering in the affirmative a question by Feng Liu and Qingying Xue. We also use computational assistance to find sharp constants in…
We study the problem concerning the variation of the Hardy-Littlewood maximal function in higher dimensions. As the main result, we prove that the variation of the non-centered Hardy-Littlewood maximal function of a radial function is…
We give an alternative proof of a sharp generalization of an integral inequality for the dyadic maximal operator due to which the evaluation of the Bellman function of this operator with respect to two variables, is possible. This last…
Let $M$ denote the centered Hardy--Littlewood operator on $\mathbb{R}$. We prove that \[ {\rm Var} (Mf)\le {\rm Var} (f) - \frac12\big| |f(\infty)|-|f(-\infty)|\big| \] for piecewise constant functions $f$ with nonzero and zero values…
We consider the maximal operator with respect to uncentered cubes on Euclidean space with arbitrary dimension. We prove that for any function with bounded variation, the variation of its maximal function is bounded by the variation of the…
To estimate the optimal constant in Hardy-type inequalities, some variational formulas and approximating procedures are introduced. The known basic estimates are improved considerably. The results are illustrated by typical examples. It is…
In this note we give simple proofs of several results involving maximal truncated Calde\'on-Zygmund operators in the general setting of rearrangement invariant quasi-Banach function spaces by sparse domination. Our techniques allow us to…
The aim of this short note is to give an alternative proof, which applies to functions of bounded variation in arbitrary domains, of an inequality by Maz'ya that improves Friedrichs inequality. A remarkable feature of such a proof is that…
The inverse tangent function can be bounded by different inequalities, for example by Shafer's inequality. In this publication, we propose a new sharp double inequality, consisting of a lower and an upper bound, for the inverse tangent…
In this paper, we will establish the best constants for certain classes of weighted Moser-Trudinger inequalities on the entire Euclidean spaces $\mathbb{R}^N$. We will also prove the existence of maximizers of these sharp weighted…
We obtain concentration and large deviation for the sums of independent and identically distributed random variables with heavy-tailed distributions. Our concentration results are concerned with random variables whose distributions satisfy…
In this paper we prove sharp Hardy inequalities by using Maximal function theory. Our results improve and extend the well-known results of G.Hardy \cite{Ha04}, T.Cazenave \cite {Ca03}, J.-Y.Chemin\cite {Ch06} and T.Tao\cite {TT06}.
For a real c\`{a}dl\`{a}g function f and a positive constant c we find another c\`{a}dl\`{a}g function, which has the smallest total variation pos- sible among all functions uniformly approximating f with accuracy c/2. The solution is…
The focus of this article is the approximation of functions which are analytic on a compact interval except at the endpoints. Typical numerical methods for approximating such functions depend upon the use of particular conformal maps from…
In this paper, new upper and lower bounds for the Trapezoid inequality of absolutely continuous functions are obtained. Applications to some special means are provided as well.
We study in this paper the function approximation error of linear interpolation and extrapolation. Several upper bounds are presented along with the conditions under which they are sharp. All results are under the assumptions that the…