arXiv subjects
Publications and source records attributed to Eleftherios N. Nikolidakis.
We introduce the space of dyadic bounded mean oscillation functions $f$ defined on $[0,1]^n$ and study the behavior of the nonincreasing rearrangement of $f$, as an element of the space $BMO((0,1])$. We also study the analogous class of functions that satisfy the dyadic Gurov-Reshetnyak condition and look upon their integrability properties.
We prove that for any weight $ϕ$ defined on $[0,1]^n$ that satisfies a reverse Holder inequality with exponent p > 1 and constant $c\ge1$ upon all dyadic subcubes of $[0,1]^n$, it's non increasing rearrangement satisfies a reverse Holder inequality with the same exponent and constant not more than $2^nc-2^n + 1$, upon all subintervals of $[0; 1]$ of the form $[0; t]$. This gives as a consequence, according to the results in [8], an interval $[p; p_0(p; c)) = I{p,c}$, such that for any $q \in I{p,c}$, we have that $ϕ$ is in $L^q$.
We prove a generalization of a Hardy type inequality for negative exponents valid for non-negative functions defined on $(0,1]$. As an application we find the exact best possible range of $p$ such that $1<p\le q$ such that any non-decreasing $ϕ$ which satisfies the Muckenhoupt $A_q$ condition with constant $c$ upon all open subintervals of $(0,1]$ should additionally satisfy the $A_p$ condition for another possibly real constant $c'$. The result have been treated in \cite{9} based on \cite{1}, but we give in this paper an alternative proof which relies on the above mentioned inequality.
For p>1 we find the Bellman function of two variables associated with the dyadic maximal operator on Rn.Actually we do that in the more general setting of tree-like maximal operators.We provide a simple and elementary proof,different from those in [4] and [9].
We will prove that if $ϕ$ belongs to the class $A_1(\R)$ with constant $c\ge1$ then the decreasing rearrangement of $ϕ$, belongs to the same class with constant not more than $c$. We also find for such $ϕ$ the exact best possible range of those $p>1$ for which $ϕ\in L^p$. In this way we provide alternative proofs of the results that appear in \cite{1}.
We obtain sharp estimates for the quasi norm of the maximal function of f when it satisfies certain conditions.