Searcharxiv⌕ Search

arXiv subjects

Andrei K. Lerner

Publications and source records attributed to Andrei K. Lerner.

40 records · Page 3Linked to original sources
arXiv2011

Mixed $A_p$-$A_r$ inequalities for classical singular integrals and Littlewood-Paley operators

We prove mixed $A_p$-$A_r$ inequalities for several basic singular integrals, Littlewood-Paley operators, and the vector-valued maximal function. Our key point is that $r$ can be taken arbitrary big. Hence such inequalities are close in spirit to those obtained recently in the works by T. Hytönen and C. Pérez, and M. Lacey. On one hand, the "$A_p$-$A_{\infty}$" constant in these works involves two independent suprema. On the other hand, the "$A_p$-$A_r$" constant in our estimates involves a joint supremum but of a bigger expression. We show in simple examples that both such constants are incomparable. This leads to a natural conjecture that the estimates of both types can be further improved.

math.CA↗
arXiv2010

Sharp weighted norm inequalities for Littlewood-Paley operators and singular integrals

We prove sharp $L^p(w)$ norm inequalities for the intrinsic square function (introduced recently by M. Wilson) in terms of the $A_p$ characteristic of $w$ for all $1<p<\infty$. This implies the same sharp inequalities for the classical Lusin area integral $S(f)$, the Littlewood-Paley $g$-function, and their continuous analogs $S_ψ$ and $g_ψ$. Also, as a corollary, we obtain sharp weighted inequalities for any convolution Calderón-Zygmund operator for all $1<p\le 3/2$ and $3\le p<\infty$, and for its maximal truncations for $3\le p<\infty$.

math.CA↗
arXiv2004

On modular inequalities in variable L^p spaces

We show that the Hardy-Littlewood maximal operator and a class of Calderón-Zygmund singular integrals satisfy the strong type modular inequality in variable $L^p$ spaces if and only if the variable exponent $p(x)\sim const$.

math.CA↗