Mixed weak estimates of Sawyer type for commutators of generalized singular integrals and related operators

FABIO BERRA; MARILINA CARENA; GLADIS PRADOLINI

MICHIGAN MATHEMATICAL JOURNAL

MICHIGAN MATHEMATICAL JOURNAL

Lugar: Michigan; Año: 2019 vol. 68 p. 527 - 564

0026-2285

We study mixed weak type inequalities for the commutator $[b,T]$, where $b$ is a BMO function and $T$ is a Calder´on-Zygmund operator. More precisely, we prove that for every $t>0$egin{equation*}uvleft(left{xinR^n:left|rac{[b,T](fv)(x)}{v(x)}ight|>tight}ight)leqCint_{R^n}Phileft(rac{|f(x)|}{t}ight)u(x)v(x),dx,end{equation*}where $Phi(t)=t(1+log^{+}{t})$, $uin A_1$ and $vinA_{infty}(u)$. Our technique involves the classical Calder´on-Zygmund decomposition, which allow us to give a direct proof without taking into account the associated maximal operator. We use this result to prove an analogous inequality for higher order commutators.For a given Young function $arphi$ we also consider singular integral operators $T$ whose kernels satisfy a $L^{arphi}$-H"{o}rmander property,and we find sufficient conditions on $arphi$ such that a mixed weak estimate holds for $T$ and also for its higher order commutators $T^m_b$.We also obtain a mixed estimation for a wide class of maximal operators associated to certain Young functions of $Llog L$ type which are in intimate relation with the commutators. This last estimate involves an arbitrary weight $u$ and a radial function $v$ which is not even locally integrable.