Lp(.) - Lq(.) estimates for convolution operators with singular measures supported on surfaces of half the ambient dimension

URCIUOLO, MARTA SUSANA; VALLEJOS, LUCAS ALEJANDRO

GEORGIAN MATHEMATICAL JOURNAL

HELDERMANN VERLAG

Lugar: Lemgo; Año: 2022

1072-947X

Let $alpha _{i},$ $eta _{i}>0,$ $1leq ileq n$ and, for $t>0$ and $%x=left( x_{1},...,x_{n}ight) in mathbb{R}^{n}$, let $tcdot x=left(t^{alpha _{1}}x_{1},...,t^{alpha _{n}}x_{n}ight) ,$ $tcirc x=left(t^{eta _{1}}x_{1},...,t^{eta _{n}}x_{n}ight) $ and let $alpha =alpha_{1}+...+alpha _{n},$ $Vert xVert _{alpha}=sumlimits_{i=1}^{n}leftert x_{i}ightert ^{rac{1}{alpha _{i}}}$%. Let $arphi _{1},...,arphi _{n}$ be real functions in $C^{infty}left( mathbb{R}^{n}setminus left{ 0ight} ight) $ such that $%arphi =left( arphi _{1},...,arphi _{n}ight) $ is a homogeneousfunction with respect to these groups of dilations i.e, egin{equation*}arphi left( tcdot xight) =tcirc arphi left( xight) .end{equation*}%Let $gamma >0$ and let $mu $ be the Borel measure in $mathbb{R}^{2n}$given by egin{equation*}mu left( Eight) =int chi _{E}left( x,arphi left( xight) ight)leftVert xightVert _{alpha }^{gamma -alpha }dx.end{equation*}%Let $T_{mu }f=mu ast f$, $fin S(mathbb{R}^{2n})$. In this paper westudy the boundedness of $T_{mu }$ from $L^{p(cdot )}(mathbb{R}^{2n})$into $L^{q(cdot )}(mathbb{R}^{2n})$ for certain variable exponents $%p(cdot )$ and $q(cdot )$.