The Prague seminar on function spaces

forthcoming program



June 13, 2018

Abdulhamit Küçükaslan (Mathematical Institute, Czech Academy of Sciences, Prague):

Boundedness of Hilbert transform in local Morrey-Lorentz spaces
Abstract:

In this talk, we investigate the boundedness of the Hilbert transform $H$ in the local Morrey-Lorentz spaces $M_{p,q;\lambda}^{loc}$
$1\le q \le \i$, $\frac{q}{q+\lambda} \leq p \leq \frac{q}{\lambda}$. We prove that the operator $H$ is bounded in $M_{p,q;\lambda}^{loc}$ under the condition $1< q <\i$, $\frac{q}{q+\lambda} < p < \frac{q}{\lambda}$.

In the limiting case $p =\frac{q}{q+\lambda}, 1<q<\i$, we prove that the operator $H$ is bounded from the space $M_{p,q;\lambda}^{loc}$ to the weak local Morrey-Lorentz space $WM_{p,q;\lambda}^{loc}$. Also we show that for the limiting case  $p =\frac{q}{\lambda}, 0<q\leq \i$, the modified Hilbert transform $\widetilde{H}$ is bounded from the space $M_{p,q;\lambda}^{loc}$ to the bounded mean oscillation space $BMO$.



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