room: A1-33, 10:00–12:00
Vakhtang Kokilashvili (Tbilisi State University)
Integral operators in fully measurable grand Lebesgue spaces and applications
Summary. The goal of talk is to present our recent results dealing with the mapping behavior of integral operators in new function spaces. We plan to discuss the following topics:
- Boundedness criteria of integral operators in weighted fully measurable grand Lebesgue spaces de fined on quasi-metric measure spaces
- Boundedness criteria of integral operators in weighted fully measurable grand Lebesgue spaces defi ned on quasi-metric measure spaces
- One-sided operators
- Integral transforms de fined of product spaces
- Application to the Riemann–Hilbert boundary value problem for analytic function.
Alexander Meskhi (Tbilisi State University and Georgian Technical University)
Weighted extrapolation in grand Lebesgue spaces defined on product sets
Summary. Our aim is present our recent results regarding weighted extrapolation in grand Lebesgue spaces, generally speaking, defined on product sets. In particular, we show that ifthe one-weight inequality holds in the classical weighted Lebesgue space \(L^{p_0}_w (X_1\times \dots\times X_n)\) for a class of pairs of functions \((f, g)\) defined on \(X_1 \times\dots\times X_n\) and for all weights \(w\) from the strong Muckenhoupt class \(A^{(S)}_{p_0} (X_1 \times \dots\times X_n)\), then the one-weight estimate also holdsin grand Lebesgue spaces \(L^{p}_w (X_1\times\dots\times X_n)\) for the same pairs of functions \((f, g)\) and for all Muckenhoupt weights \(w\in A^{(S)}_{p} (X_1\times\dots\times X_n)\). Our results cover both diagonal and off diagonal cases. We deal also with extrapolation from \(A^{(S)}_\infty (X_1\times\dots\times X_n)\) weights. Based on these results we prove new one-weight estimates for single as well as multiple integral operators such as strong maximal, Calderón–Zygmund and fractional integral operators with product kernels in these spaces. Commutators of singular and fractional integrals will be also discussed.
Coffee and the at 9:30 in the professors’ club.