Speaker: Alexander Meskhi, Kutaisi International University and A. Razmadze Mathematical Institute

Time : 15:00 - 16.00 CEST (Rome/Paris)

Hosted at: SISSA, International School of Advanced Studies, Trieste, Italy

Zoom :  A zoom meeitng link will appear here, one hour before the talk

Organizers : Pavan Pranjivan Mehta* (pavan.mehta@sissa.it) and Arran Fernandez** (arran.fernandez@emu.edu.tr)

* SISSA, International School of Advanced Studies, Italy

** Eastern Mediterranean University, Northern Cyprus

Keywords: trace inequality, two-weight inequality, fractional integral operator

Abstract: Trace inequality for fractional integral operator K_α:
∥K_αf ∥_{L_μ^p}  ≤  C∥f ∥_L^p        (1)

plays an important role in harmonic analysis and PDEs. There exist various criteria on a Borel measure $\mu$ governing inequality \eqref{star}. Transparent necessary and sufficient conditions on $\mu$ ensuring the trace inequality will be discussed.

One of our aims is to discuss the problem of finding an appropriate Lorentz space $L^{p,s}$ such that the well-known D. Adams-type condition on a measure $\mu$ is both necessary and sufficient for the validity of the trace inequality

∥I_αf ∥_{L_μ^p}  ≤  C∥f ∥_L^{p,s}      (2)

where I_α is the Riesz potential. We show that the desired space is $L^{p,1}$. To study this problem was motivated by the fact that the inequality fails for p=s under the D. Adams-type condition on μ. The latter condition is necessary and sufficient for the validity of the trace inequality (1) for K_α = I_α if and only if 1 < p < q < ∞.

Furthermore, we give a complete characterization of the trace inequality for some multilinear fractional integral operators $T_{\alpha}$:
\[
\|T_{\alpha}(f_1,\cdots,f_m)\|_{L_{\mu}^p}\leq C\prod_{k=1}^{m}\|f_k\|_{L^{p_k}}       (3)
\]
Some related inequalities will also be discussed. For example, necessary and sufficient condition on a (non-doubling) measure $\mu$ for which the following inequality holds

∥J_γ,μ f ∥_{L_μ^{p,q}}  ≤  C∥f ∥_L^{r,s}         (4)

for a fractional integral operator J_γ,μ defined with respect to μ . The latter problem for the classical Lebesgue spaces was studied in [1].

The talk is mainly based on the papers [2-6].

Biography:  A. Meskhi is a President of the Georgian Mathematician Union since 2022. In 2024 he was elected as a member of the Georgian National Academy of Sciences. In 1998 A. Meskhi depended his PhD thesis. In 2001 he got degree of Dr. of Science. In 2003-2005 he was the Postdoc of ``Scuola Normale Superiore'' of Pisa. A. Meskhi was awarded by Euler Premium for young scientists established by the German Mathematical Association (2000), Award of the Georgian Mathematical Union for the Best Scientific Works (2002, 2009); Andrea Razmadze Prize of the Georgian National Academy of Sciences (2012). In 2017-2022 he was a Scholar of the Georgian National Academy of Sciences. Since 2022 A. Meskhi is Chair of the Organizing committee of the Annual International Conference of the Georgian Mathematical Union, Batumi, Georgia. A. Meskhi is an Editor in Chief of Transactions of A. Razmadze Mathematical Institute. He is head of the department of Mathematical Analysis at A. Razmadze Mathematical Institute of I. Javakhishvili Tbilisi State University and Professor at Kutaisi International University. Since 2016 he is an Invited Professor at San Diego State University (Georgian Branch). He is an author/co-author of 7 monographs and about 150 scientific papers. He was supervisor of 6 PhD theses.

Bibliography

[1]  D. E. Edmunds, V. Kokilashvili and A. Meskhi, Bounded and Compact Integral Operators, Kluwer Academic Publishers, Dordrecht, Boston, London, 2002.
[2] A.Meskhi, Measure of Non-compactness for Integral Operators in Weighted Lebesgue Spaces. Nova Science Publishers, New York, 2009.
[3] V. Kokilashvili, A. Meskhi and L. E. Persson, Weighted Norm Inequalities for Integral Transforms with Product Kernels,Nova Science Publishers, New York, 2010.
[4] V. Kokilashvili, A. Meskhi, H. Rafeiro and S. Samko, Integral operators in non- standard function spaces, Volume I: Variable exponent Lebesgue and amalgam spaces, Birkh¨auser/ Springer, Heidelberg, 2016, pp. 1-567.
[5] V. Kokilashvili, A. Meskhi, H. Rafeiro and S. Samko, Integral operators in non-standard function spaces, Volume II: Variable exponent H¨older, Morrey-Campanato and grand spaces, Birkh¨auser/Springer, Heidelberg, 2016, pp. 568-1003.
[6] V. Kokilashvili, A. Meskhi, H. Rafeiro and S. Samko, Integral operators in non-standard function spaces, Volume III: Advances in Grand Function Spaces, Birkh¨auser/ Springer, Helm, 2024, pp 1004-1526.
[7] V. Kokilashvili, A. Meskhi, H. Rafeiro and S. Samko, Integral operators in non-standard function spaces , Volume IV : Progress in Morrey-type spaces and related topic, Birkh¨auser/ Springer, Helm, 1527-2077, 2025.