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Kanat Tulenov

Publications and source records attributed to Kanat Tulenov.

6 recordsLinked to original sources
arXiv2026

Fujita exponents on quantum Euclidean spaces

We study the well-posedness of a non-linear heat equation with power nonlinearity with positive initial data on quantum Euclidean spaces. We prove a noncommutative analogue of the classical Fujita theorem by identifying the critical exponent separating finite-time blow-up from global existence for small initial data. Moreover, we establish a fundamental inequality in general semifinite von Neumann algebras that is of independent interest and plays a crucial role in the study of global existence and local well-posedness of solutions of nonlinear equations in noncommutative setting.

math.AP↗
arXiv2026

Hormander-Mikhlin type theorem on non-commutative spaces

In this paper, we introduce a Fourier-type formalism on non-commutative spaces. As a result, we obtain two versions of Hormander-Mikhlin Lp-multiplier theorem: on locally compact Kac groups and on semi-finite von Neumann algebras, respectively. In the simplest case our result coincides with a sharp version of the classical Hormander Lp-multiplier theorem, which was obtained by Grafakos and Slavikova in [11]. Finally, we present some applications to the evolution equation in non-commutative setting.

math.OA↗
arXiv2025

Hörmander type Fourier multiplier theorem and Nikolskii inequality on quantum tori, and applications

In this paper, we study Hörmander type Fourier multiplier theorem and the Nikolskii inequality on quantum tori. On the way to obtain these results, we also prove some classical inequalities such as Paley type, Hausdorff-Young-Paley, Hardy-Littlewood, and Logarithmic Sobolev inequalities on quantum tori. As applications we establish embedding theorems between Sobolev, Besov spaces as well as embeddings between Besov and Wiener and Beurling spaces on quantum tori. We also analyse $β$-versions of Wiener and Beurling spaces and their embeddings, and interpolation properties of all these spaces on quantum tori. As an applications of the analysis, we also derive a version of the Nash inequality, and the time decay for solutions of a heat type equation.

math.FA↗
arXiv2025

Sobolev inequality and its applications to nonlinear PDE on noncommutative Euclidean spaces

In this work, we study the Sobolev inequality on noncommutative Euclidean spaces. As a simple consequence, we obtain the Gagliardo-Nirenberg type inequality and as its application we show global well-posedness of nonlinear PDEs in the noncommutative Euclidean space. Moreover, we show that the logarithmic Sobolev inequality is equivalent to the Nash inequality for possibly different constants in this noncommutative setting by completing the list in noncommutative Varopoulos's theorem in [37]. Finally, we present a direct application of the Nash inequality to compute the time decay for solutions of the heat equation in the noncommutative setting.

math.AP↗
arXiv2025

A note on $L^p$-$L^q$ boundedness of Fourier multipliers on noncommutative spaces

In this work, we study Fourier multipliers on noncommutative spaces. In particluar, we show a simple proof of $L^p$-$L^q$ estimate of Fourier multipliers on general noncommutative spaces associated with semi-finite von Neumann algebras. This includes the case of Fourier multipliers on general locally compact unimodular groups.

math.FA↗