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K. Tulenov

Publications and source records attributed to K. Tulenov.

9 recordsLinked to original sources
arXiv2026

Trigonometric bases in noncommutative $L_p(\mathbb{T}^d_\theta)$ spaces and associated partial sum operators

We develop a harmonic-analytic method for constructing a generalized trigonometric system in noncommutative $L_p(\mathbb{T}^d_\theta)$ spaces arising from the strongly continuous representation of $\mathbb{T}^d$ and show that the generalized trigonometric system is a Schauder basis in $L_p(\mathbb{T}^d_\theta)$ for $1<p<\infty.$ In particular, we prove that this trigonometric system forms an RUC-basis in $L_p(\mathbb{T}^d_\theta)$ for $2<p<\infty.$ Our results provide a noncommutative counterpart of the classical trigonometric basis in $L_p(\mathbb{T}^d)$. Further, we obtain a weak $(1,1)$ type estimate of partial sum operators associated with noncommutative trigonometric systems. This allows us to study uniformly boundedness of partial sum operators between pairs of symmetric spaces that do not necessarily possess nontrivial Boyd indices, extending known results in this direction to the setting of quasi-Banach symmetric spaces.

math.OA↗
arXiv2026

Fractional Sobolev embeddings on noncommutative torus

In this paper, we study the noncommutative fractional symmetric Sobolev spaces on noncommutative torus. We prove noncommutative distributional fractional Sobolev inequality and as its application, we obtain Sobolev embeddings. In order to obtain these results, we first prove a noncommutative version of the famous O'Neil inequality for the convolution. As a first application of our main results, we obtain a Cwikel-Solomyak-type estimate. As an another application, we show a $L_2$-time decay for the mild solution of the Cauchy problem for the diffusion equation in this noncommutative setting. When $\theta=0,$ our results recover many known results on Sobolev embedding on the torus.

math.AP↗
arXiv2026

Optimal distributional estimates of the multiple Hilbert transform

In this paper, we study optimal distributional estimates for the multiple Hilbert transform. We obtain pointwise upper and lower distributional estimates of the multiple Hilbert transform in terms of the $d$-fold composition of the Calder\'{o}n operator with itself. This extends the fundamental results by A. P. Calder\'{o}n, D. Boyd, and Ch. Fefferman for arbitrary $d\in\mathbb N.$

math.FA↗
arXiv2025

Fourier multipliers and their applications to PDE on the quantum Euclidean space

In this work, we present some applications of the $L^p$-$L^q$ boundedness of Fourier multipliers to PDEs on the noncommutative (or quantum) Euclidean space. More precisely, we establish $L^p$-$L^q$ norm estimates for solutions of heat, wave, and Schr\"odinger type equations with Caputo fractional derivative in the case $1 < p \leq 2 \leq q < \infty.$ Moreover, we obtain well-posedness of nonlinear heat and wave equations on the noncommutative Euclidean space.

math.AP↗
arXiv2023

$L^p -L^q$ boundedness of Fourier multipliers on quantum Euclidean spaces

In this paper, we study Fourier multipliers on quantum Euclidean spaces and obtain results on their $L^p -L^q$ boundedness. On the way to get these results, we prove Paley, Hausdorff-Young-Paley, and Hardy-Littlewood inequalities on the quantum Euclidean space. As applications, we establish the $L^p -L^q$ estimate for the heat semigroup and Sobolev embedding theorem on quantum Euclidean spaces. We also obtain quantum analogues of the logarithmic Sobolev and Nash type inequalities.

math.FA↗
arXiv2019

The boundedness of the Hilbert transformation from one rearrangement invariant Banach space into another and applications

In this paper, we study the boundedness of the Hilbert transformation in Lorentz function spaces, thereby complementing classical results of Boyd. We also characterize the optimal range of a triangular truncation operator in Schatten-Lorentz ideals. These results further entail sharp commutator estimates and applications to operator Lipschitz functions in Schatten-Lorentz ideals.

math.FA↗
arXiv2019

The optimal range of the Calderòn operator and its applications

We identify the optimal range of the Calderòn operator and that of the classical Hilbert transform in the class of symmetric quasi-Banach spaces. Further consequences of our approach concern the optimal range of the triangular truncation operator, operator Lipschitz functions and commutator estimates in ideals of compact operators.

math.FA↗