arXiv subjects
Publications and source records attributed to Erlan Nursultanov.
We introduce new classes of general monotone sequences and study their properties. For functions whose Fourier coefficients belong to these classes, we establish Hardy-Littlewood-type theorems.
This paper studies Hausdorff-Young-type inequalities within the framework of Lorentz spaces $L_{p,q}$. Focusing on the dependence of the associated constants on the integrability parameter $p$, we derive optimal bounds in the limiting case $p\rightarrow 2$, addressing the Herz-Bochkarev problem. The results obtained refine the pioneering estimates in [3] and are comparable to recent advances in [16]. The main ingredients of our approach are new grand Lorentz space techniques.
We study norm inequalities for the Fourier transform, namely, \begin{equation}\label{introduction} \|\widehat f\|_{X_{p,q}^λ} \lesssim \|f\|_{Y}, \end{equation} where $X$ is either a Morrey or Campanato space and $Y$ is an appropriate function space. In the case of the Morrey space we sharpen the estimate $ \|\widehat f\|_{M_{p,q}^λ} \lesssim \|f\|_{L_{s',q}},$ $ s\geq 2,$ $\frac{1}{s} = \frac{1}{p}-\fracλ{n}.$ We also show that \eqref{introduction} does not hold when both $X$ and $Y$ are Morrey spaces. If $X$ is a Campanato space, we prove that \eqref{introduction} holds for $Y$ being the truncated Lebesgue space.
We obtain Fourier inequalities in the weighted $L_p$ spaces for any $1<p<\infty$ involving the Hardy-Cesàro and Hardy-Bellman operators. We extend these results to product Hardy spaces for $p\le 1$. Moreover, boundedness of the Hardy-Cesàro and Hardy-Bellman operators in various spaces (Lebesgue, Hardy, BMO) is discussed. One of our main tools is an appropriate version of the Hardy-Littlewood-Paley inequality $ \|\widehat{f}\|_{L_{p',q}} \lesssim \left\|f\right\|_{L_{p,q}}$.
In this paper we prove a noncommutative version of Hardy-Littlewood inequalities relating a function and its Fourier coefficients on the group $SU(2)$. As a consequence, we use it to obtain lower bounds for the $L^p-L^q$ norms of Fourier multipliers on the group $SU(2)$, for $1 < p \leq 2 \leq q < 1$. In addition, we give upper bounds of a similar form, analogous to the known results on the torus, but now in the noncommutative setting of $SU(2)$.
In this paper we prove new inequalities describing the relationship between the "size" of a function on a compact homogeneous manifold and the "size" of its Fourier coefficients. These inequalities can be viewed as noncommutative versions of the Hardy-Littlewood inequalities obtained by Hardy and Littlewood on the circle. For the example case of the group SU(2) we show that the obtained Hardy-Littlewood inequalities are sharp, yielding a criterion for a function to be in $L^p$ on SU(2) in terms of its Fourier coefficients. We also establish Paley and Hausdorff-Young-Paley inequalities on general compact homogeneous manifolds. The latter is applied to obtain conditions for the $L^p$-$L^q$ boundedness of Fourier multipliers for $1<p\leq 2\leq q<\infty$ on compact homogeneous manifolds as well as the $L^p$-$L^q$ boundedness of general (non-invariant) operators on compact Lie groups. We also record an abstract version of the Marcinkiewicz interpolation theorem on totally ordered discrete sets, to be used in the proofs with different Plancherel measures on the unitary duals.
In this paper we prove Nikolskii's inequality on general compact Lie groups and on compact homogeneous spaces with the constant interpreted in terms of the eigenvalue counting function of the Laplacian on the space, giving the best constant for certain indices, attained on the Dirichlet kernel. Consequently, we establish embedding theorems between Besov spaces on compact homogeneous spaces, as well as embeddings between Besov spaces and Wiener and Beurling spaces. We also analyse Triebel-Lizorkin spaces and $β$-versions of Wiener and Beurling spaces and their embeddings, and interpolation properties of all these spaces.
In this paper, we prove analogues of O'Neil's inequalities for the convolution in the weighted Lebesgue spaces. We also establish the weighted two-sided norm inequalities for the potential operator.