On Hardy q-inequalities

Maligrańda, Lech; Oinarov, Ryskul; Persson, Lars‐Erik

doi:10.1007/s10587-014-0125-6

Cited by 13 publications

(5 citation statements)

References 23 publications

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“…A number of articles address the further developments and recent results in the q-deformed calculus (see e.g. [3], [6], [10], [26], [37], [38] and we refer to the books [11] and [13]).…”

Section: Introductionmentioning

confidence: 99%

Multidimensional van der Corput-Type Estimates Involving Mittag-Leffler Functions

Ruzhansky

Torebek

2020

Fract Calc Appl Anal

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The paper is devoted to study multidimensional van der Corput-type estimates for the intergrals involving Mittag-Leffler functions. The generalisation is that we replace the exponential function with the Mittag-Leffler-type function, to study multidimensional oscillatory integrals appearing in the analysis of time-fractional evolution equations. More specifically, we study two types of integrals with functions Eα, β(i λ ϕ(x)), x ∈ ℝN and Eα, β(iαλ ϕ(x)), x ∈ ℝN for the various range of α and β. Several generalisations of the van der Corput-type estimates are proved. As an application of the above results, the Cauchy problem for the multidimensional time-fractional Klein-Gordon and time-fractional Schrödinger equations are considered.

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“…A number of articles address the further developments and recent results in the q-deformed calculus (see e.g. [3], [6], [10], [26], [37], [38] and we refer to the books [11] and [13]).…”

Section: Introductionmentioning

confidence: 99%

Multidimensional van der Corput-Type Estimates Involving Mittag-Leffler Functions

Ruzhansky

Torebek

2020

Fract Calc Appl Anal

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“…One interesting topic, q-analogues of the many inequalities derived from classical analysis, has been established. Its use can be seen in works such as [3,8,13,15,17]. These integral inequalities can be used for the study of qualitative and quantitative properties of integrals, see [1,14,19].…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, they proved that the constant C 0 is the sharp one. On the other hand, in [13] Maligranda, Oinarov and Persson derived some q-analysis variants of the classical Hardy inequality and obtained their corresponding best constants. Motivated by their work, a natural question raised is whether the q-analogue of a multivariate Hausdorff operator enjoys the same properties as the classical multivariate Hausdorff operator defined in (1.1).…”

Section: Introductionmentioning

confidence: 99%

Sharp Constants for Multivariate Hausdorff -Inequalities

Fan

Zhao²

2018

J. Aust. Math. Soc.

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In this paper, we focus on the multivariate Hausdorff operator of the form $$\begin{eqnarray}\mathbf{H}_{\unicode[STIX]{x1D6F7}}(f)(x)=\int _{(0,+\infty )^{n}}{\displaystyle \frac{\unicode[STIX]{x1D6F7}\big(\frac{x_{1}}{t_{1}},\frac{x_{2}}{t_{2}},\ldots ,\frac{x_{n}}{t_{n}}\big)}{t_{1}t_{2}\cdots t_{n}}}f(t_{1},t_{2},\ldots ,t_{n})\,\mathbf{dt},\end{eqnarray}$$ where $\mathbf{dt}=dt_{1}\,dt_{2}\cdots \,dt_{n}$ or $\mathbf{dt}=d_{q}t_{1}\,d_{q}t_{2}\cdots d_{q}t_{n}$ is the discrete measure in $q$-analysis. The sharp bounds for the multivariate Hausdorff operator on spaces $L^{p}$ with power weights are calculated, where $p\in \mathbb{R}\backslash \{0\}$.

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“…Both inequalities were proved independently by Gao [30, Corollary 3.1–3.2] (see also [31, Theorem 1.1] and [32, Theorem 6.1]) for and some special cases of α (this means that there are still some regions of parameters with no proof of (1.1)). Moreover, in [33, Theorems 2.1 and 2.3] proved another sharp discrete analogue of inequality (1.1) in the following form: and for , and , where .…”

Section: Introductionmentioning

confidence: 99%