We provide a comprehensive study of interrelations between different measures of smoothness of functions on various domains and smoothness properties of approximation processes. Two general approaches to this problem have been developed: The first based on geometric properties of Banach spaces and the second on Littlewood–Paley and Hörmander-type multiplier theorems. In particular, we obtain new sharp inequalities for measures of smoothness given by the $K$ -functionals or moduli of smoothness. As examples of approximation processes we consider best polynomial and spline approximations, Fourier multiplier operators on $𝕋^{d}$ , $ℝ^{d}$ , $[- 1, 1]$ , nonlinear wavelet approximation, etc.

Open Access

Krawtchouk transforms and convolutions

Philip Feinsilver and
René Schott

1950009

https://doi.org/10.1142/S1664360719500097

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We present an operator calculus based on Krawtchouk polynomials, including Krawtchouk transforms and corresponding convolution structure which provides an inherently discrete alternative to Fourier analysis. This approach is well suited for applications such as digital image processing. This paper includes the theoretical aspects and some basic examples.

Open Access

$W^{1, p}$ versus $C^{1}$ : The nonsmooth case involving critical growth

Yunru Bai,
Leszek Gasiński,
Patrick Winkert, and
Shengda Zeng

2050009

https://doi.org/10.1142/S1664360720500095

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In this paper, we study a class of generalized and not necessarily differentiable functionals of the form

$J (u) = \int_{Ω} G (x, \nabla u) d x - \int_{Ω} j_{1} (x, u) d x - \int_{\partial Ω} j_{2} (x, u) d σ$

with functions

$j_{1} : Ω \times ℝ \to ℝ$ ,

$j_{2} : \partial Ω \times ℝ \to ℝ$ that are only locally Lipschitz in the second argument and involving critical growth for the elements of their generalized gradients

$\partial j_{k} (x, \cdot), k = 1, 2$ even on the boundary

$\partial Ω$ . We generalize the famous result of Brezis and Nirenberg [

$H^{1}$ versus

$C^{1}$ local minimizers, C. R. Acad. Sci. Paris Sér. I Math. 317(5) (1993) 465–472] to a more general class of functionals and extend all the other generalizations of this result which has been published in the last decades.

Open Access

Ziegler partial morphisms in additive exact categories

Manuel Cortés-Izurdiaga,
Pedro A. Guil Asensio,
Berke Kalebog̃az, and
Ashish K. Srivastava

2050012

https://doi.org/10.1142/S1664360720500125

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We develop a general theory of partial morphisms in additive exact categories which extends the model theoretic notion introduced by Ziegler in the particular case of pure-exact sequences in the category of modules over a ring. We relate partial morphisms with (co-)phantom morphisms and injective approximations and study the existence of such approximations in these exact categories.

Open Access

Geometric Hardy and Hardy–Sobolev inequalities on Heisenberg groups

Michael Ruzhansky,
Bolys Sabitbek, and
Durvudkhan Suragan

2050016

https://doi.org/10.1142/S1664360720500162

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In this paper, we present geometric Hardy inequalities for the sub-Laplacian in half-spaces of stratified groups. As a consequence, we obtain the following geometric Hardy inequality in a half-space of the Heisenberg group with a sharp constant:

$\int_{ℍ^{+}} | \nabla_{H} u |^{p} d ξ \geq {(\frac{p - 1}{p})}^{p} \int_{ℍ^{+}} \frac{𝒲 {(ξ)}^{p}}{dist {(ξ, \partial ℍ^{+})}^{p}} | u |^{p} d ξ, p > 1,$

which solves a conjecture in the paper [S. Larson, Geometric Hardy inequalities for the sub-elliptic Laplacian on convex domain in the Heisenberg group, Bull. Math. Sci. 6 (2016) 335–352]. Here,

$𝒲 (ξ) = {(\sum_{i = 1}^{n} {〈 X_{i} (ξ), ν 〉}^{2} + {〈 Y_{i} (ξ), ν 〉}^{2})}^{\frac{1}{2}}$

is the angle function. Also, we obtain a version of the Hardy–Sobolev inequality in a half-space of the Heisenberg group:

${(\int_{ℍ^{+}} | \nabla_{H} u |^{p} d ξ - {(\frac{p - 1}{p})}^{p} \int_{ℍ^{+}} \frac{𝒲 {(ξ)}^{p}}{dist {(ξ, \partial ℍ^{+})}^{p}} | u |^{p} d ξ)}^{\frac{1}{p}} \geq C {(\int_{ℍ^{+}} | u |^{p^{*}} d ξ)}^{\frac{1}{p^{*}}},$

where

$dist (ξ, \partial ℍ^{+})$ is the Euclidean distance to the boundary,

$p^{*} : = Q p / (Q - p)$ , and

$2 \leq p < Q$ . For

$p = 2$ , this gives the Hardy–Sobolev–Maz’ya inequality on the Heisenberg group.

Open Access