Weighted and endpoint estimates for commutators of bilinear pseudo-differential operators

<abstract><p>In this paper, by applying the accurate estimates of the Hörmander class, the authors consider the commutators of bilinear pseudo-differential operators and the operation of multiplication by a Lipschitz function. By establishing the pointwise estimates of the corresponding sharp maximal function, the boundedness of the commutators is obtained respectively on the products of weighted Lebesgue spaces and variable exponent Lebesgue spaces with $ \sigma \in\mathcal{B}BS_{1, 1}^{1} $. Moreover, the endpoint estimate of the commutators is also established on $ L^{\infty}\times L^{\infty} $.</p></abstract>

Approximation by Bézier Variant of Baskakov-Durrmeyer-Type Hybrid Operators

We give a Bézier variant of Baskakov-Durrmeyer-type hybrid operators in the present article. First, we obtain the rate of convergence by using Ditzian-Totik modulus of smoothness and also for a class of Lipschitz function. Then, weighted modulus of continuity is investigated too. We study the rate of point-wise convergence for the functions having a derivative of bounded variation. Furthermore, we establish the quantitative Voronovskaja-type formula in terms of Ditzian-Totik modulus of smoothness at the end.

Approximation properties of the new type generalized Bernstein-Kantorovich operators

<abstract><p>In this paper, we introduce new type of generalized Kantorovich variant of $ \alpha $-Bernstein operators and study their approximation properties. We obtain estimates of rate of convergence involving first and second order modulus of continuity and Lipschitz function are studied for these operators. Furthermore, we establish Voronovskaya type theorem of these operators. The last section is devoted to bivariate new type $ \alpha $-Bernstein-Kantorovich operators and their approximation behaviors. Also, some graphical illustrations and numerical results are provided.</p></abstract>

Some New Inequalities Involving κ-Fractional Integral for Certain Classes of Functions and Their Applications

In this article, we present several new inequalities involving the κ-fractional integral for the integrable function ℱ which satisfies one of the following conditions: aℱq is preinvex for some q>1; bℱ′ is bounded; cℱ′ is a Lipschitz function. As applications, we establish new inequalities for the weighted arithmetic and generalized logarithmic means.

Non-occurrence of the Lavrentiev phenomenon for a class of convex nonautonomous Lagrangians

We consider the classical functional of the Calculus of Variations of the form $$\begin{array}{} \displaystyle I(u)=\int\limits_{{\it\Omega}}F(x, u(x), \nabla u(x))\,dx, \end{array}$$ where Ω is a bounded open subset of ℝn and F : Ω × ℝ × ℝn → ℝ is a Carathéodory convex function; the admissible functions u coincide with a prescribed Lipschitz function ϕ on ∂Ω. We formulate some conditions under which a given function in ϕ + $\begin{array}{} \displaystyle W^{1,p}_0 \end{array}$(Ω) with I(u) < +∞ can be approximated in the W1,p norm and in energy by a sequence of smooth functions that coincide with ϕ on ∂Ω. As a particular case we obtain that the Lavrentiev phenomenon does not occur when F(x, u, ξ) = f(x, u) + h(x, ξ) is convex and x ↦ F(x, 0, 0) is sufficiently smooth.

A dichotomy of sets via typical differentiability

We obtain a criterion for an analytic subset of a Euclidean space to contain points of differentiability of a typical Lipschitz function: namely, that it cannot be covered by countably many sets, each of which is closed and purely unrectifiable (has a zero-length intersection with every $C^1$ curve). Surprisingly, we establish that any set failing this criterion witnesses the opposite extreme of typical behaviour: in any such coverable set, a typical Lipschitz function is everywhere severely non-differentiable.

An Integralgeometric Approach to Dorronsoro Estimates

A theorem of Dorronsoro from 1985 quantifies the fact that a Lipschitz function $f \colon \mathbb{R}^{n} \to \mathbb{R}$ can be approximated by affine functions almost everywhere, and at sufficiently small scales. This paper contains a new, purely geometric, proof of Dorronsoro’s theorem. In brief, it reduces the problem in $\mathbb{R}^{n}$ to a problem in $\mathbb{R}^{n - 1}$ via integralgeometric considerations. For the case $n = 1$, a short geometric proof already exists in the literature. A similar proof technique applies to parabolic Lipschitz functions $f \colon \mathbb{R}^{n - 1} \times \mathbb{R} \to \mathbb{R}$. A natural Dorronsoro estimate in this class is known, due to Hofmann. The method presented here allows one to reduce the parabolic problem to the Euclidean one and to obtain an elementary proof also in this setting. As a corollary, I deduce an analogue of Rademacher’s theorem for parabolic Lipschitz functions.