We show that the Hölder exponent and the chirp exponent of a function can be prescribed simultaneously on a set of full measure, if they are both lower limits of continuous functions. We also show that this result is optimal: In general, Hölder and chirp exponents cannot be prescribed outside a set of Hausdorff dimension less than one. The direct part of the proof consists in an explicit construction of a function determined by its orthonormal wavelet coefficients; the optimality is the direct consequence...

Given a smooth family of vector fields satisfying Chow-Hörmander’s condition of step 2 and a regularity assumption, we prove that the Sobolev spaces of fractional order constructed by the standard functional analysis can actually be “computed” with a simple formula involving the sub-riemannian distance.Our approach relies on a microlocal analysis of translation operators in an anisotropic context. It also involves classical estimates of the heat-kernel associated to the sub-elliptic Laplacian.

If $f$ is a Henstock-Kurzweil integrable function on the real line, the Alexiewicz norm of $f$ is $\parallel f\parallel ={sup}_I\left|{\int }_{I}f\right|$ where the supremum is taken over all intervals $I\subset ℝ$. Define the translation ${\tau }_x$ by ${\tau }_{x}f\left(y\right)=f(y-x)$. Then $\parallel {\tau }_{x}f-f\parallel$ tends to $0$ as $x$ tends to $0$, i.e., $f$ is continuous in the Alexiewicz norm. For particular functions, $\parallel {\tau }_{x}f-f\parallel$ can tend to 0 arbitrarily slowly. In general, $\parallel {\tau }_{x}f-f\parallel \ge \text{osc}f\left|x\right|$ as $x\to 0$, where $\text{osc}f$ is the oscillation of $f$. It is shown that if $F$ is a primitive of $f$ then $\parallel {\tau }_{x}F-F\parallel \le \parallel f\parallel \left|x\right|$. An example shows that the function $y\mapsto {\tau }_{x}F\left(y\right)-F\left(y\right)$ need not be in $L^1$. However, if $f\in L^1$ then $\parallel {\tau }_x{F-F\parallel }_1\le {\parallel f\parallel }_1\left|x\right|$....

The problem of continuous dependence for inverses of fundamental matrices in the case when uniform convergence is violated is presented here.

We review the known facts and establish some new results concerning continuous-restrictions, derivative-restrictions, and differentiable-restrictions of Lebesgue measurable, universally measurable, and Marczewski measurable functions, as well as functions which have the Baire properties in the wide and restricted senses. We also discuss some known examples and present a number of new examples to show that the theorems are sharp.

In this paper we use a generalized version of absolute continuity defined by J. Kurzweil, J. Jarník, Equiintegrability and controlled convergence of Perron-type integrable functions, Real Anal. Exch. 17 (1992), 110–139. By applying uniformly this generalized version of absolute continuity to the primitives of the Henstock-Kurzweil-Pettis integrable functions, we obtain controlled convergence theorems for the Henstock-Kurzweil-Pettis integral. First, we present a controlled convergence theorem for...

We establish a connection between the L² norm of sums of dilated functions whose jth Fourier coefficients are $\left(j^{-\alpha }\right)$ for some α ∈ (1/2,1), and the spectral norms of certain greatest common divisor (GCD) matrices. Utilizing recent bounds for these spectral norms, we obtain sharp conditions for the convergence in L² and for the almost everywhere convergence of series of dilated functions.

It is shown that a uniform version of Sklyarenko's integrability condition for Perron integrals together with pointwise convergence of a sequence of integrable functions are sufficient for a convergence theorem for Perron integrals.