High-order adaptive multiresolution wavelet upwind schemes for hyperbolic conservation laws

We prove a version of the Lebesgue differentiation theorem for mappings that are defined on a measure space and take values into a metric space, with respect to the differentiation basis induced by a von Neumann lifting. As a consequence, we obtain a lifting theorem for the space of sections of a measurable Banach bundle and a disintegration theorem for vector measures whose target is a Banach space with the Radon–Nikodým property.

Density topologies for strictly positive Borel measures

Many physical phenomena give rise to mathematical models in terms of fractal, non-differentiable functions. The paper introduces a broad generalization of the derivative in terms of the maximal modulus of continuity of the primitive function. These derivatives are called indicial derivatives. As an application, the indicial derivatives are used to characterize the nowhere monotonous functions. Furthermore, the non-differentiability set of such derivatives is proven to be of measure zero. As a second application, the indicial derivative is used in the proof of the Lebesgue differentiation theorem. Finally, the connection with the fractional velocities is demonstrated.

We study a version of the Lebesgue differentiation theorem in which the integral averages are replaced with medians over Busemann–Feller differentiation bases. Our main result gives several characterizations for the differentiation property in terms of the corresponding median maximal function. As an application, we study pointwise behaviour in Besov and Triebel–Lizorkin spaces, where functions are not necessarily locally integrable. Most of our results apply also for functions defined on metric measure spaces.

The Lebesgue differentiation theorem revisited

A version of the Lebesgue differentiation theorem is offered, where the [Formula: see text] norm is replaced with any rearrangement-invariant norm. Necessary and sufficient conditions for a norm of this kind to support the Lebesgue differentiation theorem are established. In particular, Lorentz, Orlicz and other customary norms for which Lebesgue’s theorem holds are characterized.

Time-Inconsistent Stochastic LinearrQuadratic Control: Characterization and Uniqueness of Equilibrium

In this paper, we continue our study on a general time-inconsistent stochastic linear--quadratic (LQ) control problem originally formulated in [6]. We derive a necessary and sufficient condition for equilibrium controls via a flow of forward--backward stochastic differential equations. When the state is one dimensional and the coefficients in the problem are all deterministic, we prove that the explicit equilibrium control constructed in \cite{HJZ} is indeed unique. Our proof is based on the derived equivalent condition for equilibria as well as a stochastic version of the Lebesgue differentiation theorem. Finally, we show that the equilibrium strategy is unique for a mean--variance portfolio selection model in a complete financial market where the risk-free rate is a deterministic function of time but all the other market parameters are possibly stochastic processes.

In paper [1] Ahmad et al. investigated the use of sharp function, known from functional analysis, in image processing. The sharp function gives a measure of variations of a function and can be used as an edge detector [2]. We extend the classical notion of sharp function to prove the classical Lebesgue differentiation theorem and Marcinkiewicz theorem for the sublinear operator T(x;y).

We exhibit a close correspondence between $L_1$-computable functions and Schnorr tests. Using this correspondence, we prove that a point $x\in [0,1]^d$ is Schnorr random if and only if the Lebesgue Differentiation Theorem holds at $x$ for all $L_1$-computable functions $f\in L_1([0,1]^d)$.

Using elementary arguments based on the Fourier transform we prove that for $${1 \leq q < p < \infty}$$ and $${s \geq 0}$$ with s > n(1/2 − 1/p), if $${f \in L^{q,\infty} (\mathbb{R}^n) \cap \dot{H}^s (\mathbb{R}^n)}$$ , then $${f \in L^p(\mathbb{R}^n)}$$ and there exists a constant c p,q,s such that $$\| f \|_{L^{p}} \leq c_{p,q,s} \| f \|^\theta _{L^{q,\infty}} \| f \|^{1-\theta}_{\dot{H}^s},$$ where 1/p = θ/q + (1−θ)(1/2−s/n). In particular, in $${\mathbb{R}^2}$$ we obtain the generalised Ladyzhenskaya inequality $${\| f \| _{L^4} \leq c \| f \|^{1/2}_{L^{2,\infty}} \| f \|^{1/2}_{\dot{H}^1}}$$ .We also show that for s = n/2 and q > 1 the norm in $${\| f \|_{\dot{H}^{n/2}}}$$ can be replaced by the norm in BMO. As well as giving relatively simple proofs of these inequalities, this paper provides a brief primer of some basic concepts in harmonic analysis, including weak spaces, the Fourier transform, the Lebesgue Differentiation Theorem, and Calderon–Zygmund decompositions.

Integration in quasi-Banach spaces and the fundamental theorem of calculus

This special issue of Theory or Computing Systems consists of papers associated with the 6th Annual CCR conference held in beautiful Cape Town, 31st January– February 4th 2011. The conference series is devoted to issues around algorithmic information theory, Kolmogorov complexity, and their relationship with computability theory, complexity theory, logic and reverse mathematics. In 2011, the conference was co-located with the 8th Annual Computability and Analysis conference. This co-location reflects the increasing interactions between computable analysis and algorithmic randomness through effective analysis of almost everywhere behavior in analysis such as the Lebesgue Differentiation Theorem and things like Brownian motion. Both conferences were admirably overseen by Vasco Brattka and his program committees. The last 10–15 years has seen huge progress in the areas represented by CCR with two long monographs in the area, recognition at the International Congress of Mathematicians, and a thoroughly thriving research community. Whilst the CCR conference uses a model which does not have an actual proceedings at the meeting, the papers in this special issue reflect the meeting’s business. These papers reflect the high caliber of both the researchers and the intellectual merit of the investigations. The work spans issues from applications in biology, computer science to fundamental issues about Kolmogorov complexity. The papers all went through the full Theory of Computing Systems journal reviewing process and several of the initial submissions were rejected. Enjoy.

The best constant approximant operators in Lorentz spaces [formula omitted] and their applications

Given an L1 -computable function, f , we identify a canonical represen- tative of the equivalence class of f , where f and g are equivalent if and only if R jf gj is zero. Using this representative, we prove a modified version of the Lebesgue Differentiation Theorem. Our theorem is stated in terms of Martin-L¨ random points in Euclidean space. 2000 Mathematics Subject Classification 03D80 (primary); 26A24 (secondary)

In this paper we are concerned with the pointwise behaviour of functions in certain classes of weakly differentiable functions. The ancestor of all modern results dealing with pointwise properties of nonsmooth functions is certainly the Lebesgue differentiation theorem, which asserts that if ft is an open subset of R ~, n > l , and u is a locally integrable function in f~, then lim~_~0+ 3CB,.(x) u(y) dy exists and is finite for a.e. xEf~, and

The standard techniques used to prove the Lebesgue differentiation theorem (that monotonic functions are a.e.~differentiable) are presented in an unusual way that reveals more about their nature and allows greater generality.

A complete version of Lebesgue's differentiation theorem, including the image of the exceptional set, is proved in an elementary way.

A short proof is given for the Lebesgue Differentiation Theorem using a variation of the Heine-Borel covering property, without reliance on sophisticated approaches such as Vitali covers and the rising sun lemma.