Riesz and Wolff potentials and elliptic equations in variable exponent weak Lebesgue spaces

We prove optimal integrability results for solutions of the p(x)-Laplace equation in the scale of (weak) Lebesgue spaces. To obtain this, we show that variable exponent Riesz and Wolff potentials map L1 to variable exponent weak Lebesgue spaces

Pointwise estimates to the modified Riesz potential

In a smooth domain a function can be estimated pointwise by the classical Riesz potential of its gradient. Combining this estimate with the boundedness of the classical Riesz potential yields the optimal Sobolev-Poincar, inequality. We show that this method gives a Sobolev-Poincar, inequality also for irregular domains whenever we use the modified Riesz potential which arise naturally from the geometry of the domain. The exponent of the Sobolev-Poincar, inequality depends on the domain. The Sobolev-Poincar, inequality given by this approach is not sharp for irregular domains, although the embedding for the modified Riesz potential is optimal. In order to obtain the results we prove a new pointwise estimate for the Hardy-Littlewood maximal operator.Peer reviewe

Maximal operator in variable exponent generalized morrey spaces on quasi-metric measure space

We consider generalized Morrey spaces on quasi-metric measure spaces , in general unbounded, with variable exponent p(x) and a general function defining the Morrey-type norm. No linear structure of the underlying space X is assumed. The admission of unbounded X generates problems known in variable exponent analysis. We prove the boundedness results for maximal operator known earlier only for the case of bounded sets X. The conditions for the boundedness are given in terms of the so called supremal inequalities imposed on the function , which are weaker than Zygmund-type integral inequalities often used for characterization of admissible functions . Our conditions do not suppose any assumption on monotonicity of in r

Interpolation in variable exponent spaces

In this paper we study both real and complex interpolation in the recently introduced scales of variable exponent Besov and Triebel–Lizorkin spaces. We also take advantage of some interpolation results to study a trace property and some pseudodifferential operators acting in the variable index Besov scale

Congenital microcephaly: Case definition & guidelines for data collection, analysis, and presentation of safety data after maternal immunisation.

Need for developing case definitions and guidelines for data collection, analysis, and presentation for congenital microcephaly as an adverse event following maternal immunisation Congenital microcephaly, also referred to as primary microcephaly due to its presence in utero or at birth, is a descriptive term for a structural defect in which a fetus or infant’s head (cranium) circumference is smaller than expected when compared to other fetuses or infants of the same gestational age, sex and ethnic background

Uniform convexity and associate spaces

Abstract We prove that the associate space of a generalized Orlicz space $L^{\varphi(\cdot)}$ is given by the conjugate modular $\varphi^*$ even without the assumption that simple functions belong to the space. Second, we show that every weakly doubling $\Phi$-function is equivalent to a doubling $\Phi$-function. As a consequence, we conclude that $L^{\varphi(\cdot)}$ is uniformly convex if $\varphi$ and $\varphi^*$ are weakly doubling