Singular integral operators in some variable exponent Lebesgue spaces

The paper deals with the exploration of those subclasses of the variable exponent Lebesgue space {L^{p(\,\cdot\,)}} with {\min p(\,\cdot\,)=1}, which are invariant with respect to Cauchy singular integral operators.

Modular Uniform Convexity of Lebesgue Spaces of Variable Integrability

We analyze the modular geometry of the variable exponent Lebesgue space Lp(.). We show that Lp(.) possesses a modular uniform convexity property. Part of the novelty is that the property holds even in the case supp(x) = ∞ . We present specific applications to fixed point theory. xÆΩ

The Hardy--Littlewood maximal operator and BLO1/log class of exponents

It is well known that if the Hardy–Littlewood maximal operator is bounded in the variable exponent Lebesgue space ${L^{p(\,\cdot\,)}[0;1]}$, then ${p(\,\cdot\,)\in\mathrm{BMO}^{1/\log}}$. On the other hand, there exists an exponent ${p(\,\cdot\,)\in\mathrm{BMO}^{1/\log}}$, ${1<p_{-}\leq p_{+}<\infty}$, such that the Hardy–Littlewood maximal operator is not bounded in ${L^{p(\,\cdot\,)}[0;1]}$. But for any exponent ${p(\,\cdot\,)\in\mathrm{BMO}^{1/\log}}$, ${1<p_{-}\leq p_{+}<\infty}$, there exists a constant ${c>0}$ such that the Hardy–Littlewood maximal operator is bounded in ${L^{p(\,\cdot\,)+c}[0;1]}$. In this paper, we construct an exponent ${p(\,\cdot\,)}$, ${1<p_{-}\leq p_{+}<\infty}$, ${1/p(\,\cdot\,)\in\mathrm{BLO}^{1/\log}}$ such that the Hardy–Littlewood maximal operator is not bounded in ${L^{p(\,\cdot\,)}[0;1]}$.

Approximation by trigonometric polynomials in the framework of variable exponent grand Lebesgue spaces

In this paper we establish direct and inverse theorems on approximation by trigonometric polynomials for the functions of the closure of the variable exponent Lebesgue space in the variable exponent grand Lebesgue space.

Local Characterizations of Besov and Triebel-Lizorkin Spaces with Variable Exponent

We introduce new Besov and Triebel-Lizorkin spaces with variable integrable exponent, which are different from those introduced by the second author early. Then we characterize these spaces by the boundedness of the local Hardy-Littlewood maximal operator on variable exponent Lebesgue space. Finally the completeness and the lifting property of these spaces are also given.