Variable Lebesgue Spaces Lp Research Articles

New Gaussian Riesz transforms on variable Lebesgue spaces

We give sufficient conditions on the exponent p: ℝd → [1, ∞) for the boundedness of the non-centered Gaussian maximal function on variable Lebesgue spaces Lp(·) (γd), as well as of the new higher order Riesz transforms associated with the Ornstein—Uhlenbeck semigroup, which are the natural extensions of the supplementary first order Gaussian Riesz transforms defined by A. Nowak and K. Stempak in [25].

Boundedness of the Maximal Function of the Ornstein-Uhlenbeck semigroup on variable Lebesgue spaces with respect to the Gaussian measure and consequences

The main result of this paper is the proof of the boundedness of the Maximal Function T* of the Ornstein-Uhlenbeck semigroup {Tt}t≥ 0 in Rd, on Gaussian variable Lebesgue spaces Lp(.) (γd); under a condition of regularity on p(.) following [5] and [8]. As an immediate consequence of that result, the Lp(.) (γd)-boundedness of the Ornstein-Uhlenbeck semigroup {Tt}t≥ 0 in Rd is obtained. Another consequence of that result is the Lp(.) (γd)-boundedness of the Poisson-Hermite semigroup and the Lp(.) (γd)- boundedness of the Gaussian Bessel potentials of order β > 0.

Weak- and strong-type inequality for the cone-like maximal operator in variable Lebesgue spaces

The classical Hardy-Littlewood maximal operator is bounded not only on the classical Lebesgue spaces L p(Rd) (in the case p > 1), but (in the case when 1/p(·) is log-Holder continuous and p- = inf{p(x): x ∈ Rd > 1) on the variable Lebesgue spaces L p(·)(Rd), too. Furthermore, the classical Hardy-Littlewood maximal operator is of weak-type (1, 1). In the present note we generalize Besicovitch’s covering theorem for the so-called γ-rectangles. We introduce a general maximal operator Ms γδ, and with the help of generalized Φ-functions, the strong- and weak-type inequalities will be proved for this maximal operator. Namely, if the exponent function 1/p(·) is log-Holder continuous and p- ≥ s, where 1 ≤ s ≤ ∞ is arbitrary (or p- ≥ s), then the maximal operator Ms γδ is bounded on the space L p(·)(Rd) (or the maximal operator is of weak-type (p(·), p(·))).

Weighted norm inequalities for the maximal operator on variable Lebesgue spaces

We prove weighted strong and weak-type norm inequalities for the Hardy–Littlewood maximal operator on the variable Lebesgue space Lp(⋅). Our results generalize both the classical weighted norm inequalities on Lp and the more recent results on the boundedness of the maximal operator on variable Lebesgue spaces.

Convergence in variable Lebesgue spaces

We consider the relationship in the variable Lebesgue space Lp(·)(Ω) between convergence in norm, convergence in modular, and convergence in measure, for both bounded and unbounded exponent functions.