Alberto Fiorenza

Grand and Small Lebesgue Spaces and Their Analogs

We give the following, equivalent, explicit expressions for the norms of the small and grand Lebesgue spaces, which depend only on the non-decreasing rearrangement (we assume here that the underlying measure space has measure 1):

The fractional maximal operator and fractional integrals on variable

We prove that if the exponent function p(·) satisfies log-Hölder continuity conditions locally and at infinity, then the fractional maximal operator Mα, 0 <α< n, maps Lp(·) to Lq(·), where 1 p(x)− 1 q(x) = αn . We also prove a weak-type inequality corresponding to the weak (1, n/(n − α)) inequality for Mα. We build upon earlier work on the Hardy-Littlewood maximal operator by Cruz-Uribe, Fiorenza and Neugebauer [3]. As a consequence of these results for Mα, we show that the fractional integral operator Iα satisfies the same norm inequalities. These in turn yield a generalization of the Sobolev embedding theorem to variable Lp spaces.

L^p

We consider a generalized version of the small Lebesgue spaces, introduced in [5] as the associate spaces of the grand Lebesgue spaces. We find a simplified expression for the norm, prove relevant properties, compute the fundamental function and discuss the comparison with the Orlicz spaces.

spaces.

We prove that the commutator [b, Iα], b ∈ BMO, Iα the fractional integral operator, satisfies the sharp, modular weak-type inequality f(x) tdx, where B(t) = tlog(e + t) and Ψ(t)=[tlog(e + tα/n)]n/(n−α). These commutators were first considered by Chanillo, and our result complements his. The heart of our proof consists of the pointwise inequality, M#([b, Iα]f)(x) ≤ CbBMO [Iαf(x) + Mα,Bf(x)], where M# is the sharp maximal operator, and Mα,B is a generalization of the fractional maximal operator in the scale of Orlicz spaces. Using this inequality we also prove one-weight inequalities for the commutator; to do so we prove one and two-weight norm inequalities for Mα,B which are of interest in their own right.[b, Iα]f(x)

On small Lebesgue spaces

We generalize the classical L log L inequalities of Wiener and Stein for the Hardy-Littlewood maximal operator to variable L P spaces where the exponent function p(-) approaches 1 in value. We prove a modular inequality with no assumptions on the exponent function, and a strong norm inequality if we assume the exponent function is log-Holder continuous. As an application of our approach we give another proof of a related endpoint result due to Hasto.

Endpoint estimates and weighted norm inequalities for commutators of fractional integrals

For functions f in Sobolev spaces W1,p(x)(Ω) with exponent lower semicontinuous, bounded away from 1 and ∞ and with the property of the density of smooth functions, it is shown that for each open set ω ⊂⊂ Ω, for each h ∈ RN such that ω+th ⊂ Ω ∀ t∈ [0,1], the following inequality holds where min p(x, x + h) denotes the minimum of p along the segment whose endpoints are x, x + h. As a consequence, if p(x) is also continuous, for mollifiers (ρn){n ∈ N the liminf and the limsup of are respectively minorized and majorized by expressions equivalent to ‖|∇f|‖Lp(x)(Ω).

L log L RESULTS FOR THE MAXIMAL OPERATOR IN VARIABLE LP SPACES

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A MEAN CONTINUITY TYPE RESULT FOR CERTAIN SOBOLEV SPACES WITH VARIABLE EXPONENT

Abstract We extend the results of [1–5] on the uniqueness of solutions of parabolic equations. Our results give also some regularity results which complete the existence results made in [6–8].

On extrapolation blowups in the

We consider the well-known characterization of the Golden ratio as limit of the ratio of consecutive terms of the Fibonacci sequence, and we give an explanation of this property in the framework of the Difference Equations Theory. We show that the Golden ratio coincides with this limit not because it is the root with maximum modulus and multiplicity of the characteristic polynomial, but, from a more general point of view, because it is the root with maximum modulus and multiplicity of a restricted set of roots, which in this special case coincides with the two roots of the characteristic polynomial. This new perspective is the heart of the characterization of the limit of ratio of consecutive terms of all linear homogeneous recurrences with constant coefficients, without any assumption on the roots of the characteristic polynomial, which may be, in particular, also complex and not real.

Regularity and comparison results in grand Sobolev spaces for parabolic equations with measure data

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.