A Liouville-type theorem for very weak solutions of nonlinear partial differential equations.
Duality and reflexivity in grand Lebesgue spaces.
A summability condition on the gradient ensuring BMO.
Categories of results in variable Lebesgue spaces theory
Variable (exponent) Lebesgue spaces represent a relevant research area within the theory of Banach function spaces. Much attention is devoted to look for sufficient conditions on the variable exponent to establish the assertions within the theory. In this Note we try to show the beauty of the research in this field, mainly quoting some known results organized into “categories", each of them characterized by a common typology of conditions on the variable exponent. New results involve the failure of...
Orlicz capacities and applications to some existence questions for elliptic PDEs having measure data
We study the sequence , which is solution of in an open bounded set of and on , when tends to a measure concentrated on a set of null Orlicz-capacity. We consider the relation between this capacity and the -function , and prove a non-existence result.
Indices of Orlicz spaces and some applications
We study connections between the Boyd indices in Orlicz spaces and the growth conditions frequently met in various applications, for instance, in the regularity theory of variational integrals with non-standard growth. We develop a truncation method for computation of the indices and we also give characterizations of them in terms of the growth exponents and of the Jensen means. Applications concern variational integrals and extrapolation of integral operators.
Orlicz capacities and applications to some existence questions for elliptic pdes having measure data
We study the sequence , which is solution of in an open bounded set of and = 0 on ∂Ω, when tends to a measure concentrated on a set of null Orlicz-capacity. We consider the relation between this capacity and the -function , and prove a non-existence result.
The maximal theorem for weighted grand Lebesgue spaces
We study the Hardy inequality and derive the maximal theorem of Hardy and Littlewood in the context of grand Lebesgue spaces, considered when the underlying measure space is the interval (0,1) ⊂ ℝ, and the maximal function is localized in (0,1). Moreover, we prove that the inequality holds with some c independent of f iff w belongs to the well known Muckenhoupt class , and therefore iff for some c independent of f. Some results of similar type are discussed for the case of small Lebesgue spaces....
Weighted endpoint estimates for commutators of fractional integrals
Given , , and , we give sufficient conditions on weights for the commutator of the fractional integral operator, , to satisfy weighted endpoint inequalities on and on bounded domains. These results extend our earlier work [3], where we considered unweighted inequalities on .
BMO regularity for one-dimensional minimizers of some Lagrange problems.
Regularity results for vector fields of bounded distortion and applications.
On an optimal decomposition in Zygmund spaces.
On extrapolation blowups in the scale.
A topology on inequalities.
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