The maximal theorem for weighted grand Lebesgue spaces

Abstract

Abstract. 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 M f p),w ≤ c f p),w holds with some c independent of f iff w belongs to the well known Muckenhoupt class Ap, and therefore iff M f p,w ≤ c f p,w for some c independent of f .Some results of similar type are discussed for the case of small Lebesgue spaces.

Boundedness of Weighted Singular Integral Operators on a Carleson Curve in Grand Lebesgue Spaces

“…, see A. Fiorenza, B. Gupta and P. Jain [6] for the Euclidean setting. Note that the boundedness of the maximal operator in weighted grand Lebesgue spaces remains to be equivalent to the Muckenhoupt condition as proved in [6].…”

“…In the theory in PDE, it turns out that they are right spaces in which some nonlinear equations have to be considered. We refer for instance, to papers [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13].…”

Boyd Indices in Generalized Grand Lebesgue Spaces and Applications

“…The computation of the Boyd indices of generalized grand Lebesgue spaces L p),δ introduced in [7] is a problem of interest in Function Spaces Theory and in Harmonic analysis (see e.g., [17]). …”

Sobolev embeddings for Riesz potentials of functions in grand Morrey spaces of variable exponents over non-doubling measure spaces

“…The generalized grand Lebesgue spaces appeared in [20], where the existence and uniqueness of the non-homogeneous N -harmonic equations div (|∇u| N −2 ∇u) = µ were studied. The boundedness of the maximal operator on the grand Lebesgue spaces was studied in [14]. The boundedness of the maximal operator and Sobolev's inequality for grand Morrey spaces with doubling measure were also studied in [32].…”