A Survey on Some Variable Function Spaces

Intrinsic Square Function Characterizations of Variable Hardy–Lorentz Spaces

“…where the infimum is taken over all the decompositions of f as (15). e following result is eorem 5.4 of [4].…”

“…is space has been extensively studied by many researchers due to its wide use in different fields such as harmonic analysis and partial differential equations, see, for example, [12][13][14][15]. e real variable theory of Hardy spaces H p (R n ), introduced by Stein and Weiss in [16], is a well-known generalization of the Lebesgue spaces L p (R n ).…”

Intrinsic Square Function Characterizations of Variable Weak Hardy Spaces

“…Recall that the variable Lebesgue space L p(•) (R n ), with a variable exponent function p(•) : R n → (0, ∞), is a generalization of the classical Lebesgue space L p (R n ), via replacing the constant exponent p by the exponent function p(•), which consists of all functions f such that R n |f (x)| p(x) dx < ∞. The study of variable Lebesgue spaces can be traced back to Birnbaum-Orlicz [7] and Orlicz [27], however, they have been the subject of more intensive study since the early 1990s because of their intrinsic interest for applications into harmonic analysis, partial differential equations and variational integrals with nonstandard growth conditions (see [8,11,19,41] and their references).…”

Multilinear fractional type operators and their commutators on Hardy spaces with variable exponents

“…The atomic decomposition of Hardy spaces with variable exponents was first established independently in [11,28]. Moreover, Yang et al [40,41,43] established some equivalent characterizations of Hardy spaces with variable exponents. Recently, the author revisited the atomic decomposition for H p(•) via the Littlewood-Paley-Stein theory in [35].…”