Riesz potential operator in continual variable exponents Herz spaces

Abstract

We find conditions on the variable parameters p(x),q(t) and α(t), defining the Herz space Hp(·),q(·),α(·)(Rn), for the validity of Sobolev type theorem for the Riesz potential operator to be bounded within the frameworks of such variable exponents Herz spaces. We deal with a “continual” version of Herz spaces (which coincides with the “discrete” one when q is constant).

Weak Type Estimate of Singular Integral Operators on Variable Weak Herz-Type Hardy Spaces

“…Let f ∈ H Kα(•) p(•),q(•) . Due to Theorem 4, we have decomposition (19) and estimate (20). We will show that…”

“…In the classical Herz spaces, Theorems 1 and 2 can be found in [16]. We refer the reader to [19] for the boundedness of Riesz potential operator in continual variable exponents Herz spaces.…”

“…Let f ∈ H Kα(•) p(•),q(•) . Then we have decomposition (19) and estimate (20). As before, we decompose f as in (21).…”

Higher order commutators of fractional integrals on Morrey type spaces with variable exponents

“…Following the fundamental work of Kováčik and Rákosník [24], the theory of function spaces with variable exponent has made an explosive growth over the past 25 years. In particular, after the boundedness of the Hardy-Littlewood maximal operator has been proved in [6,10,28], Lebesgue spaces and various other function spaces arising in analysis and PDE, such as Morrey spaces, Herz spaces and Hardy spaces and so on, have been intensively studied in the variable exponent setting, see [2,9,12,23,29,[35][36][37][38][39].…”