Some new weak-( H p - L p )-type inequalities for weighted maximal operators of Fejér means of Vilenkin–Fourier series

In this paper, we introduce new weighted maximal operators of the Fejér means of the Vilenkin–Fourier series. We prove that, for some “optimal” weights, these operators are bounded from the martingale Hardy space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>H</m:mi> <m:mi>p</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>G</m:mi> <m:mi>m</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {Hₚ(Gₘ)} to the space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mtext>weak-</m:mtext> <m:mo>⁢</m:mo> <m:msub> <m:mi>L</m:mi> <m:mi>p</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>G</m:mi> <m:mi>m</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {weak-Lₚ(Gₘ)} for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>0</m:mn> <m:mo><</m:mo> <m:mi>p</m:mi> <m:mo><</m:mo> <m:mfrac> <m:mn>1</m:mn> <m:mn>2</m:mn> </m:mfrac> </m:mrow> </m:math> {0<p<1/2} . Moreover, we also establish the sharpness of this result. As a consequence, we obtain a number of new and well-known results.