A note on maximal operators related to Laplace-Bessel differential operators on variable exponent Lebesgue spaces

Abstract

In this paper, we consider the maximal operator related to the Laplace-Bessel differential operator ( B B -maximal operator) on L p ( ⋅ ) , γ ( R k , + n ) {L}_{p\left(\cdot ),\gamma }\left({{\mathbb{R}}}_{k,+}^{n}) variable exponent Lebesgue spaces. We will give a necessary condition for the boundedness of the B B -maximal operator on variable exponent Lebesgue spaces. Moreover, we will obtain that the B B -maximal operator is not bounded on L p ( ⋅ ) , γ ( R k , + n ) {L}_{p\left(\cdot ),\gamma }\left({{\mathbb{R}}}_{k,+}^{n}) variable exponent Lebesgue spaces in the case of p − = 1 {p}_{-}=1 . We will also prove the boundedness of the fractional maximal function associated with the Laplace-Bessel differential operator (fractional B B -maximal function) on L p ( ⋅ ) , γ ( R k , + n ) {L}_{p\left(\cdot ),\gamma }\left({{\mathbb{R}}}_{k,+}^{n}) variable exponent Lebesgue spaces.

Some results concerning Riesz–Bessel transforms of high order

“…Proof By using $$$ \mid {f}^{\sharp}\mid \le 2{M}_{\gamma }f $$$ and the boundedness of the $$$ B $$$‐maximal operators $$$ {M}_{\gamma } $$$ on $$$ {L}_{p\left(\cdotp \right),\gamma}\left({\mathrm{\mathbb{R}}}_{k,+}^n\right) $$$ variable Lebesgue spaces (see [2, 8]), the desired result is obtained. □…”

“…The problem of boundedness of these operators and their versions, which are generated by Laplace-Bessel differential operators, is important in harmonic analysis. Over the years, this problem has been discussed by Aliev, Ekincioglu, Gadjiev, Guliyev, Kaya, Kipriyanov, Klyuchantsev, Lyakhov, Serbetci, Stempak, and others [1][2][3][4][5][6][7][8][9][10][11][12][13].…”

“…While these assumptions are sufficient, we prefer to work with a weaker hypothesis, which is the boundedness of B-maximal operator M 𝛾 on variable Lebesgue spaces L p(•),𝛾 . In [2,8,20], it has been obtained that B-maximal operators defined on variable Lebesgue spaces are bounded. These studies and the results obtained therein motivate us to investigate the boundedness of Riesz-Bessel transforms of high order on variable…”

“…While these assumptions are sufficient, we prefer to work with a weaker hypothesis, which is the boundedness of $$$ B $$$‐maximal operator $$$ {M}_{\gamma } $$$ on variable Lebesgue spaces $$$ {L}_{p\left(\cdotp \right),\gamma } $$$. In [2, 8, 20], it has been obtained that $$$ B $$$‐maximal operators defined on variable Lebesgue spaces are bounded. These studies and the results obtained therein motivate us to investigate the boundedness of Riesz–Bessel transforms of high order on variable Lebesgue spaces $$$ {L}_{p\left(\cdotp \right),\gamma}\left({\mathrm{\mathbb{R}}}_{k,+}^n\right) $$$.…”

A different approach to boundedness of the B-maximal operators on the variable Lebesgue spaces

“…In [8], Guliyev has obtained the L p,γ −boundedness of the B−maximal operator. Moreover, in [6,12], it has been shown that the B−maximal operator is L p(•),γ −bounded by using the L p(•) − boundedness of a maximal operator whose domain is a homogeneous space.…”