Riesz Potential and Fractional Maximal Function

In this article, we begin with Riesz potential. We then discuss some properties of the Riesz potential. Finally we discuss a relation of Riesz Potential with fractional maximal function in the sense that fractional maximal function can be controlled by Riesz potential and the fractional maximal function maps the space Lp to Lq whenever the Riesz potential does.

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

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.

An extension of the Muckenhoupt–Wheeden theorem to generalized weighted Morrey spaces

In this paper, we find the condition on a function ω and a weight v which ensures the equivalency of norms of the Riesz potential and the fractional maximal function in generalized weighted Morrey spaces {{\mathcal{M}}_{p,\omega}({\mathbb{R}}^{n},v)} and generalized weighted central Morrey spaces {\dot{\mathcal{M}}_{p,\omega}({\mathbb{R}}^{n},v)}, when v belongs to the Muckenhoupt {A_{\infty}}-class.

Endpoint Sobolev Continuity of the Fractional Maximal Function in Higher Dimensions

We establish continuity mapping properties of the noncentered fractional maximal operator $M_{\beta }$ in the endpoint input space $W^{1,1}({\mathbb R}^d)$ for $d \geq 2$ in the cases for which its boundedness is known. More precisely, we prove that for $q=d/(d-\beta )$ the map $f \mapsto |\nabla M_\beta f|$ is continuous from $W^{1,1}({\mathbb R}^d)$ to $L^{q}({\mathbb R}^d)$ for $ 0 < \beta < 1$ if $f$ is radial and for $1 \leq \beta < d$ for general $f$. The results for $1\leq \beta < d$ extend to the centered counterpart $M_\beta ^c$. Moreover, if $d=1$, we show that the conjectured boundedness of that map for $M_\beta ^c$ implies its continuity.