No Arabic abstract
In this note we prove the estimate $M^{sharp}_{0,s}(Tf)(x) le c,M_gamma f(x)$ for general fractional type operators $T$, where $M^{sharp}_{0,s}$ is the local sharp maximal function and $M_gamma$ the fractional maximal function, as well as a local version of this estimate. This allows us to express the local weighted control of $Tf$ by $M_gamma f$. Similar estimates hold for $T$ replaced by fractional type operators with kernels satisfying H{o}rmander-type conditions or integral operators with homogeneous kernels, and $M_gamma $ replaced by an appropriate maximal function $M_T$. We also prove two-weight, $L^p_v$-$L^q_w$ estimates for the fractional type operators described above for $1<p< q<infty$ and a range of $q$. The local nature of the estimates leads to results involving generalized Orlicz-Campanato and Orlicz-Morrey spaces.
The relationship between the operator norms of fractional integral operators acting on weighted Lebesgue spaces and the constant of the weights is investigated. Sharp boundsare obtained for both the fractional integral operators and the associated fractional maximal functions. As an application improved Sobolev inequalities are obtained. Some of the techniques used include a sharp off-diagonal version of the extrapolation theorem of Rubio de Francia and characterizations of two-weight norm inequalities.
We prove in this note one weight norm inequalities for some positive Bergman-type operators.
In this note the weak type estimates for fractional integrals are studied. More precisely, we adapt the arguments of Domingo-Salazar, Lacey, and Rey to obtain improvements for the endpoint weak type estimates for regular fractional sparse operators.
We prove the endpoint weak type estimate for square functions of Marcinkiewicz type with fractional integrals associated with non-isotropic dilations. This generalizes a result of C. Fefferman on functions of Marcinkiewicz type by considering fractional integrals of mixed homogeneity in place of the Riesz potentials of Euclidean structure.
Let $Omega$ be homogeneous of degree zero and have mean value zero on the unit sphere ${S}^{n-1}$, $T_{Omega}$ be the convolution singular integral operator with kernel $frac{Omega(x)}{|x|^n}$. For $bin{rm BMO}(mathbb{R}^n)$, let $T_{Omega,,b}$ be the commutator of $T_{Omega}$. In this paper, by establishing suitable sparse dominations, the authors establish some weak type endpoint estimates of $Llog L$ type for $T_{Omega,,b}$ when $Omegain L^q(S^{n-1})$ for some $qin (1,,infty]$.