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We study the inequalities of the type $|int_{mathbb{R}^d} Phi(K*f)| lesssim |f|_{L_1(mathbb{R}^d)}^p$, where the kernel $K$ is homogeneous of order $alpha - d$ and possibly vector-valued, the function $Phi$ is positively $p$-homogeneous, and $p = d/(d-alpha)$. Under mild regularity assumptions on $K$ and $Phi$, we find necessary and sufficient conditions on these functions under which the inequality holds true with a uniform constant for all sufficiently regular functions $f$.
Many different types of fractional calculus have been defined, which may be categorised into broad classes according to their properties and behaviours. Two types that have been much studied in the literature are the Hadamard-type fractional calculus
An important class of fractional differential and integral operators is given by the theory of fractional calculus with respect to functions, sometimes called $Psi$-fractional calculus. The operational calculus approach has proved useful for understa
We prove in this note one weight norm inequalities for some positive Bergman-type operators.
Let ${mathbb{P}_t}_{t>0}$ be the classical Poisson semigroup on $mathbb{R}^d$ and $G^{mathbb{P}}$ the associated Littlewood-Paley $g$-function operator: $$G^{mathbb{P}}(f)=Big(int_0^infty t|frac{partial}{partial t} mathbb{P}_t(f)|^2dtBig)^{frac12}.