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Register a new userFractional integration of summable functions: Mazyas $Phi$-inequalities
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Authors
Dmitriy Stolyarov
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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 $f$.
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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 and tempered fractional calculus. This paper establishes a connection between these two definitions, writing one in terms of the other by making use of the theory of fractional calculus with respect to functions. By extending this connection in a natural way, a generalisation is developed which unifies several existing fractional operators: Riemann--Liouville, Caputo, classical Hadamard, Hadamard-type, tempered, and all of these taken with respect to functions. The fundamental calculus of these generalised operators is established, including semigroup and reciprocal properties as well as application to some example functions. Function spaces are constructed in which the new operators are defined and bounded. Finally, some formulae are derived for fractional integration by parts with these operators.
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 understanding and extending this topic of study. Motivated by fractional differential equations, we present an operational calculus approach for Laplace transforms with respect to functions and their relationship with fractional operators with respect to functions. This approach makes the generalised Laplace transforms much easier to analyse and to apply in practice. We prove several important properties of these generalised Laplace transforms, including an inversion formula, and apply it to solve some fractional differential equations, using the operational calculus approach for efficient solving.
We establish improv
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: $$G^{mathbb{P}}(f)=Big(int_0^infty t|frac{partial}{partial t} mathbb{P}_t(f)|^2dtBig)^{frac12}.$$ The classical Littlewood-Paley $g$-function inequality asserts that for any $1<p<infty$ there exist two positive constants $mathsf{L}^{mathbb{P}}_{t, p}$ and $mathsf{L}^{mathbb{P}}_{c, p}$ such that $$ big(mathsf{L}^{mathbb{P}}_{t, p}big)^{-1}big|fbig|_{p}le big|G^{mathbb{P}}(f)big|_{p} le mathsf{L}^{mathbb{P}}_{c,p}big|fbig|_{p},,quad fin L_p(mathbb{R}^d). $$ We determine the optimal orders of magnitude on $p$ of these constants as $pto1$ and $ptoinfty$. We also consider similar problems for more general test functions in place of the Poisson kernel. The corresponding problem on the Littlewood-Paley dyadic square function inequality is investigated too. Let $Delta$ be the partition of $mathbb{R}^d$ into dyadic rectangles and $S_R$ the partial sum operator associated to $R$. The dyadic Littlewood-Paley square function of $f$ is $$S^Delta(f)=Big(sum_{RinDelta} |S_R(f)|^2Big)^{frac12}.$$ For $1<p<infty$ there exist two positive constants $mathsf{L}^{Delta}_{c,p, d}$ and $ mathsf{L}^{Delta}_{t,p, d}$ such that $$ big(mathsf{L}^{Delta}_{t,p, d}big)^{-1}big|fbig|_{p}le big|S^Delta(f)big|_{p}le mathsf{L}^{Delta}_{c,p, d}big|fbig|_{p},quad fin L_p(mathbb{R}^d). $$ We show that $$mathsf{L}^{Delta}_{t,p, d}approx_d (mathsf{L}^{Delta}_{t,p, 1})^d;text{ and }; mathsf{L}^{Delta}_{c,p, d}approx_d (mathsf{L}^{Delta}_{c,p, 1})^d.$$ All the previous results can be equally formulated for the $d$-torus $mathbb{T}^d$. We prove a de Leeuw type transference principle in the vector-valued setting.
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