Fractional Leibniz rules for bilinear spectral multipliers on Carnot groups

Let \mathscr{L} be a smooth second-order real differential operator in divergence form on a manifold of dimension n . Under a bracket-generating condition, we show that the ranges of validity of spectral multiplier estimates of Mikhlin–Hörmander type and wave propagator estimates of Miyachi–Peral type for \mathscr{L} cannot be wider than the corresponding ranges for the Laplace operator on \mathbb{R}^n . The result applies to all sub-Laplacians on Carnot groups and more general sub-Riemannian manifolds, without restrictions on the step. The proof hinges on a Fourier integral representation for the wave propagator associated with \mathscr{L} and nondegeneracy properties of the sub-Riemannian geodesic flow.

(1971). Fractional Derivatives and Leibniz Rule. The American Mathematical Monthly: Vol. 78, No. 6, pp. 645-649.

The fractional Leibniz rules on the product space of Carnot groups

We show that in the Kato–Ponce inequality \|J^s(fg)-fJ^s g\|_p \lesssim \| \partial f \|_{\infty} \,\| J^{s-1} g \|_p + \| J^s f \|_p \,\|g\|_{\infty} , the J^s f term on the right-hand side can be replaced by J^{s-1} \partial f . This solves a question raised in Kato–Ponce [14]. We propose a new fractional Leibniz rule for D^s=(-\Delta)^{s/2} and similar operators, generalizing the Kenig–Ponce–Vega estimate [15] to all s > 0 . We also prove a family of generalized and refined Kato–Ponce type inequalities which include many commutator estimates as special cases. To showcase the sharpness of the estimates at various endpoint cases, we construct several counterexamples. In particular, we show that in the original Kato–Ponce inequality, the L^{\infty} -norm on the right-hand side cannot be replaced by the weaker BMO norm. Some divergence-free counterexamples are also included.

In this short communication, we show that the validity of the Leibniz rule for a fractional derivative on a coarse-grained medium brings about a modied chain rule, in agreement with alternative versions of fractional calculus. We compare our results to those of a recent article on this matter.

Leibniz rules and Gauss–Green formulas in distributional fractional spaces

We obtain a fractional Leibniz rule associated to bilinear Hermite pseudo-multipliers in the context of Hermite Besov and Hermite Triebel–Lizorkin spaces. As a byproduct, we show that the classes of bounded functions in these spaces (which include Hermite Sobolev and Hermite Hardy–Sobolev spaces) are algebras under pointwise multiplication. To obtain these results we develop appropriate decompositions for bilinear pseudo-multipliers and molecular estimates for certain families of functions in the Hermite setting.

We develop a general framework for the analysis of operator-valued multilinear multipliers acting on Banach-valued functions. Our main result is a Coifman-Meyer type theorem for operator-valued multilinear multipliers acting on suitable tuples of UMD spaces. A concrete case of our theorem is a multilinear generalization of Weis' operator-valued H\"ormander-Mihlin linear multiplier theorem. Furthermore, we derive from our main result a wide range of mixed $L^p$-norm estimates for multi-parameter multilinear paraproducts, leading to a novel mixed norm version of the partial fractional Leibniz rules of Muscalu et. al.. Our approach works just as well for the more singular tensor products of a one-parameter Coifman-Meyer multiplier with a bilinear Hilbert transform, extending results of Silva. We also prove several operator-valued $T (1)$-type theorems both in one parameter, and of multi-para\-meter, mixed-norm type. A distinguishing feature of our $T(1)$ theorems is that the usual explicit assumptions on the distributional kernel of $T$ are replaced with testing-type conditions. Our proofs rely on a newly developed Banach-valued version of the outer $L^p$ space theory of Do and Thiele.

The fractional Leibniz rule is generalized by the Coifman-Meyer estimate. It is shown that the arbitrary redistribution of fractional derivatives for higher order with the corresponding correction terms.

The purpose of this paper is to give a definition and prove the fundamental properties of Besov spaces generated by the Neumann Laplacian. As a by-product of these results, the fractional Leibniz rule in these Besov spaces is obtained.

On bivariate fractional calculus with general univariate analytic kernels

Pointwise decay of solutions to the energy critical nonlinear Schrödinger equations

The goal of the work is to verify the fractional Leibniz rule for Dirichlet Laplacian in the exterior domain of a compact set. The key point is the proof of gradient estimates for the Dirichlet problem of the heat equation in the exterior domain. Our results describe the time decay rates of the derivatives of solutions to the Dirichlet problem.

Discrete fractional calculus and the Saalschutz theorem

We consider various versions of fractional Leibniz rules (also known as Kato-Ponce inequalities) with polynomial weights \(\langle x\rangle ^a = (1+|x|^2)^{a/2}\) for \(a\ge 0\). We show that the weighted Kato-Ponce estimate with the inhomogeneous Bessel potential \(J^s = (1- \Delta )^{{s}/{2}}\) holds for the full range of bilinear Lebesgue exponents, for all polynomial weights, and for the sharp range of the degree s. This result, in particular, demonstrates that neither the classical Muckenhoupt weight condition nor the more general multilinear weight condition is required for the weighted Kato-Ponce inequality. We also consider a few other variants such as commutator and mixed norm estimates, and analogous conclusions are derived. Our results contain strong-type inequalities for both \(L^1\) and \(L^\infty \) endpoints, which extend several existing results.