Hormander Multiplier Research Articles

Boundedness of Sparse and Rough Operators on Weighted Lorentz Spaces

We present new estimates in the setting of weighted Lorentz spaces for important operators in Harmonic Analysis such as sparse operators, Bochner–Riesz at the critical index, Hormander multipliers and rough singular integrals among others.

The Hörmander multiplier theorem for n-linear operators

In this paper, we study the Hormander multiplier theorem for multilinear operators. We generalize the result of Tomita (J Funct Anal 259(8):2028–2044, 2010) to wider target spaces and extend that of Grafakos and Van Nguyen (Monatsh Math 190(4):735–753, 2019) to multilinear operators. We indeed give two different proofs: The first proof is based on the results of Grafakos et al. (Can J Math 65(2):299–330, 2013; II J Math Soc Jpn 69(2):529–562, 2017), Grafakos and Van Nguyen (Colloq Math 144(1):1–30, 2016; Monatsh Math 190(4):735–753, 2019), Miyachi and Tomita (Rev Mat Iberoam 29(2):495–530, 2013) and for the second one we provide a new and original approach, inspired by Muscalu et al. (Acta Math 193(2):269–296, 2004). We also give an application and discuss the sharpness of the result.

Some Remarks on the Mikhlin–Hörmander and Marcinkiewicz Multiplier Theorems: A Short Historical Account and a Recent Improvement

We present a short historical overview of the Mikhlin–Hormander and Marcinkiewicz multiplier theorems. We discuss different versions of them and provide comparisons. We also present a recent improvement of the Marcinkiewicz multiplier theorem in the two-dimensional case.

BMO multilinear multiplier theorem of Mikhlin–Hörmander type

In this paper we provide an improved BMO version of the Mikhlin–Hormander multiplier theorem for multilinear operators.

Equivalence of (quasi-)norms on a vector-valued function space and its applications to multilinear operators

In this paper we present (quasi-)norm equivalence on a vector-valued function space $L^p_A(l^q)$ and extend the equivalence to $p=\infty$ and $0<q<\infty$ in the scale of Triebel-Lizorkin space, motivated by Fraizer-Jawerth. By applying the results, we improve the multilinear Hormander's multiplier theorem of Tomita, that of Grafakos-Si, and the boundedness results for bilinear pseudo-differential operators, given by Koezuka-Tomita.

The Hörmander multiplier theorem, III: the complete bilinear case via interpolation

We develop a special multilinear complex interpolation theorem that allows us to prove an optimal version of the bilinear Hormander multiplier theorem concerning symbols that lie in the Sobolev space $$L^r_s({\mathbb {R}}^{2n})$$ , $$2\le r<\infty $$ , $$rs>2n$$ , uniformly over all annuli. More precisely, given such a symbol with smoothness index s, we find the largest open set of indices $$(1/p_1,1/p_2 )$$ for which we have boundedness for the associated bilinear multiplier operator from $$L^{p_1}({\mathbb {R}}^{ n})\times L^{p_2} ({\mathbb {R}}^{ n})$$ to $$ L^p({\mathbb {R}}^{ n})$$ when $$1/p=1/p_1+1/p_2$$ , $$1<p_1,p_2<\infty $$ .

The Hörmander multiplier theorem, II: The bilinear local $$L^2$$ L 2 case

We use wavelets of tensor product type to obtain the boundedness of bilinear multiplier operators on $$\mathbb R^n\times \mathbb R^n$$ associated with Hormander multipliers on $$\mathbb R^{2n}$$ with minimal smoothness. We focus on the local $$L^2$$ case and we obtain boundedness under the minimal smoothness assumption of n / 2 derivatives. We also provide counterexamples to obtain necessary conditions for all sets of indices.

Hörmander Functional Calculus for Poisson Estimates

The aim of the article is to show a Hormander spectral multiplier theorem for an operator A whose kernel of the semigroup exp(−zA) satisfies certain Poisson estimates for complex times z. Here exp(−zA) acts on $${L^p(\Omega),\,1 < p < \infty}$$ , where $${\Omega}$$ is a space of homogeneous type with the additional conditions that the volume of balls grows polynomially of exponent d and the measure of annuli is controlled by the corresponding euclidean term. In most of the known Hormander type theorems in the literature, Gaussian bounds and self-adjointness for the semigroup are needed, whereas here the new feature is that the assumptions are the to some extent weaker Poisson bounds, and $${H^\infty}$$ calculus in place of self-adjointness. The order of derivation in our Hormander multiplier result is typically $${\frac{d}{2}}$$ , d being the dimension of the space $${\Omega}$$ . Moreover the functional calculus resulting from our Hormander theorem is shown to be $${R}$$ -bounded. Finally, the result is applied to some examples.

A new class of spaces and the application in the Hormander multipliers

We introduce a new class of spaces. The properties are discussed sufficiently. And the Wiener spaces embedded into the spaces. We present the application in the Hormander multipliers. Keyword: Hormander multiplierss α-Modulation Spaces Wiener spaces MSC(2010): 42B15 42B35

Eigenfunction estimates for Neumann Laplacian and applications to multiplier problems

On compact Riemannian manifolds with boundary, the L∞ estimates and gradient estimates for the eigenfunctions of the Neumann Laplacian are proved. Applying the Lp estimates and gradient estimates to multiplier problems on eigenfunction expansions for the Neumann Laplacian, some new estimates for Bochner Riesz means and the sharp Hormander Multiplier Theorem are obtained.

Spectral multipliers for Schrödinger operators

We prove a sharp Hormander multiplier theorem for Schrodinger operators $H=-\Delta+V$ on $\mathbb{R}^n$. The result is obtained under certain condition on a weighted $L^\infty$ estimate, coupled with a weighted $L^2$ estimate for $H$, which is a weaker condition than that for nonnegative operators via the heat kernel approach. Our approach is elaborated in one dimension with potential $V$ belonging to certain critical weighted $L^1$ class. Namely, we assume that $\int(1+|x|) |V(x)|\,dx$ is finite and $H$ has no resonance at zero. In the resonance case, we assume $\int(1+|x|^2) |V(x)|\, dx$ is finite.

Gradient estimates for the eigenfunctions on compact manifolds with boundary and Hörmander Multiplier Theorem

On compact Riemannian manifolds (M, g) of dimension n ≥ 2 with smooth boundary, the gradient estimates for the eigenfunctions of the Dirichlet Laplacian are proved by the maximum principle. Using the L∞ estimates and gradient estimates, the Hormander multiplier theorem is showed for the eigenfunction expansion of Dirichlet Laplacian on compact manifolds with boundary.

Convolution operators on Lorentz spaces over locally compact Vilenkin groups

Let G be a locally compact Vilenkin group. Using Herz spaces, we give sufficient conditions for a distribution on G to be a convolution operator on certain Lorentz spaces. Our results generalize Hormander's multiplier theorem on G. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Lp-Spectral Multipliers for the Hodge Laplacian Acting on 1-Forms on the Heisenberg Group

We prove that, if \(\Delta_{1}\) is the Hodge Laplacian acting on differential 1-forms on the (2n + 1)-dimensional Heisenberg group, and if m is a Mihlin–Hormander multiplier on the positive half-line, with L2-order of smoothness greater than \(n + \frac{1}{2}\) , then m(Δ1) is Lp-bounded for 1 < p < ∞. Our approach leads to an explicit description of the spectral decomposition of \(\Delta_{1}\) on the space of L2-forms in terms of the spectral analysis of the sub-Laplacian L and the central derivative T, acting on scalar-valued functions.

The Compactness of Commutators on the Heisenberg Group

We consider the commutators of the Hormander multiplier with CMO–functions on the Heisenberg group. The result of compactness on Lp spaces is proved.

Maximal functions associated with Fourier multipliers of Mikhlin-H�rmander type

We show that maximal operators formed by dilations of Mikhlin- Hormander multipliers are typically not bounded on Lp(ℝd). We also give rather weak conditions in terms of the decay of such multipliers under which Lp boundedness of the maximal operators holds.

Fourier Multipliers for L p on Chébli-Trimèche Hypergroups

In this paper we consider Fourier multipliers for $L^p$ $(p>1)$ on Chebli-Trimeche hypergroups and establish a version of Hormander's multiplier theorem. As applications we give some results concerning the Riesz potentials and oscillating multipliers.

Variants of the Calderón-Zygmund Theory for $L^p$-spaces

The purposes of this paper may be described as follows: (i) to provide a useful substitute for the Cotlar-Stein lemma for Lp-spaces (the orthogonality conditions are replaced by certain fairly weak smoothness asumptions); (ii) to investigate the gap between the Hormander multiplier theorem and the Littman-McCarthy-Riviere example - just how little regularity is really needed? (iii) to simplify and extend the work of Duoandikoetxea and Rubio de Francia and Christ and Stein, which sometimes has unnecessarily strong assumptions, and to introduce a sensitivity to different Lp-spaces which does not appear in their work.