Fourier Multipliers Research Articles

Mikhlin-Type Hp Multiplier Theorem on the Heisenberg Group

The classical Fourier multiplier theorem by Mikhlin in Hardy spaces is extended to the Heisenberg group. The proof relies on the theories of atom and molecule functions and the property of special Hermite functions. The main result is a Mikhlin-type Hp multiplier theorem on the Heisenberg group. If an operator-valued function M(λ) satisfies certain conditions, the right-multiplier operator TM is bounded on the Hardy space Hp(Hn), which is defined in terms of maximal functions, and elements can be decomposed into atoms or molecules. The paper also discusses the relationship with other results and open problems.

A Novel Approach to the Fractional Laplacian via Generalized Spherical Means

Although at least ten equivalent definitions of the fractional Laplacian exist in unbounded domains, we introduce an additional equivalent definition based on the generalized spherical mean-value operator—a Fourier multiplier operator involving the normalized Bessel function. Specifically, we demonstrate that this new definition allows us to reduce any n-dimensional fractional Laplacian to a one-dimensional operator, which simplifies computation and enhances efficiency. We propose two methods for computing the generalized spherical means of a given function: one by solving standard wave equations and the other by solving Darboux’s equations.

On Fourier multipliers with rapidly oscillating symbols

Asymptotically sharp bounds are provided for the L p L_p norms of the Fourier multipliers with symbols e i λ φ ( ξ / | ξ | ) e^{i\lambda \varphi (\xi /|\xi |)} , where λ ∈ R \lambda \in \mathbb {R} is a large parameter.

Pointwise Convergence of Noncommutative Fourier Series

This paper is devoted to the study of pointwise convergence of Fourier series for group von Neumann algebras and quantum groups. It is well-known that a number of approximation properties of groups can be interpreted as summation methods and mean convergence of the associated noncommutative Fourier series. Based on this framework, this paper studies the refined counterpart of pointwise convergence of these Fourier series. As a key ingredient, we develop a noncommutative bootstrap method and establish a general criterion of maximal inequalities for approximate identities of noncommutative Fourier multipliers. Based on this criterion, we prove that for any countable discrete amenable group, there exists a sequence of finitely supported positive definite functions tending to 1 1 pointwise, so that the associated Fourier multipliers on noncommutative L p L_p -spaces satisfy the pointwise convergence for all p > 1 p>1 . In a similar fashion, we also obtain results for a large subclass of groups (as well as quantum groups) with the Haagerup property and the weak amenability. We also consider the analogues of Fejér and Bochner-Riesz means in the noncommutative setting. Our approach heavily relies on the noncommutative ergodic theory in conjunction with abstract constructions of Markov semigroups, inspired by quantum probability and geometric group theory. Finally, we also obtain as a byproduct the dimension free bounds of the noncommutative Hardy-Littlewood maximal inequalities associated with convex bodies.

Hp Estimates for multilinear pseudo-differential operators with flag symbols

In this paper, we establish the hp1×⋯×hpN→Lp (0<pi≤∞ for i=1,2,…,N) estimate for N-linear pseudo-differential operators, where the symbols are given by the product of two standard symbols in multilinear Hörmander class N-S1,δ0. Such product-type symbols are motivated by the flag Fourier multipliers studied by Muscalu [35,36], and we consider its multilinear analogue in pseudo-differential setting. Our arguments are also inspired by Miyachi, Tomita [34] and Chen, Huang, Lu in [6].

Matrix weighted $$\alpha $$-modulation spaces

AbstractGiven a matrix-weight W in the Muckenhoupt class $$\textbf{A}_p({{\mathbb {R}}}^n)$$ A p ( R n ) , $$1\le p&lt;\infty $$ 1 ≤ p &lt; ∞ , we introduce corresponding vector-valued continuous and discrete $$\alpha $$ α -modulation spaces $$M^{s,\alpha }_{p,q}(W)$$ M p , q s , α ( W ) and $$m^{s,\alpha }_{p,q}(W)$$ m p , q s , α ( W ) and prove their equivalence through the use of adapted tight frames. Compatible notions of molecules and almost diagonal matrices are also introduced, and an application to the study of Fourier multipliers on vector valued spaces is given.

Quantitative weighted estimates for the multilinear pseudo-differential operators in function spaces

Abstract In this paper, the weighted estimates for multilinear pseudo-differential operators were systematically studied in rearrangement invariant Banach and quasi-Banach spaces. These spaces contain the Lebesgue space, the classical Lorentz space and Marcinkiewicz space as typical examples. More precisely, the weighted boundedness and weighted modular estimates, including the weak endpoint case, were established for multilinear pseudo-differential operators and their commutators. As applications, we show that the above results also hold for the multilinear Fourier multipliers, multilinear square functions, and a class of multilinear Calderón–Zygmund operators.

Multi-scale Sparse Domination

We prove a bilinear form sparse domination theorem that applies to many multi-scale operators beyond Calderón–Zygmund theory, and also establish necessary conditions. Among the applications, we cover large classes of Fourier multipliers, maximal functions, square functions and variation norm operators.

Operator-valued multiplier theorems for causal translation-invariant operators with applications to control theoretic input-output stability

AbstractWe prove an operator-valued Laplace multiplier theorem for causal translation-invariant linear operators which provides a characterization of continuity from $$H^\alpha ({\mathbb {R}},U)$$ H α ( R , U ) to $$H^\beta ({\mathbb {R}},U)$$ H β ( R , U ) (fractional U-valued Sobolev spaces, U a complex Hilbert space) in terms of a certain boundedness property of the transfer function (or symbol), an operator-valued holomorphic function on the right-half of the complex plane. We identify sufficient conditions under which this boundedness property is equivalent to a similar property of the boundary function of the transfer function. Under the assumption that U is separable, the Laplace multiplier theorem is used to derive a Fourier multiplier theorem. We provide an application to mathematical control theory, by developing a novel input-output stability framework for a large class of causal translation-invariant linear operators which refines existing input-output stability theories. Furthermore, we show how our work is linked to the theory of well-posed linear systems and to results on polynomial stability of operator semigroups. Several examples are discussed in some detail.

An extension of the Liouville theorem for Fourier multipliers to sub-exponentially growing solutions

We study the equation m(D)f=0 in a large class of sub-exponentially growing functions. Under appropriate restrictions on m \in C(\mathbb{R}^{n}) , we show that every such solution can be analytically continued to a sub-exponentially growing entire function on \mathbb{C}^{n} if, and only if, m(\xi) \neq 0 for \xi \neq 0 .

The Liouville theorem for a class of Fourier multipliers and its connection to coupling

AbstractThe classical Liouville property says that all bounded harmonic functions in , that is, all bounded functions satisfying , are constant. In this paper, we obtain necessary and sufficient conditions on the symbol of a Fourier multiplier operator , such that the solutions to are Lebesgue a.e. constant (if is bounded) or coincide Lebesgue a.e. with a polynomial (if is polynomially bounded). The class of Fourier multipliers includes the (in general non‐local) generators of Lévy processes. For generators of Lévy processes, we obtain necessary and sufficient conditions for a strong Liouville theorem where is positive and grows at most exponentially fast. As an application of our results above, we prove a coupling result for space‐time Lévy processes.

Non-symmetric differentially subordinate martingales and sharp weak-type bounds for Fourier multipliers

Non-symmetric differentially subordinate martingales and sharp weak-type bounds for Fourier multipliers

Bent-half space model problem for Lame equation with surface tension

The study of fluid flow is a very fascinating area of fluid dynamics. Fluid motion has received more and more attention in recent years and numerous researchers have looked into this topic. However, they rarely used a mathematical analysis approach to analyse fluid motion; instead, they used numerical analysis. This serves as a significant justification for the researcher's decision to study fluid flow from the perspective of mathematical analysis. In this paper, we consider the ${\mathcal R}$-boundedness of the solution operator families of the Lam\'e equation with surface tension in bent half-space model problem by taking into account the surface tension in a bounded domain of {\it N}-dimensional Euclidean space ($N \geq 2$). The motion of the model problem can be described by linearizing an equation system of a model problem. This research is a continuation of [13]. They investigated the ${\mathcal R}$-boundedness of the solution operator families in the half-space case for the model problem of the Lam\'e equation with surface tension. First of all, by using Laplace transformation we consider the resolvent of the model problem, then treat the problem in bent half-space case. By using Weis's operator-valued Fourier multiplier theorem, we know that ${\mathcal R}$-boundedness implies the maximal $L_p$-$L_q$ regularity for the initial boundary value. This regularity is an essential tool for the partial differential equation problem.

Reproducing property of bounded linear operators and kernel regularized least square regressions

We consider bounded linear operators from the view of functional reproducing property. For some bounded linear operators associated with orthogonal polynomials we define an inner product space associated with a kernel constructed with orthogonal polynomials, show that it is a functional reproducing kernel Hilbert space (FRKHS) associated with these bounded linear operators and give decay rate for the best FRKHS approximation with a [Formula: see text]-functional associated with the FRKHS. On this basis, we provide a learning rate for kernel regularized regression whose hypothesis space is the defined FRKHS. As applications, we define some concrete FRKHSs associated with polynomial operators such as the Bernstein–Durrmeyer operators, the de la Vallée Poussin operators on both the unit sphere [Formula: see text] and the unit ball [Formula: see text]. We show that these polynomial operators have reproducing property with respect to the corresponding concrete FRKHSs and show the learning rate for the kernel regularized regression. In short, we provide a way of constructing FRKHS operators with Fourier multipliers and show a learning framework from the view of operator approximation.

Generic norm growth of powers of homogeneous unimodular Fourier multipliers

Generic norm growth of powers of homogeneous unimodular Fourier multipliers

Generalized Multiple Multiplicative Fourier Transform and Estimates of Integral Moduli of Continuity

Generalized Multiple Multiplicative Fourier Transform and Estimates of Integral Moduli of Continuity

Coercive estimates for multilayer degenerate di erential operators

We obtain the conditions under which a given multilayer differential operator \(P(D)\) (polynomial \(P(\xi)\)) is more powerful than operator \(Q(D)\) (polynomial \(Q(\xi)\)). This is used to obtain estimates of monomials, which, in turn, using the theory of Fourier multipliers, is used to obtain coercive estimates of derivatives of functions through the differential operator \(P(D)\) applied to these functions.

Estimates for some bilinear wave operators

We consider some bilinear Fourier multiplier operators and give a bilinear version of Seeger, Sogge, and Stein’s result for Fourier integral operators. Our results improve, for the case of Fourier multiplier operators, Rodríguez-López, Rule, and Staubach’s result for bilinear Fourier integral operators. The sharpness of the results is also considered.

TRILINEAR FOURIER MULTIPLIERS ON HARDY SPACES

Abstract In this paper, we obtain the $H^{p_1}\times H^{p_2}\times H^{p_3}\to H^p$ boundedness for trilinear Fourier multiplier operators, which is a trilinear analogue of the multiplier theorem of Calderón and Torchinsky [4]. Our result improves the trilinear estimate in [22] by additionally assuming an appropriate vanishing moment condition, which is natural in the boundedness into the Hardy space $H^p$ for $0&lt;p\le 1$ .

On the interpolation of the spaces $W^{l,1}( \mathbb{R}^{d})$ and $W^{r,\infty }( \mathbb{R}^{d})$

We study some properties of spaces obtained by interpolation of the Sobolev spaces W^{k,1}(\mathbb{R}^{d}) and W^{l,\infty}(\mathbb{R}^{d}) , where l and r are nonnegative integers, and {d\geq 2} . We are concerned with the standard real and complex methods of interpolation. In the case of the real method, an old result of De Vore and Scherer (1979) gives that (W^{l,1}(\mathbb{R}^{d}),W^{l,\infty }(\mathbb{R}^{d}))_{\theta,p_{\theta}}= W^{l,p_{\theta}}(\mathbb{R}^{d}), where \theta\in (0,1) and 1/p_{\theta}=1-\theta . We complement this result by considering the case l\neq r . We prove that, when l\neq r , (W^{l,1}(\mathbb{R}^{d}),W^{r,\infty }(\mathbb{R}^{d}))_{\theta,q}=B_{q}^{\sigma ,q}(\mathbb{R}^{d}), \tag{$\star$} where \sigma :=( 1-\theta ) l+\theta r and 1/q= 1-\theta , if and only if l-r\in \mathbb{R}\setminus [1,d] . Also, we prove a similar fact when W^{l,1} is replaced in (\star) by a space W^{s,p} where s\neq r is a real number and p\in (1,\infty) . Several other problems like the boundedness of the Riesz transforms on interpolation spaces are also considered. In the case of the complex method, it was proved by M. Milman (1983) that, for any 1&lt;p&lt;\infty , (W^{l,1}(\mathbb{R}^{d}),W^{l,p }(\mathbb{R}^{d}))_{\theta}=W^{l,p_{\theta}}(\mathbb{R}^{d}), \tag{$\star\star$} where 1/p_{\theta}=(1-\theta)+\theta/p . We show by simple arguments that (\star\star) fails when p=\infty and l\geq 1 , answering a question of P. W. Jones (1984). As an immediate consequence of these arguments, we show that certain closed subspaces of (C(\mathbb{T}^{d}))^{N} (with N\in \mathbb{N}^{*} ) that are described by Fourier multipliers are not complemented in (C(\mathbb{T}^{d}))^{N} .