Inequalities For Fourier Research Articles

Fourier inequalities for averaged integral transforms

The classical Hardy–Littlewood result states that ‖|⋅|1−2/p|fˆ|‖p≲‖f‖p for 1<p⩽2. The similar inequality for orthonormal Fourier series has been obtained by Paley. We extend these results for any p>1 considering different integral averages of f in the left-hand side of the estimates.

Three inequalities for quadratic-phase Fourier transform

Three inequalities for quadratic-phase Fourier transform

Hardy–Littlewood-type theorems for Fourier transforms in [formula omitted

We obtain Fourier inequalities in the weighted Lp spaces for any 1<p<∞ involving the Hardy–Cesàro and Hardy–Bellman operators. We extend these results to product Hardy spaces for p⩽1. Moreover, boundedness of the Hardy-Cesàro and Hardy-Bellman operators in various spaces (Lebesgue, Hardy, BMO) is discussed. One of our main tools is an appropriate version of the Hardy–Littlewood–Paley inequality ‖fˆ‖Lp′,q≲‖f‖Lp,q.

Spectral multipliers in group algebras and noncommutative Calderón-Zygmund theory

In this paper we solve three problems in noncommutative harmonic analysis which are related to endpoint inequalities for singular integrals. In first place, we prove that an L2-form of Hörmander's kernel condition suffices for the weak type (1,1) of Calderón-Zygmund operators acting on matrix-valued functions. To that end, we introduce an improved CZ decomposition for martingale filtrations in von Neumann algebras, and apply a very simple unconventional argument which notably avoids pseudolocalization. In second place, we establish as well the weak L1 endpoint for matrix-valued CZ operators over nondoubling measures of polynomial growth, in the line of the work of Tolsa and Nazarov/Treil/Volberg. The above results are valid for other von Neumann algebras and solve in the positive two open problems formulated in 2009. An even more interesting problem is the lack of L1 endpoint inequalities for singular Fourier and Schur multipliers over nonabelian groups. Given a locally compact group G equipped with a conditionally negative length ψ:G→R+, we prove that Herz-Schur multipliers with symbol m∘ψ satisfying a Mikhlin condition in terms of the ψ-cocycle dimension are of weak type (1,1). Our result extends to Fourier multipliers for amenable groups and imposes sharp regularity conditions on the symbol. The proof crucially combines our new CZ methods with novel forms of recent transference techniques. This L1 endpoint gives a very much expected inequality which complements the L∞→BMO estimates proved in 2014 by Junge, Mei and Parcet.

Weighted Fourier Inequalities and Boundedness of Variation

Weighted Fourier Inequalities and Boundedness of Variation

On weighted Fourier inequalities - some new scales of equivalent conditions

On weighted Fourier inequalities - some new scales of equivalent conditions

Sharp approximation theorems and Fourier inequalities in the Dunkl setting

In this paper we study direct and inverse approximation inequalities in Lp(Rd), 1<p<∞, with the Dunkl weight. We obtain these estimates in their sharp form substantially improving previous results. We also establish new estimates of the modulus of smoothness of a function f via the fractional powers of the Dunkl Laplacian of approximants of f. Moreover, we obtain new Lebesgue type estimates for moduli of smoothness in terms of Dunkl transforms. Needed Pitt-type and Kellogg-type Fourier–Dunkl inequalities are derived.

Weighted Fourier Inequalities in Lebesgue and Lorentz Spaces

In this paper, we obtain sufficient conditions for the weighted Fourier-type transforms to be bounded in Lebesgue and Lorentz spaces. Two types of results are discussed. First, we review the method based on rearrangement inequalities and the corresponding Hardy’s inequalities. Second, we present Hormander-type conditions on weights so that Fourier-type integral operators are bounded in Lebesgue and Lorentz spaces. Both restricted weak- and strong-type results are obtained. In the case of regular weights necessary and sufficient conditions are given.

Some new Fourier inequalities for unbounded orthogonal systems in Lorentz\u2013Zygmund spaces

In this paper we prove some essential complements of the paper (J. Inequal. Appl. 2019:171, 2019) on the same theme. We prove some new Fourier inequalities in the case of the Lorentz–Zygmund function spaces L_{q,r}(log L)^{alpha } involved and in the case with an unbounded orthonormal system. More exactly, in this paper we prove and discuss some new Fourier inequalities of this type for the limit case L_{2,r}(log L)^{alpha }, which could not be proved with the techniques used in the paper (J. Inequal. Appl. 2019:171, 2019).

Sharp inequalities for geometric Fourier transform and associated ambiguity function

The geometric Fourier transform defined using Clifford algebra plays an increasingly active role in modern data analysis, in particular for color image processing. However, due to the non-commutativity, it is hard to obtain many important analytic properties. In this paper, we prove several important sharp inequalities, including the Hausdorff-Young inequality and its converse, Pitt's inequality and Lieb's inequality for Clifford ambiguity functions. With these inequalities, several uncertainty principles with optimal constants are derived for the geometric Fourier transform. Most results in this paper are new even in the easier case, i.e. the quaternion setting. The method developed here also works for more general geometric Fourier transforms.

Hardy-type theorems on Fourier transforms revised

We obtain new conditions on periodic integrable functions so that their transformed Fourier series belong to Lp. This improves the classical Hardy and Bellman results. A counterpart for the Fourier transforms is also established. Our main tool is a new extension of the Hausdorff–Young–Paley inequality for Fourier transforms.

The Weighted Fourier Inequality, Polarity, and Reverse Hölder Inequality

We examine the problem of the Fourier transform mapping one weighted Lebesgue space into another, by studying necessary conditions and sufficient conditions which expose an underlying geometry. In the necessary conditions, this geometry is connected to an old result of Mahler concerning the the measure of a convex body and its geometric polar being essentially reciprocal. An additional assumption, that the weights must belong to a reverse Holder class, is used to formulate the sufficient condition.

Weighted Fourier Inequalities via Rearrangements

The method of using rearrangements to give sufficient conditions for Fourier inequalities between weighted Lebesgue spaces is revisited, a comparison between two known sufficient conditions is completed, and the method is extended to provide sufficient conditions for a new range of indices. When $$1<q<p<\infty $$ , simple conditions on weights ensure that the Fourier transform will map a weighted $$L^p$$ space into a weighted $$L^q$$ space. These are established in Theorems 1 and 4 of Benedetto and Heinig (J Fourier Anal Appl 9(1):1–37, 2003). The proofs apply when $$2<q<p$$ and $$1<q<p<2$$ but not in the remaining case, $$1<q<2<p$$ . Here, counterexamples are given to show that these simple conditions are no longer sufficient when $$1<q<2<p$$ . Also, various additional conditions are presented, any of which will restore sufficiency in that case.

On Lipschitz solutions for some forward–backward parabolic equations. II: the case against Fourier

As a sequel to the paper [9], we study the existence and properties of Lipschitz solutions to the initial-boundary value problem of some forward-backward parabolic equations with diffusion fluxes violating Fourier's inequality.

Uncertainty Principle in Terms of Entropy for the Riemann–Liouville Operator

We prove Hausdorff–Young inequality for the Fourier transform connected with Riemann–Liouville operator. We use this inequality to establish the uncertainty principle in terms of entropy. Next, we show that we can derive the Heisenberg–Pauli–Weyl inequality for the precedent Fourier transform.

On the homogeneous Boltzmann equation with soft-potential collision kernels

We consider the well-posedness problem for the space-homogeneous Boltzmann equation with soft-potential collision kernels. By revisiting the classical Fourier inequalities and fractional integrals, we deduce a set of bilinear estimates for the collision operator on the space of integrable functions possessing certain degree of smoothness and we apply them to prove the local-in-time existence of a solution to the Boltzmann equation in both integral form and the original one. Uniqueness and stability of solutions are also established.

Pitt and Boas inequalities for Fourier and Hankel transforms

We prove Pitt and Boas inequalities for products of radial functions and spherical harmonics in Rn. In the process, we obtain upper and lower estimates of the operator norm of the Hankel transform with power weights. Our inequalities are sharp in some specific cases.

Uncertainty inequalities for the Heisenberg group

We establish the Heisenberg–Pauli–Weyl uncertainty inequalities for Fourier transform and the continuous wavelet transform on the Heisenberg group.

Two-sided weighted Fourier inequalities

Fourier transform estimates for ‖ f ‖Lq,w via ‖ f ‖L p,w from above and from below are studied. For p = q, equivalence results, i.e., C1‖ f ‖L p,w ≤ ‖ f ‖L p,w ≤ C2‖ f ‖L p,w , w(x) = w(1/x)x p−2, 1 ≤ p <∞, are shown to be valid for functions from certain classes under the Muckenhoupt conditions: w ∈ Ap or w ∈ A2p . Sharpness of these conditions is proved. Mathematics Subject Classification (2010): 42A38 (primary); 26D15, 46E30 (secondary).

On inequalities for the Fourier transform of functions from Lorentz spaces

where Lp,q(R) is the classical Lorentz space. These inequalities are called the Hausdorff–Young and Hardy–Littlewood–Stein inequalities, respectively, [1]. There are similar inequalities for Fourier transform on the torus [0, 1], i.e., for f = { fn}, where fn are the Fourier coefficients with respect to some orthonormal system. Suppose that ω is a nonnegative function on [0,∞]. The generalized Lorentz space Λq(ω,R) is the set of all measurable functions f on R such that: