Sharp continuity results for the short-time Fourier transform and for localization operators

We completely characterize the boundedness on Wiener amalgam spaces of the short-time Fourier transform (STFT), and on both L ^p and Wiener amalgam spaces of a special class of pseudodifferential operators, called localization operators. Precisely, sufficient conditions for the STFT to be bounded on the Wiener amalgam spaces W(L ^p , L ^q ) are given and their sharpness is shown. Localization operators are treated similarly: using different techniques from those employed in the literature, we relax the known sufficient boundedness conditions for these operators to be bounded on L ^p spaces and prove the optimality of our results. Next, we exhibit sufficient and necessary conditions for such operators to be bounded on Wiener amalgam spaces.     

Inversion formulas for the short-time Fourier transform

The inversion formula for the short-time Fourier transform is usually considered in the weak sense, or only for specific combinations of window functions and function spaces such as L₂. In the present article the so-called θ-summability (with a function parameter θ) is considered which induces norm convergence for a large class of function spaces. Under some conditions on θ we prove that the summation of the short-time Fourier transform of ƒ converges to ƒ in Wiener amalgam norms, hence also in the L_p sense for L_p functions, and pointwise almost everywhere.     

Sharp boundedness and continuity results for the singular porous medium equation

This paper generalizes Shelah’s generic pair conjecture (now theorem) for the measurable cardinal case from first order theories to finite diagrams. We use homogeneous models in the place of saturated models.     

Time-frequency localization for the short time Fourier transform

We use some estimating of orthogonal projection in a reproducing kernel Hilbert space, to prove a sharp quantitaive form of Shapiro's mean dispersion theroem with generalized dispersion for the short time Fourier transform. Other forms of localization of orthonormal sequences in L²?^d) notably the umbrella theorem, are also proved for the short time Fourier transform.     

Differential operators on toric varieties and Fourier transform

We show that Fourier transforms on the Weyl algebras have a geometric counterpart in the framework of toric varieties, namely they induce isomorphisms between twisted rings of differential operators on regular toric varieties, whose fans are related to each other by reflections of one-dimensional cones. The simplest class of examples is provided by the toric varieties related by such reflections to projective spaces. It includes the blow-up at a point of the affine space and resolution of singularities of varieties appearing in the study of the minimal orbit of .     

On the Fourier transform of distributions and differential operators in Clifford analysis

In Brackx et al., 2004 (F. Brackx, R. Delanghe and F. Sommen (2004). Spherical means and distributions in Clifford analysis. In: Tao Qian, Thomas Hempfling, Alan McIntosch and Frank Sommen (Eds.), Advances in Analysis and Geometry: New Developments Using Clifford Algebra, Trends in Mathematics, pp. 65–96. Birkhäuser, Basel.), some fundamental higher dimensional distributions have been reconsidered within the framework of Clifford analysis. Here, the Fourier transforms of these distributions are calculated, revealing a.o. the Fourier symbols of some important translation invariant (convolution) operators, which can be interpreted as members of the considered families. Moreover, these results are the incentive for calculating the Fourier symbols of some differential operators which are at the heart of Clifford analysis, but do not show the property of translation invariance and hence, can no longer be interpreted as convolution operators.     

Sharp rate of average decay of the Fourier transform of a bounded set

Estimates for the decay of Fourier transforms of measures have extensive applications in numerous problems in harmonic analysis and convexity including the distribution of lattice points in convex domains, irregularities of distribution, generalized Radon transforms and others. Here we prove that the spherical L ²-average decay rate of the Fourier transform of the Lebesgue measure on an arbitrary bounded convex set in $\mathbb{R}^{d}$ is $${\bigg(\int_{S^{d-1}}{\big|\widehat{\chi}_B(R\omega)\big|}^2d\omega \bigg)}^{{1}/{2}} \lesssim R^{-\frac{d+1}{2}}.\eqno(*)$$ This estimate is optimal for any convex body and in particular it agrees with the familiar estimate for the ball. The above estimate was proved in two dimensions by Podkorytov, and in all dimensions by Varchenko under additional smoothness assumptions. The main result of this paper proves (*) in all dimensions under the convexity hypothesis alone. We also prove that the same result holds if the boundary of is C^3/2.     

On a method of defining the Fourier transform of nonselfadjoint operators

We give a generalization of the study of nonselfadjoint perturbations of selfadjoint operators and the Fourier transforms of such perturbations to the case when the unperturbed operator is nonselfadjoint and has a simple structure. In defining and studying the Fourier transform of the perturbed operator we use smoothness of Kato type of the perturbation with respect to the unperturbed operator.Translated fromMatematicheskie Melody i Fiziko-Mekhanicheskie Polya, Issue 34, 1991, pp. 19–21.     

Sharp inequalities for convolution-type operators

Let μ be a probability measure on [− a, a], a > 0, and let x₀ε[− a, a], f ε Cⁿ([−2a, 2a]), n 0 even. Using moment methods we derive best upper bounds to ¦∫_−a^a ([f(x₀ + y) + f(x₀ − y)]/2) μ(dy) − f(x₀)¦, leading to sharp inequalities that are attainable and involve the second modulus of continuity of f⁽ⁿ⁾ or an upper bound of it.     

Domains of convolutions and of some operators related to Fourier transform

It is shown that the extended domain of the convolution operator with an almost periodic function on a locally compact group coincides with the proper domain of the operator. Examples are given of integral operators with the proper domain L¹ and with the extended domain containing functions of arbitrary growth.     

Sharp bounds for fractional one-sided operators

In this paper,we characterize the sharp boundedness of the one-sided fractional maximal function for one-weight and two-weight inequalities.Also a new two-weight testing condition for the one-sided fractional maximal function is introduced extending the work of Martín-Reyes and de la Torre.Improving some extrapolation result for the one-sided case,we get weak sharp bounded estimates for one-sided fractional maximal function and weak and strong sharp bounded estimates for one-sided fractional integral.     

The boundedness of maximal operators and singular integrals via Fourier transform estimates

In this paper, the author studies the mapping properties for some general maximal operators and singular integrals on certain function spaces via Fourier transform estimates. Also, some concrete maximal operators and singular integrals are studied as applications.