Mixed Norm Estimates Research Articles

Estimates for the $X$-ray transform restricted to 2-manifolds

We prove almost sharp mixed-norm estimates for the X -ray transform restricted to lines whose directions lie on certain well-curved two dimensional manifolds.

Mixed Norm Estimates of Schrödinger Waves and Their Applications

In this paper we establish mixed norm estimates of interactive Schrödinger waves and apply them to study smoothing properties and global well-posedness of the nonlinear Schrödinger equations with mass critical nonlinearity.

On the boundary convergence of solutions to the Hermite–Schrödinger equation

In a half-space we consider the Hermite Schrodinger equation with given boundary values. We prove a formula that links the solution of this problem to that of the classical Schrodinger equation. It shows that mixed norm estimates for the Hermite- Schrodinger equation can be obtained immediately from those known in the classical case. In one space dimension, we deduce sharp pointwise convergence results at the boundary, by means of this link.

Fourier and Radon transform on harmonic 𝑁𝐴 groups

In this article we study the Fourier and the horocyclic Radon transform on harmonic N A NA groups (also known as Damek-Ricci spaces). We consider the geometric Fourier transform for functions on L p L^p -spaces and prove an analogue of the L 2 L^2 -restriction theorem. We also prove some mixed norm estimates for the Fourier transform generalizing the Hausdorff-Young and Hardy-Littlewood-Paley inequalities. Unlike Euclidean spaces the domains of the Fourier transforms are various strips in the complex plane. All the theorems are considered on these entire domains of the Fourier transforms. Finally we deal with the existence of the Radon transform on L p L^p -spaces and obtain its continuity property.

Fractional Integrals on Product Manifolds

In this paper, the authors give a mixed norm estimate for the multi-parameter fractional integrals on product measurable spaces. This estimate is applied to obtain the boundedness for the fractional integrals of Nagel-Stein type on product manifolds, the fractional integral of Folland-Stein type with rough convolution kernels on product homogeneous groups, and the discrete fractional integrals of Stein-Wainger type.

Strichartz estimates via the Schrödinger maximal operator

We consider the Schrodinger operator $${e^{it\Delta}}$$ acting on initial data f in $${\dot{H}^s}$$ . We show that an affirmative answer to a question of Carleson, concerning the sharp range of s for which $${\lim_{t\to 0}e^{it\Delta}f(x)=f(x)}$$ a.e. $${x\in \mathbb {R}^n}$$ , would imply an affirmative answer to a question of Planchon, concerning the sharp range of q and r for which $${e^{it\Delta}}$$ is bounded in $${L_x^q(\mathbb {R}^n,L^r_t(\mathbb {R}))}$$ . When n = 2, we unconditionally improve the range for which the mixed norm estimates hold.

Mixed-norm estimates for a class of nonisotropic directional maximal operators and Hilbert transforms

For all d ⩾ 2 and p ∈ ( 1 , max ( 2 , ( d + 1 ) / 2 ) ] , we prove sharp L p to L p ( L q ) estimates (modulo an endpoint) for a directional maximal operator associated to curves generated by the dilation matrices exp ( ( log t ) P ) , where P has real entries and eigenvalues with positive real part. For the corresponding Hilbert transform we prove an analogous result for all d ⩾ 2 and p ∈ ( 1 , 2 ] . As corollaries, we prove L p bounds for variable kernel singular integral operators and Nikodym-type maximal operators taking averages over certain families of curved sets in R d .

Mixed norm estimates for certain generalized Radon transforms

We obtain essentially optimal local mixed norm inequalities for certain generalized Radon transforms defined by integration over curves.

MIXED NORM ESTIMATES FOR POTENTIAL OPERATORS RELATED TO THE RADON TRANSFORM

AbstractWe define potential operators on hyperplanes and give sharp mixed norm inequalities for them. One of the operators coincides with the Radon transform for which mixed norm estimates are known but in reverse order. Those inequalities will be crucial in our proofs.

L p Mixed-norm estimates for convolution operator defined by singular measures

Let Γ be a C∞ curve in ℝn and µ be the measure induced by Lebesgue measure on Γ, multiplied by a smooth cut-off function. In this paper, we will prove some mixed-norm estimates based on the average decay estimates of the Fourier transform of µ.

Bounds for Kakeya-type maximal operators associated with $k$-planes

A $(d,k)$ set is a subset of $\rea^d$ containing a translate of every $k$-dimensional plane. Bourgain showed that for $k \geq \kcrit(d)$, where $\kcrit(d)$ solves $2^{\kcrit-1}+\kcrit = d$, every $(d,k)$ set has positive Lebesgue measure. We give a short proof of this result which allows for an improved $L^p$ estimate of the corresponding maximal operator, and which demonstrates that a lower value of $\kcrit$ could be obtained if improved mixed-norm estimates for the $x$-ray transform were known.

Mixed norm estimates for some averaging operators in ℝ2

We obtain mixed norm estimates for averaging operators on R 2 defined by convolution with μ, where either μ is arclength measure on certain curves in R 2 or d μ = dx/xβ on the positive x-axis.

Long-Time Existence of Quasilinear Wave Equations Exterior to Star-Shaped Obstacles via Energy Methods

We establish long-time existence results for quasilinear wave equations in the exterior of star-shaped obstacles. To do so, we prove an analogue of the mixed-norm estimates of Keel, Smith, and Sogge for the perturbed wave equation. The arguments that are presented rely only upon the invariance of the wave operator under translations and spatial rotations.

On a Hyper Hilbert Transform

The authors define the directional hyper Hilbert transform and give its mixed norm estimate. The similar conclusions for the directional fractional integral of one dimension are also obtained in this paper. As an application of the above results, the authors give the Lp-boundedness for a class of the hyper singular integrals and the fractional integrals with variable kernel. Moreover, as another application of the above results, the authors prove the dimension free estimate for the hyper Riesz transform. This is an extension of the related result obtained by Stein.

Mixed norm estimates for a restricted X-ray transform

In this paper, we establish optimal mixed norm inequalities, except for endpoints, for a certain restricted X-ray transform in arbitrary dimensions. In doing so, we demonstrate that a method [2] applied heretofore only to simpler nonmixed norm estimates can also be adapted to the mixed norm case.

Mixed-norm estimates for a restricted X-ray transform in ℝ4 and ℝ5

The Radon transform found important applications in integral geometry and in the study of PDE’s. R1,d is often called the X-ray transform due to its applications in radiology, we denote it by Xfull. It is well-known [19], [12] that the sharp mixed norm estimates for the full X-ray transform implies the Kakeya conjecture and it is related to some of the main problems in the summability of Fourier transform, Fourier restriction and more generally to oscillatory integrals, non-linear P.D.E’s and number theory [7], [1], [2], [20], [3], [14]. For some mapping properties of Xfull, see, e.g., [6], [5], [19] and [12]. Note that G1,d is a 2d− 2-dimensional manifold, thus Xfull is overdetermined for d ≥ 3, and it is of interest to consider its restrictions to lower dimensional subspaces of G1,d. For the definition of the restricted X-ray transforms as part of a more general class of transformations and some of its properties, see [11]. One particular example is the restriction of Xfull to the space of light rays (lines in R making a 45 degree angle with the plane xd = 0). Recently, Wolff [21] obtained mixed norm estimates for this operator (almost sharp in R) and used this information to prove almost sharp bilinear cone restriction estimates in all dimensions. We are interested in the restriction of Xfull to d dimensional line complexes in R. Let d ≥ 3; the subspace Gd of G1,d we are interested in is defined as follows: Let γd be the curve {γd(t) : γd(t) = (1, t, t, ..., td−1), t ∈ (−1, 1)} in R. Let l(t, x) denote

Traces of Besov Spaces Revisited

For the trace of Besov spaces B^s_{p,q} onto a hyperplane, the borderline case with s = \frac {n}{p} – (n–1) and 0 < p < 1 is analysed and a new dependence on the sum-exponent q is found. Through examples the restriction operator defined for s down to \frac {1}{p} , and valued in L_p , is shown to be distinctly different and, moreover, unsuitable for elliptic boundary problems. All boundedness properties (both new and previously known) are found to be easy consequences of a simple mixed-norm estimate, which also yields continuity with respect to the normal coordinate. The surjectivity for the classical borderline s = \frac {1}{p} (1 ≤ p < \infty) is given a simpler proof for all q \in ]20, 1] , using only basic functional analysis. The new borderline results are based on corresponding convergence criteria for series with spectral conditions.

A mixed norm estimate for the X-ray transform

A mixed norm estimate for the X-ray transform

Radial weights and mixed norm inequalities for the disc multiplier

In this paper we show that the L 2 -weighted inequalities with radial weights of the multiplier for the ball are controlled by the maximal function defined by averages on arbitrary rectangles. This solves the case of radial weights of a problem proposed by E. Stein in 1978. It also gives an elegant and simple proof of the boundedness of the multiplier for the ball on the spaces of mixed norm L rad p ( L ang 2 ) for 2n (n + 1) < p < 2n (n − 1) , obtained by A. Córdoba and by G. Mockenhaupt. Additional work is presented in order to extend these mixed norm estimates to the end points, 2n (n + 1) and 2n (n − 1) , in the (sharp) form of restricted weak-type inequalities.

Regularity of averages over hypersurfaces

Averages over smooth measures on smooth compact hypersurfaces inR n are studied. With assumptions on the decay of the Fourier transform of the measure we obtain mixed norm estimates for these means, for exampleL p estimates of multiparameter maximal functions over compact hypersurfaces.