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A Note on Weighted Estimates for the Schrödinger Operator
Abstract
We study two problems closely related to each other. The first one is concerned with some smoothing weighted estimates with weights in a certain Morrey-Campanato spaces, for the solution of the free Schrödinger equation. The second one is a weighed trace inequality.
... At this point, it should be noted that this local smoothing estimate can be written in terms of a weighted L 2 setting as in (1). Indeed, if ρ is any function such that j∈Z ρ 2 L ∞ (|x|∼2 j ) < ∞, one has Figure 1. ...
... The smoothing effect in the setting (1) can occur due to the decay of the weight |x| −α in the spatial direction. As shown in Figure 1, the smoothing factor σ is indeed related with the decay factor α. At this point, we naturally further ask how much regularity we can expect when considering decay not in the spatial direction but in the space-time direction like |(x, t)| −α . ...
... One is spectral methods based on resolvent estimates, starting from Kato's work [6], and the other is to use Fourier restriction estimates from harmonic analysis (see e.g. [1,18] and references therein). A completely different approach was also recently developed by Ruzhansky and Sugimoto [17] using canonical transforms. ...
... At this point, it is worth noting that this local smoothing estimate can be written in terms of a weighted L 2 norm as in (2). Indeed, if ρ is any function such that j∈Z ρ 2 L ∞ (|x|∼2 j ) < ∞, one has ...
... for V ∈ F p with p > (n − 1)/2, n ≥ 3, which can be found in [5,4] (see also [2,14]). Indeed, applying the re-scaled estimate of (16), ...
... for some − n−2 2 < s < 1 2 and 1 ≤ p < n 2(1−s) . Indeed, Barceló et al [1] obtained this generalized smoothing estimate (1.2) 1 for ...
... 1 Estimate (1.2) was also shown in [1] for the border line p = n 2(1−s) , but in this case L 2(1−s),p = L p . In view of generalization of (1.1) with a singular power weight, we therefore exclude it in (1.2). ...
The Kato-Yagima smoothing estimate is a smoothing weighted estimate with a singular power weight for the Schr\"odinger propagator. The weight has been generalized to Morrey-Campanato weights. In this paper we make this generalization more sharp in terms of the so-called Kerman-Sawyer weights. Our result is based on a more sharpened Fourier restriction estimate than previously known ones.
... for some −1 < s ≤ n−2 2 and 1 ≤ p ≤ n 2s+2 . Indeed, Barceló et al [1] obtained the following equivalent estimates ...
... We shall now give an outline of the main ideas in our work. The known approach to Strichartz and smoothing estimates with time-independent weights is based on weighted L 2 bounds for the resolvent operator as well as for the Fourier restriction (see, for example, [31,37,1,3,33]), but it is no longer available in the case of time-dependent weights. Our method that works for this case is completely different from it and is a more fruitful approach which is based on a combination between two kind of arguments, one from the theory of dispersive estimates and the other one from weighted inequalities. ...
The primary objective in this paper is to give an answer to an open question posed by J. A. Barcel\'o, J. M. Bennett, A. Carbery, A. Ruiz and M. C. Vilela concerning the problem of determining the optimal range on and for which the following Strichartz estimate with time-dependent weights w in Morrey-Campanato type classes holds: \begin{equation}\label{absset} \|e^{it\Delta}f\|_{L_{x,t}^2(w(x,t))}\leq C\|w\|_{\mathfrak{L}^{2s+2,p}_2}^{1/2}\|f\|_{\dot{H}^s}. \end{equation} Beyond the case , we further ask in this paper how much regularity we can expect on this setting. But interestingly, it turns out that this estimate is false whenever , which shows that the smoothing effect cannot occur in this time-dependent setting and the dispersion in the Schr\"odinger flow is not strong enough to have the effect. This naturally leads us to consider the possibility of having the effect at best in higher-order versions of this estimate with () whose dispersion is more strong. We do obtain a smoothing effect exactly for these higher-order versions. More generally, we will obtain the estimates where in a unified manner and also their corresponding inhomogeneous estimates to give applications to the global well-posedness in a weighted setting for Schr\"odinger and wave equations (as well as linearized KdV equations) with small time-dependent perturbations. This is our secondary objective in this paper.
... One is spectral methods based on resolvent estimates, starting from Kato's work [7], and the other is to make use of Fourier restriction estimates from harmonic analysis (see e.g. [1,19] and references therein). A completely different approach was also recently developed by Ruzhansky and Sugimoto [18] using canonical transforms. ...
We obtain some new Morawetz estimates for the Klein-Gordon flow of the form \begin{equation*} \big\||\nabla|^{\sigma} e^{it \sqrt{1-\Delta}}f \big\|_{L^2_{x,t}(|(x,t)|^{-\alpha})} \lesssim \|f\|_{H^s} \end{equation*} where and . The conventional approaches to Morawetz estimates with are no longer available in the case of time-dependent weights . Here we instead apply the Littlewood-Paley theory with Muckenhoupt weights to frequency localized estimates thereof that are obtained by making use of the bilinear interpolation between their bilinear form estimates which needs to carefully analyze some relevant oscillatory integrals according to the different scaling of for low and high frequencies.
... for V ∈ F p with p > (n − 1)/2, n ≥ 3, which can be found in [5,4] (see also [2,14]). Indeed, applying the re-scaled estimate of (2.4), ...
We prove weighted estimates for the Klein-Gordon equation perturbed with singular potentials such as the inverse-square potential. We then deduce the well-posedness of the Cauchy problem for this perturbed equation. We go on to discuss local smoothing and Strichartz estimates which improve previously known ones.
... One can also see [1], [2], [14] for related work. An explicit weight, |x| s dx dt, is discussed in [10] (see (2.6) there). ...
We prove that if the Hausdorff dimension of , is greater than , there exists such that the pinned distance set has positive Lebesgue measure. This improves a result of Peres and Schlag. The key new ingredient in our proof is the following identity. Using a group action argument, we show that for any Schwartz function f on and any , where is the normalized surface measure on . An interesting remark is that the right hand side can be easily seen equals where is the standard Laplacian and .
... but we can use Lemma 2.1 to shows that this inequality is equivalent to (2.8). See also [1]. Special cases of the restriction inequality in [23] are in [8] and [9]. ...
We prove new Pitt inequalities for the Fourier transforms with radial and
non-radial weights using weighted restriction inequalities for the Fourier
transform on the sphere. We also prove new Riemann-Lebesgue estimates and
versions of the uncertainty principle for the Fourier transform.
... Taking the sup in t was sometimes used for time-dependent potentials in other problems ( [39,40,4,2,3]) concerning Schrödinger equations. Since |V (x, t)| ≤ W (x) for almost every t, the following corollary is an immediate consequence of Theorem 1.4 and is expected to lead to unique continuation for time-dependent potentials. ...
In this paper we develop a new method to handle the problem of unique
continuation for the Schr\"odinger equation. In general the problem is to find
a class of potentials which allow the unique continuation. The key point of our
work is to develop a direct link between the problem and weighted
resolvent estimates with potentials as weights. We carry out it in an abstract
way, and thereby we do not need to deal with each of the potential classes. In
doing so, we will make use of limiting absorption principle and Kato
H-smoothing theorem in spectral theory, and employ some tools from harmonic
analysis. Once the resolvent estimate is set up for a potential class, from our
abstract theory the unique continuation would follow from the same potential
class.
In this regard, another main issue for us is to know which class of
potentials allows the resolvent estimate. We establish such a new class which
contains the previously known ones. As a consequence, we obtain new results on
the unique continuation. Our resolvent estimates may have further applications
for other related problems such as well-posedness. We will also deal with that
problem.
Combining TT* argument and bilinear interpolation, this paper obtains the Strichartz and smoothing estimates of dispersive semi-group e−itP(D) in weighted L2 spaces. Among other things, we recover the results in [1]. Moreover, the application of these results to the well-posedness of some equations are shown in the last section.
The Kato–Yajima smoothing estimate is a smoothing weighted L2 estimate with a singular power weight for the Schrödinger propagator. The weight has been generalized relatively recently to Morrey–Campanato weights. In this paper we make this generalization more sharp in terms of the so‐called Kerman–Sawyer weights. Our result is based on a more sharpened Fourier restriction estimate in a weighted L2 space. Obtained results are also extended to the fractional Schrödinger propagator.
We prove some weighted refinements of the classical Strichartz inequalities for initial data in the Sobolev spaces Ḣ
s
(ℝ
n
). We control the weightedL
2-norm of the solution of the free Schrödinger equation whenever the weight is in a Morrey-Campanato type space adapted to
that equation. Our partial positive results are complemented by some necessary conditions based on estimates for certain particular
solutions of the free Schrödinger equation.
KeywordsSchrodinger equations-Morrey-Campanato Spaces
MSC200034A12-42B35
We show that the strong unique continuation property holds for the inequality |Δ u| ≤ |υ|u|, where the potential υ(x) satisfies the C. Fefferman-Phong condition in a certain range of p values. We also deal with the situation of u(x) vanishing at infinity. These are all consequences of appropriate Carleman inequalities.
We describe a certain "self-similar" family of solutions to the free
Schroedinger equation in all dimensions, and derive some consequences of such
solutions for two specific problems.
We prove the inequality (1) for weights w in a class which contains the class , introduced by C. Fefferman and D. H. Phong in studying eigenvalues of Schrödinger operators. In our case, C is independent of the lower order terms of P. As a consequence we prove unique continuation theorem for solutions of Δ + V, V in the same class.
The purpose of this note is twofold. First it is a corrigenda of our paper \cite{RV1}. And secondly we make some remarks concerning the interpolation properties of Morrey spaces
We describe a certain "self-similar" family of solutions to the free Schroedinger equation in all dimensions, and derive some consequences of such solutions for two specific problems.
We study Strichartz estimates for the solution of the Cauchy problem associated with the inhomogeneous free Schrödinger equation in the case when the initial data is equal to zero, proving some new estimates for certain exponents and giving counterexamples for some others.
We prove non interpolation results for the family of Morrey spaces. We introduce a scale of block spaces, which are preduals of Morrey spaces in some range. Negative interpolation results are also obtained in this case.
Let us consider in a domain O of Rn solutions of the differential inequality
|?u(x)| = V(x)|u(x)|, x I O,
where V is a non smooth, positive potential.
We are interested in global unique continuation properties. That means that u must be identically zero on O if it vanishes on an open subset of O.
Unique continuation for Schrödinger operators with potential in Morrey spaces " and a remark on interpolation of Morrey spaces [7] , Local regularity of solutions to wave equations with time-dependent potentials, Duke Math
[6]
, Corrigenda to: " Unique continuation for Schrödinger operators with potential in Morrey
spaces " and a remark on interpolation of Morrey spaces, Publ. Mat. 39 (1995), no. 2, 405–411.
[7]
, Local regularity of solutions to wave equations with time-dependent potentials, Duke
Math. J. 76 (1994), no. 3, 913–940.
Unique continuation for Schrödinger operators with potential in Morrey spaces" and a remark on interpolation of Morrey spaces
- Corrigenda To
, Corrigenda to: "Unique continuation for Schrödinger operators with potential in Morrey
spaces" and a remark on interpolation of Morrey spaces, Publ. Mat. 39 (1995), no. 2, 405-411.