Local well-posedness of nonlinear dispersive equations on modulation spaces

By using tools of time-frequency analysis, we obtain some improved local well-posedness results for the NLS, NLW and NLKG equations with Cauchy data in modulation spaces $M{p, 1}_{0,s}$.Comment: 11 page

Local well-posedness for the nonlinear Schr\"odinger equation in the intersection of modulation spaces Mp,qs(Rd)∩M∞,1(Rd)M_{p, q}^s(\mathbb{R}^d) \cap M_{\infty, 1}(\mathbb{R}^d)

We introduce a Littlewood-Paley characterization of modulation spaces and use it to give an alternative proof of the algebra property, somehow implicitly contained in Sugimoto (2011), of the intersection $M^s_{p,q}(\mathbb{R}^d) \cap M_{\infty, 1}(\mathbb{R}^d)$ for $d \in \mathbb{N}$, $p, q \in [1, \infty]$ and $s \geq 0$. We employ this algebra property to show the local well-posedness of the Cauchy problem for the cubic nonlinear Schr\"odinger equation in the above intersection. This improves Theorem 1.1 by B\'enyi and Okoudjou (2009), where only the case $q = 1$ is considered, and closes a gap in the literature. If $q > 1$ and $s > d \left(1 - \frac{1}{q}\right)$ or if $q = 1$ and $s \geq 0$ then $M^s_{p,q}(\mathbb{R}^d) \hookrightarrow M_{\infty, 1}(\mathbb{R}^d)$ and the above intersection is superfluous. For this case we also reobtain a H\"older-type inequality for modulation spaces.Comment: 14 page

Local Well-posedness of Nonlinear Dispersive Equations on Modulation Spaces

By using tools of time-frequency analysis, we obtain some improved local well-posedness results for the nonlinear Schrödinger, nonlinear wave and nonlinear Klein–Gordon equations with Cauchy data in modulation spaces ℳ0,sp,1

Smoothing of commutators for a Hörmander class of bilinear pseudodifferential operators

Commutators of bilinear pseudodifferential operators with symbols in the H\"ormander class BS_{1, 0}^1 and multiplication by Lipschitz functions are shown to be bilinear Calder\'on-Zygmund operators. A connection with a notion of compactness in the bilinear setting for the iteration of the commutators is also made.Comment: 16 page

On the probabilistic Cauchy theory for nonlinear dispersive PDEs

In this note, we review some of the recent developments in the well-posedness theory of nonlinear dispersive partial differential equations with random initial data.Comment: 26 pages. To appear in Landscapes of Time-Frequency Analysis, Appl. Numer. Harmon. Ana

Linear perturbations of the Wigner transform and the Weyl quantization

We study a class of quadratic time-frequency representations that, roughly speaking, are obtained by linear perturbations of the Wigner transform. They satisfy Moyal's formula by default and share many other properties with the Wigner transform, but in general they do not belong to Cohen's class. We provide a characterization of the intersection of the two classes. To any such time-frequency representation, we associate a pseudodifferential calculus. We investigate the related quantization procedure, study the properties of the pseudodifferential operators, and compare the formalism with that of the Weyl calculus.Comment: 38 pages. Contributed chapter for volume on the occasion of Luigi Rodino's 70th birthda

Primary care obesity management in Hungary: evaluation of the knowledge, practice and attitudes of family physicians

BACKGROUND: Obesity, a threatening pandemic, has an important public health implication. Before proper medication is available, primary care providers will have a distinguished role in prevention and management. Their performance may be influenced by many factors but their personal motivation is still an under-researched area. METHOD: The knowledge, attitudes and practice were reviewed in this questionnaire study involving a representative sample of 10% of all Hungarian family physicians. In different settings, 521 practitioners (448 GPs and 73 residents/vocational trainees) were questioned using a validated questionnaire. RESULTS: The knowledge about multimorbidity, a main consequence of obesity, was balanced.Only 51% of the GPs were aware of the diagnostic threshold for obesity; awareness being higher in cities (60%) and the highest among residents (90%). They also considered obesity an illness rather than an aesthetic issue.There were wider differences regarding attitudes and practice, influenced by the the doctors' age, gender, known BMI, previous qualification, less by working location.GPs with qualification in family medicine alone considered obesity management as higher professional satisfaction, compared to physicians who had previously other board qualification (77%vs68%). They measured their patients' waist circumference and waist/hip ratio (72%vs62%) more frequently, provided the obese with dietary advice more often, while this service was less frequent among capital-based doctors who accepted the self-reported body weight dates by patients more frequently / commonly. Similar reduced activity and weight-measurement in outdoor clothing were more typical among older doctors.Diagnosis based on BMI alone was the highest in cities (85%). Consultations were significantly shorter in practices with a higher number of enrolled patients and were longer by female providers who consulted longer with patients about the suspected causes of developing obesity (65%vs44%) and offered dietary records for patients significantly more frequently (65%vs52%). Most of the younger doctors agreed that obesity management was a primary care issue.Doctors in the normal BMI range were unanimous that they should be a model for their patients (94%vs81%). CONCLUSION: More education of primary care physicians, available practical guidelines and higher community involvement are needed to improve the obesity management in Hungary

Endpoint Estimates for N-dimensional Hardy Operators and Their Commutators

In this paper, it is proved that the higher dimensional Hardy operator is bounded from Hardy space to Lebesgue space. The endpoint estimate for the commutator generated by Hardy operator and (central) BMO function is also discussed.Comment: 8 page

Modulation Invariant Bilinear T(1) Theorem

We prove a T(1) theorem for bilinear singular integral operators (trilinear forms) with a one-dimensional modulation symmetry