Analysis & PDE 

About: Analysis & PDE is an academic journal published by Mathematical Sciences Publishers. The journal publishes majorly in the area(s): Boundary (topology) & Bounded function. It has an ISSN identifier of 1948-206X. Over the lifetime, 617 publications have been published receiving 15020 citations. The journal is also known as: APDE & Analysis and PDE.

Sharp weighted bounds involving A

Abstract: We improve on several weighted inequalities of recent interest by replacing a part of the Ap bounds by weaker A∞ estimates involving Wilson’s A∞ constant [ w ] A ∞ ′ : = sup Q 1 w ( Q ) ∫ Q M ( w χ Q ) . In particular, we show the following improvement of the first author’s A2 theorem for Calderon–Zygmund operators T: ∥ T ∥ ℬ ( L 2 ( w ) ) ≤ c T [ w ] A 2 1 ∕ 2 ( [ w ] A ∞ ′ + [ w − 1 ] A ∞ ′ ) 1 ∕ 2 . Corresponding Ap type results are obtained from a new extrapolation theorem with appropriate mixed Ap-A∞ bounds. This uses new two-weight estimates for the maximal function, which improve on Buckley’s classical bound. We also derive mixed A1-A∞ type results of Lerner, Ombrosi and Perez (2009) of the form ∥ T ∥ ℬ ( L p ( w ) ) ≤ c p p ′ [ w ] A 1 1 ∕ p ( [ w ] A ∞ ′ ) 1 ∕ p ′ , 1 < p < ∞ , ∥ T f ∥ L 1 , ∞ ( w ) ≤ c [ w ] A 1 log ( e + [ w ] A ∞ ′ ) ∥ f ∥ L 1 ( w ) . An estimate dual to the last one is also found, as well as new bounds for commutators of singular integrals.

Uniqueness of ground states for pseudorelativistic Hartree equations

Abstract: We prove uniqueness of ground states Q∈H1∕2(ℝ3) for the pseudorelativistic Hartree equation, − Δ + m 2 Q − ( x − 1 ∗ Q 2 ) Q = − μ Q , in the regime of Q with sufficiently small L2-mass. This result shows that a uniqueness conjecture by Lieb and Yau [1987] holds true at least for N= ∫ |Q|2≪1 except for at most countably many N. Our proof combines variational arguments with a nonrelativistic limit, leading to a certain Hartree-type equation (also known as the Choquard–Pekard or Schrodinger–Newton equation). Uniqueness of ground states for this limiting Hartree equation is well-known. Here, as a key ingredient, we prove the so-called nondegeneracy of its linearization. This nondegeneracy result is also of independent interest, for it proves a key spectral assumption in a series of papers on effective solitary wave motion and classical limits for nonrelativistic Hartree equations.

Scattering threshold for the focusing nonlinear Klein–Gordon equation

Abstract: We show scattering versus blow-up dichotomy below the ground state energy for the focusing nonlinear Klein–Gordon equation, in the spirit of Kenig and Merle for the H1 critical wave and Schrodinger equations. Our result includes the H1 critical case, where the threshold is given by the ground state for the massless equation, and the 2D square-exponential case, where the mass for the ground state may be modified, depending on the constant in the sharp Trudinger–Moser inequality. The main difficulty is the lack of scaling invariance in both the linear and the nonlinear terms.

The mass-critical nonlinear schrödinger equation with radial data in dimensions three and higher

Abstract: We establish global well-posedness and scattering for solutions to the mass-critical nonlinear Schrodinger equation iut+Δu=±|u|4∕du for large spherically symmetric Lx2(ℝd) initial data in dimensions d≥3. In the focusing case we require that the mass is strictly less than that of the ground state. As a consequence, we obtain that in the focusing case, any spherically symmetric blowup solution must concentrate at least the mass of the ground state at the blowup time.